Finite Group Databasebeta

← #28306933

Subgroups — #28306933

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133284C211yesmaximal, characteristic
2C22×C4×#88324C211yesmaximal, characteristic
3C2×#16811489C211yesmaximal, characteristic
4C2×#16811489C211yesmaximal, characteristic
5C2×#16811489C211yesmaximal, characteristic
6C2×#16811489C211yesmaximal, characteristic
7#95×Dic53C211yesmaximal, characteristic
8C23×#88324C2211yescharacteristic
9C23×#88324C2211yescharacteristic
10C23×#88324C2211yescharacteristic
11C23×#88324C2211yescharacteristic
12C2×#13133284—22no—
13C2×#13133284—22no—
14C2×#13133284C414yescomplemented
15C2×C4×#88324—22no—
16C2×C4×#88324—22no—
17C2×C4×#88324—22no—
18#16811489—22no—
19#16811489—22no—
20#16811489—22no—
21#16811489C414yescomplemented
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25C2×#13133124C411yescharacteristic
26C2×#13133124C411yescharacteristic
27C2×#13133124C411yescharacteristic
28C2×#95×C53C411yescharacteristic
29C22×#88324—21no—
30C22×#88324—21no—
31C22×#88324—21no—
32C22×#88324—21no—
33C22×#88324—21no—
34C22×#88324—21no—
35C22×#88324—21no—
36C22×#88324—21no—
37C22×#88324—21no—
38C22×#88324—21no—
39C22×#88324—21no—
40C22×#88324—21no—
41C22×#88324—21no—
42C22×#88324—21no—
43C22×#88324—21no—
44C22×#88324—21no—
45C22×#88324—21no—
46C22×#88324—21no—
47C22×#88324—21no—
48C22×#88324—21no—
49C22×#88324C2×C412yes—
50C22×#88324C2×C412yes—
51C22×#88324C2×C412yes—
52C22×#88324C2×C412yes—
53C22×#88324D412yes—
54C22×#88324D412yes—
55C22×#88324D412yes—
56C22×#88324Q812yes—
57#13133284—28no—
58C4×#88324—28no—
59C23×Dic53C2×C411yescharacteristic
60C23×Dic53C2×C411yescharacteristic
61C23×Dic53C2×C411yescharacteristic
62C23×Dic53C2311yescharacteristic
63C2×C4×Dic53—21no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53—21no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73C2×C4×Dic53—21no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53C2×C412yes—
78C2×C4×Dic53C2×C412yes—
79C2×C4×Dic53D412yes—
80C2×C4×Dic53Q812yes—
81#13133124—22no—
82#13133124C814yescomplemented
83#13133124C814yescomplemented
84C23×C4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C411yescharacteristic
87#95×C53—22no—
88C24×C53C22×C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53C4212yes—
106C22×C4×C53C4⋊C412yes—
107C22×C4×C53C2×C812yes—
108C22×C4×C53C2×C812yes—
109C22×C4×C53M1612yes—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—21no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C22×Dic53—21no—
118C22×Dic53—21no—
119C22×Dic53—21no—
120C22×Dic53—21no—
121C22×Dic53—21no—
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53C4212yes—
125C22×Dic53C4⋊C412yes—
126C22×Dic53C2×C812yes—
127C22×Dic53C2×C812yes—
128C22×Dic53C2×C812yes—
129C22×Dic53C2×C812yes—
130C22×Dic53M1612yes—
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×Q811yescharacteristic
138C22×Dic53C2×Q811yescharacteristic
139C22×Dic53#4111yescharacteristic
140C22×Dic53#4111yescharacteristic
141C22×Dic53#4111yescharacteristic
142C22×Dic53#4111yescharacteristic
143C22×Dic53#4111yescharacteristic
144C22×Dic53#4111yescharacteristic
145C2×#88324—22no—
146C2×#88324—22no—
147C2×#88324—22no—
148C2×#88324—22no—
149C2×#88324—22no—
150C2×#88324—22no—
151C2×#88324—22no—
152C2×#88324—22no—
153C2×#88324—22no—
154C2×#88324—22no—
155C2×#88324—22no—
156C2×#88324—22no—
157C2×#88324—22no—
158C2×#88324—22no—
159C2×#88324—22no—
160C2×#88324—22no—
161C2×#88324—22no—
162C2×#88324—22no—
163C2×#88324—41no—
164C2×#88324—41no—
165C2×#88324—41no—
166C2×#88324—41no—
167C2×#88324—41no—
168C2×#88324—41no—
169C2×#88324—41no—
170C2×#88324—41no—
171C2×#88324C22⋊C414yes—
172C2×#88324C4⋊C414yes—
173C4×Dic53—22no—
174C4×Dic53—22no—
175C4×Dic53—22no—
176C4×Dic53—22no—
177C4×Dic53—22no—
178C4×Dic53—22no—
179C4×Dic53—22no—
180C4×Dic53—22no—
181C4×Dic53—41no—
182C4×Dic53—41no—
183C4×Dic53—41no—
184C4×Dic53—41no—
185C23×C53C2×C4211yescharacteristic
186C23×C53C2×C4⋊C411yescharacteristic
187C23×C53#11711yescharacteristic
188C23×C53C4×D411yescharacteristic
189C23×C53C4×D411yescharacteristic
190C23×C53C4×D411yescharacteristic
191C23×C53C4×Q811yescharacteristic
192C23×C53C22×C811yescharacteristic
193C23×C53C22×C811yescharacteristic
194C23×C53C2×M1611yescharacteristic
195C23×C53C2×M1611yescharacteristic
196C23×C53#13111yescharacteristic
197C23×C53#13111yescharacteristic
198C23×C53#13111yescharacteristic
199C23×C53#13111yescharacteristic
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53C4⋊C814yescomplemented
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53C4×C814yescomplemented
241C2×Dic53C4⋊C814yescomplemented
242C2×Dic53C2×C22⋊C411yescharacteristic
243C2×Dic53C2×C4⋊C411yescharacteristic
244C2×Dic53#11711yescharacteristic
245C2×Dic53#11711yescharacteristic
246C2×Dic53C4×D411yescharacteristic
247C2×Dic53C4×D411yescharacteristic
248C2×Dic53C4×D411yescharacteristic
249C2×Dic53C4×D411yescharacteristic
250C2×Dic53C4×D411yescharacteristic
251C2×Dic53C4×Q811yescharacteristic
252C2×Dic53C4×Q811yescharacteristic
253C2×Dic53C4×Q811yescharacteristic
254C2×Dic53#12011yescharacteristic
255C2×Dic53#12111yescharacteristic
256C2×Dic53#12111yescharacteristic
257C2×Dic53C22⋊Q811yescharacteristic
258C2×Dic53C22⋊Q811yescharacteristic
259C2×Dic53C22⋊Q811yescharacteristic
260C2×Dic53C22⋊Q811yescharacteristic
261C2×Dic53#12311yescharacteristic
262C2×Dic53#12311yescharacteristic
263C2×Dic53#12311yescharacteristic
264C2×Dic53#12411yescharacteristic
265C2×Dic53#12511yescharacteristic
266C2×Dic53#12511yescharacteristic
267C2×Dic53#12611yescharacteristic
268C2×Dic53#12611yescharacteristic
269C2×Dic53C4⋊Q811yescharacteristic
270#88324—24no—
271#88324—24no—
272#88324—42no—
273#88324—42no—
274#10876—531nomaximal, Sylow
275C22×C53C4×C22⋊C411yescharacteristic
276C22×C53C4×C4⋊C411yescharacteristic
277C22×C53C4×C4⋊C411yescharacteristic
278C22×C53#38511yescharacteristic
279C22×C53#38711yescharacteristic
280C22×C53#38811yescharacteristic
281C22×C53#38911yescharacteristic
282C22×C53C2×C4×C811yescharacteristic
283C22×C53C4×M1611yescharacteristic
284C22×C53#40511yescharacteristic
285C22×C53#40511yescharacteristic
286C22×C53C2×C4⋊C811yescharacteristic
287C22×C53C2×C4⋊C811yescharacteristic
288C22×C53#42411yescharacteristic
289C22×C53#42411yescharacteristic
290C22×C53#43111yescharacteristic
291C22×C53#43111yescharacteristic
292C22×C53#43311yescharacteristic
293C22×C53#43311yescharacteristic
294C22×C53C8×D411yescharacteristic
295C22×C53C8×D411yescharacteristic
296C22×C53C8×D411yescharacteristic
297C22×C53C8×D411yescharacteristic
298C22×C53C8×D411yescharacteristic
299C22×C53C8×D411yescharacteristic
300C22×C53#43511yescharacteristic
301C22×C53#43511yescharacteristic
302C22×C53#43511yescharacteristic
303C22×C53#43511yescharacteristic
304C22×C53#43611yescharacteristic
305C22×C53#43611yescharacteristic
306C22×C53C8×Q811yescharacteristic
307C22×C53C8×Q811yescharacteristic
308C22×C53#44611yescharacteristic
309C22×C53#44611yescharacteristic
310C4×C53—24no—
311C4×C53—42no—
312C4×C53—42no—
313Dic53—24no—
314Dic53—24no—
315Dic53—42no—
316Dic53#38511yescharacteristic
317Dic53#38611yescharacteristic
318Dic53#38711yescharacteristic
319Dic53#38711yescharacteristic
320Dic53#38711yescharacteristic
321Dic53#38811yescharacteristic
322Dic53#38911yescharacteristic
323Dic53#39111yescharacteristic
324C22×C4⋊C8—531no—
325C22×C4×C8—531no—
326C2×#336—531no—
327C2×#336—531no—
328C2×#336—531no—
329C2×#336—531no—
330C4×#95—531no—
331C2×C53#143611yescharacteristic
332C2×C53C8×C22⋊C411yescharacteristic
333C2×C53#175311yescharacteristic
334C2×C53C4×C4⋊C811yescharacteristic
335C2×C53#176811yescharacteristic
336C2×C53C8×C4⋊C411yescharacteristic
337C2×C53#177111yescharacteristic
338C2×C53#185011yescharacteristic
339C2×C53#185011yescharacteristic
340C2×C53#191611yescharacteristic
341C2×C53#191611yescharacteristic
342C2×C53#192311yescharacteristic
343C2×C53#192311yescharacteristic
344C2×C53#193911yescharacteristic
345C2×C53#193911yescharacteristic
346C23×C8—531no—
347C23×C8—531no—
348C23×C8—531no—
349C23×C8—531no—
350C22×C42—531no—
351C22×C42—531no—
352C22×C42—531no—
353C2×C4⋊C8—534no—
354C2×C4⋊C8—1062no—
355C2×C4⋊C8—1062no—
356C2×C4×C8—1062no—
357C2×C4×C8—1062no—
358C2×C4×C8—1062no—
359C2×#95—531no—
360C2×#95—531no—
361C2×#95—531no—
362C2×#95#615711yescharacteristic
363#336—534no—
364#336—1062no—
365#336—1062no—
366#336—1062no—
367C53#1087611yescharacteristic, complemented, Sylow
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4C2×#615711yescharacteristic
373C23×C4C2×#615711yescharacteristic
374C23×C4C2×#615711yescharacteristic
375C22×C8—532no—
376C22×C8—532no—
377C22×C8—532no—
378C22×C8—532no—
379C22×C8—532no—
380C22×C8—532no—
381C22×C8—532no—
382C22×C8—532no—
383C22×C8—1061no—
384C22×C8—1061no—
385C22×C8—1061no—
386C22×C8—1061no—
387C22×C8—1061no—
388C22×C8—1061no—
389C22×C8—1061no—
390C22×C8—1061no—
391C22×C8—1061no—
392C22×C8—1061no—
393C22×C8—1061no—
394C22×C8—1061no—
395C22×C8—1061no—
396C22×C8—1061no—
397C22×C8—1061no—
398C22×C8—1061no—
399C22×C8—1061no—
400C22×C8—1061no—
401C22×C8—1061no—
402C22×C8—1061no—
403C2×C42—532no—
404C2×C42—532no—
405C2×C42—532no—
406C2×C42—532no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C4⋊C8—1068no—
422C4×C8—1068no—
423#95—22no—
424#95—534no—
425#95—534no—
426#95—1062no—
427C24C22×#615711yescharacteristic
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—532no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4D53⋊C812yes—
480C22×C4D53⋊C812yes—
481C22×C4C4×#615712yes—
482C22×C4C4⋊#615712yes—
483C22×C4#1171994012yes—
484C2×C8—534no—
485C2×C8—534no—
486C2×C8—1062no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—1062no—
491C2×C8—1062no—
492C2×C8—1062no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—1062no—
496C2×C8—1062no—
497C2×C8—1062no—
498C2×C8—1062no—
499C2×C8—1062no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—2121no—
505C2×C8—2121no—
506C2×C8—2121no—
507C2×C8—2121no—
508C2×C8—2121no—
509C2×C8—2121no—
510C2×C8—2121no—
511C2×C8—2121no—
512C42—1062no—
513C42—1062no—
514C42—1062no—
515C42—1062no—
516C42—1062no—
517C42—1062no—
518C42—1062no—
519C42—1062no—
520C42—2121no—
521C42—2121no—
522C42—2121no—
523C42—2121no—
524C23C2×C4×#615711yescharacteristic
525C23C2×C4⋊#615711yescharacteristic
526C23#1313329611yescharacteristic
527C23D4×#615711yescharacteristic
528C23D4×#615711yescharacteristic
529C23D4×#615711yescharacteristic
530C23Q8×#615711yescharacteristic
531C23C2×D53⋊C811yescharacteristic
532C23C2×D53⋊C811yescharacteristic
533C23C2×#1171994011yescharacteristic
534C23D53⋊M1611yescharacteristic
535C23#1313330411yescharacteristic
536C23#1313330411yescharacteristic
537C23#1313330411yescharacteristic
538C23#1313330511yescharacteristic
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—531no—
560C2×C4—531no—
561C2×C4—531no—
562C2×C4—531no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—531no—
566C2×C4—531no—
567C2×C4—531no—
568C2×C4—531no—
569C2×C4—531no—
570C2×C4—531no—
571C2×C4—531no—
572C2×C4—531no—
573C2×C4—531no—
574C2×C4—531no—
575C2×C4—531no—
576C2×C4—531no—
577C2×C4—531no—
578C2×C4—531no—
579C2×C4—531no—
580C2×C4—531no—
581C2×C4—531no—
582C2×C4—531no—
583C2×C4—531no—
584C2×C4—531no—
585C2×C4—531no—
586C2×C4—531no—
587C2×C4—534no—
588C2×C4—534no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4#1313328414yescomplemented
609C8—1064no—
610C8—1064no—
611C8—2122no—
612C8—2122no—
613C22C22⋊C4×#615711yescharacteristic
614C22C4×C4⋊#615711yescharacteristic
615C22C4⋊C4×#615711yescharacteristic
616C22#1681157011yescharacteristic
617C22#1681157211yescharacteristic
618C22#1681157311yescharacteristic
619C22#1681157411yescharacteristic
620C22C4×D53⋊C811yescharacteristic
621C22C4×#1171994011yescharacteristic
622C22#1681158711yescharacteristic
623C22#1681158811yescharacteristic
624C22D53⋊(C4⋊C8)11yescharacteristic
625C22C2×#1313328411yescharacteristic
626C22#1681162911yescharacteristic
627C22#1681163011yescharacteristic
628C22#1681164011yescharacteristic
629C22#1681164111yescharacteristic
630C22#1681164711yescharacteristic
631C22#1681164811yescharacteristic
632C22#1681165011yescharacteristic
633C22#1681165011yescharacteristic
634C22#1681165011yescharacteristic
635C22#1681165011yescharacteristic
636C22#1681165111yescharacteristic
637C22#1681165111yescharacteristic
638C22#1681165311yescharacteristic
639C22#1681165411yescharacteristic
640C22#1681165411yescharacteristic
641C22#1681165511yescharacteristic
642C22#1681165611yescharacteristic
643C22#1681165811yescharacteristic
644C22#1681167211yescharacteristic
645C22#1681167211yescharacteristic
646C22#1681167411yescharacteristic
647C22#1681167511yescharacteristic
648C4—24no—
649C4—42no—
650C4—42no—
651C4—531no—
652C4—531no—
653C4—531no—
654C4—531no—
655C4—531no—
656C4—531no—
657C4—531no—
658C4—531no—
659C4—1064no—
660C4—1064no—
661C4—2122no—
662C2#2028989511yescharacteristic
663C2#2029044911yescharacteristic
664C2#2029045711yescharacteristic
665C2C4×#1313328411yescharacteristic
666C2#2029050311yescharacteristic
667C2#2029050711yescharacteristic
668C2#2029051511yescharacteristic
669C2#2029065911yescharacteristic
670C2#2029066011yescharacteristic
671C2#2029077011yescharacteristic
672C2#2029077111yescharacteristic
673C2#2029078511yescharacteristic
674C2#2029078611yescharacteristic
675C2#2029081311yescharacteristic
676C2#2029081411yescharacteristic
subgroupG/H
center, FrattiniC24C22×#6157
commutatorC22×C53C2×C4×C8
FittingC2×#95×C53C4
socle, prime radicalC24×C53C22×C4
cosocleC23×Dic53C23
hypercenterC2×#95#6157
hypocenterC53#10876
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. C2×#13133284C4semidirect
21. #16811489C4semidirect
82. #13133124C8semidirect
83. #13133124C8semidirect
220. C2×C4×C53C4⋊C8semidirect
240. C2×Dic53C4×C8semidirect
241. C2×Dic53C4⋊C8semidirect
367. C53#10876semidirect
608. C2×C4#13133284semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#131332842
C22×C4×#883242
C2×#168114892
#95×Dic532
#1087653