Finite Group Databasebeta

← #29687648

Subgroups — #29687648

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840041C211yesmaximal, characteristic
2#21448390C211yesmaximal, characteristic, complemented
3C4×#14519739C211yesmaximal, characteristic
4#21446769C211yesmaximal, characteristic, complemented
5#21446581C211yesmaximal, characteristic, complemented
6#21446411C211yesmaximal, characteristic, complemented
7#21446083C211yesmaximal, characteristic
8#21446079C211yesmaximal, characteristic
9#21445818C211yesmaximal, characteristic
10#21445718C211yesmaximal, characteristic, complemented
11#21445715C211yesmaximal, characteristic, complemented
12#21445671C211yesmaximal, characteristic, complemented
13#21445643C211yesmaximal, characteristic
14#21445643C211yesmaximal, characteristic
15C61×#1921C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519742—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×C42×D61—21no—
22#16840041C418yescomplemented
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34#16839666—21no—
35#16839660C2211yescharacteristic
36#16839657—21no—
37#16839631C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42#16839613—21no—
43C4×#11758198—21no—
44#16839609—21no—
45#16839609—21no—
46C42×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776C2×C414yescomplemented
69#14519742—44no—
70C4×D244—22no—
71C4×D244—22no—
72#14519739—22no—
73#14519739—22no—
74#14519739C2×C414yes—
75#14519739Q814yescomplemented
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96C42×D61—44no—
97#14519710C2×C414yescomplemented
98#14519709—22no—
99#14519708—22no—
100#14519706—22no—
101#14519706C2×C414yescomplemented
102#14519699—22no—
103#14519699C2×C414yes—
104#14519697—22no—
105#14519697C2×C414yes—
106#14519687—22no—
107#14519687—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61C2×C414yescomplemented
123#95×C61—21no—
124#95×C61—21no—
125C23×D61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C22×C412yes—
144C2×D244C2×Q812yes—
145C2×D244#4112yes—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×Dic122—24no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61C22×C412yes—
171C4⋊C4×C61C22×C412yes—
172C4⋊C4×C61C2×Q812yes—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176#11758198—24no—
177#11758198—24no—
178#11758198—24no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61#4112yes—
185#11758196—24no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—41no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61C22×C412yes—
198C4×Dic61C2×Q812yes—
199C4×Dic61#4112yes—
200C4×Dic61#4112yes—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C22×C412yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#13712yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61C4×D412yes—
255C2×Dic61C4×D412yes—
256C2×Dic61C22⋊Q812yes—
257C2×Dic61C22⋊Q812yes—
258C2×Dic61AGL(1,ℤ/8)12yes—
259C2×Dic61AGL(1,ℤ/8)12yes—
260D244—22no—
261D244—22no—
262D244—22no—
263D244C4×Q814yescomplemented
264C4×D61—48no—
265C4×D61—48no—
266C4×D61—48no—
267Dic122—48no—
268C8×C61—22no—
269C8×C61—41no—
270#92418—22no—
271#92418—41no—
272#21613—611nomaximal, Sylow
273C22×C61C2×C4×D411yescharacteristic
274C22×C61C2×C22⋊Q811yescharacteristic
275C22×C61#52511yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—24no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61C2×C4×Q811yescharacteristic
294C4×C61C4×#4111yescharacteristic
295C4×C61#52911yescharacteristic
296C4×C61#53311yescharacteristic
297Dic61—22no—
298Dic61—22no—
299Dic61—22no—
300Dic61—42no—
301Dic61—42no—
302Dic61—81no—
303Dic61#44214yes—
304C2×#474—611no—
305C4×C4⋊Q8—611no—
306C42×D4—611no—
307#1988—611no—
308#1948—611no—
309#1921—611no—
310#1921D6111yescharacteristic, complemented
311#1864—611no—
312#1864—611no—
313#1802—611no—
314#1776—611no—
315#1776—611no—
316#1767—611no—
317#1762—611no—
318#1762—611no—
319C2×C61C4×C22⋊Q811yescharacteristic
320C2×C61C2×#44211yescharacteristic
321C2×C61C4×#13711yescharacteristic
322C2×C61#307711yescharacteristic
323C2×C61#307911yescharacteristic
324C2×C61#311011yescharacteristic
325C2×C61#311511yescharacteristic
326D61—44no—
327C2×C4⋊Q8—611no—
328C2×C4×Q8—1221no—
329C2×C4×D4—611no—
330C2×C4×D4—611no—
331C2×C4×D4—611no—
332C22×C42—1221no—
333#474—618no—
334C2×#107—611no—
335C2×#107C2×D6111yescharacteristic
336C2×#106—611no—
337C2×#106—611no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8—611no—
340C2×#102—611no—
341C2×#102—611no—
342C2×#102—611no—
343C2×#102—611no—
344C2×#97—611no—
345C2×#97—611no—
346#391—1221no—
347#389—611no—
348#389—1221no—
349#389C2×D6111yescharacteristic
350#384—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—1221no—
356C4×C4⋊C4C2×D6111yescharacteristic
357C4×C22⋊C4—1221no—
358C4×C22⋊C4—1221no—
359C4×C22⋊C4—1221no—
360C43—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#340C2×D6111yescharacteristic
368#340C2×D6111yescharacteristic
369#336—611no—
370#336C2×D6111yescharacteristic
371C61#2161311yescharacteristic, complemented, Sylow
372C22×Q8—1221no—
373C22×D4—611no—
374C23×C4—1221no—
375C23×C4—1221no—
376C23×C4—1221no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C4⋊Q8—614no—
382C4⋊Q8—1222no—
383C4×Q8—2444no—
384C4×D4—614no—
385C4×D4—614no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1222no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—21no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—2444no—
417C2×C42C22×D6111yescharacteristic
418#107—614no—
419#107C4×D6114yescomplemented
420#106—1222no—
421#106—1222no—
422C4⋊C8—614no—
423C4⋊C8—1222no—
424#102—614no—
425#102—614no—
426#102—1222no—
427#102—1222no—
428#97—1222no—
429#97—1222no—
430#95—21no—
431#95—21no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437C24—1221no—
438C2×Q8—1224no—
439C2×Q8—2442no—
440C2×Q8—2442no—
441C2×D4—612no—
442C2×D4—612no—
443C2×D4—612no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4—1224no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—612no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8C2×C4×D6112yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—612no—
489C4⋊C4—612no—
490C4⋊C4—1224no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C4⋊C4C2×C4×D6112yes—
499C4⋊C4C2×C4×D6112yes—
500C4⋊C4Q8×D6112yes—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C22⋊C4—1224no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—612no—
508C42—612no—
509C42—612no—
510C42—612no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C42—2441no—
515C42—2442no—
516C42—2442no—
517C42—2442no—
518C42—2442no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23C2×D4×D6111yescharacteristic
527Q8—2448no—
528D4—614no—
529D4—1222no—
530D4—1222no—
531D4—1222no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2448no—
564C2×C4—2448no—
565C2×C4—2448no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4#1451980912yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×Q8×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C8—22no—
581C8—41no—
582C8—1222no—
583C8—2441no—
584C22—1224no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22C4×D4×D6111yescharacteristic
592C22C22⋊Q8×D6111yescharacteristic
593C22#1684027011yescharacteristic
594C22AGL(1,ℤ/8)×D6111yescharacteristic
595C22AGL(1,ℤ/8)×D6111yescharacteristic
596C22C2×#1451980911yescharacteristic
597C22#1684055911yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—614no—
605C4—1222no—
606C4—1222no—
607C4—1222no—
608C4—4881no—
609C4—2442no—
610C4—2442no—
611C4C4×Q8×D6111yescharacteristic
612C4#1684023011yescharacteristic
613C4#1684029511yescharacteristic
614C4#1684031811yescharacteristic
615C2—2444no—
616C2#2144835311yescharacteristic
617C2#442×D6111yescharacteristic
618C2C4×#1451980911yescharacteristic
619C2#2145561411yescharacteristic
620C2#2145563311yescharacteristic
621C2#2145591511yescharacteristic
622C2#2145596511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1921C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1921D61
hypocenterC61#21613
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
2. #21448390C2semidirect
4. #21446769C2semidirect
5. #21446581C2semidirect
6. #21446411C2semidirect
10. #21445718C2semidirect
11. #21445715C2semidirect
12. #21445671C2semidirect
15. C61×#1921C2semidirect
22. #16840041C4semidirect
68. #14519776C2×C4semidirect
75. #14519739Q8semidirect
97. #14519710C2×C4semidirect
101. #14519706C2×C4semidirect
122. #107×C61C2×C4semidirect
263. D244C4×Q8semidirect
310. #1921D61semidirect
371. C61#21613semidirect
419. #107C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400412
#214483902
C4×#145197392
#214467692
#214465812
#214464112
#214460832
#214460792
#214458182
#214457182
#214457152
#214456712
#214456432
C61×#19212
#2161361