Finite Group Databasebeta

← #29730356

Subgroups — #29730356

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840048C211yesmaximal, characteristic, complemented
2#21450992C211yesmaximal, characteristic
3#21450728C211yesmaximal, characteristic
4#21447252C211yesmaximal, characteristic, complemented
5#21447252C211yesmaximal, characteristic, complemented
6#21447023C211yesmaximal, characteristic, complemented
7#21447016C211yesmaximal, characteristic, complemented
8#21446792C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446252C211yesmaximal, characteristic, complemented
11#21446242C211yesmaximal, characteristic, complemented
12#21445852C211yesmaximal, characteristic, complemented
13#21445844C211yesmaximal, characteristic, complemented
14#21445650C211yesmaximal, characteristic, complemented
15C2×C61×#435C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840048—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29#16839693—21no—
30#16839676—21no—
31#16839674—21no—
32#16839674—21no—
33#16839665—21no—
34#16839650—21no—
35#16839650—21no—
36#16839628C2211yescharacteristic
37#16839627—21no—
38#16839622C2211yescharacteristic
39#16839622C2211yescharacteristic
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839622C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C61×#435—24no—
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63#14519777—22no—
64#14519777—22no—
65#14519756—22no—
66#14519756—22no—
67#14519756—41no—
68#14519756—41no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92#14519711—41no—
93#14519709—22no—
94#14519709—22no—
95#14519709—41no—
96#14519702—22no—
97#14519702—22no—
98#14519702—22no—
99#14519702—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C22×D4×C61D411yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C23×C4×C61C2311yescharacteristic
112C2×M16×C61—22no—
113C2×M16×C61—22no—
114C2×M16×C61—41no—
115C22×C8×C61—22no—
116C22×C8×C61—22no—
117C22×C8×C61—41no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C4⋊C8×C61D414yes—
129C22⋊C8×C61—22no—
130C22⋊C8×C61—22no—
131#97×C61—22no—
132C24×C61C2×D411yescharacteristic
133C24×C61C2×D411yescharacteristic
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—41no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—42no—
168C2×Dic122—81no—
169C2×Dic122—81no—
170C2×Dic122#4112yes—
171M16×C61—44no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—44no—
182C2×C8×C61C2×D412yes—
183C2×C8×C61C2×D412yes—
184C2×C8×C61#4112yes—
185C4⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61C2×D412yes—
189C42×C61#4112yes—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—81no—
195C22⋊Dic61—81no—
196C22⋊Dic61—81no—
197C22⋊Dic61—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—42no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61#13712yes—
232C23×C61C22×D411yescharacteristic
233D4×C61—44no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61#12112yes—
253C2×C4×C61#12112yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122—84no—
289C8×C61—22no—
290C8×C61—22no—
291C8×C61—22no—
292C8×C61—41no—
293#25356—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61C2×#12111yescharacteristic
309C22×C61#54611yescharacteristic
310C22×C61#54711yescharacteristic
311C22×C61C2×#13711yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61#52911yescharacteristic
320C4×C61#54211yescharacteristic
321C4×C61#54611yescharacteristic
322C4×C61#54811yescharacteristic
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#475—611no—
332C2×#435D6111yescharacteristic
333#2563—611no—
334#2536—611no—
335#2077—611no—
336#2077—611no—
337#2036—611no—
338#2035—611no—
339#1993—611no—
340#1949—611no—
341#1894—611no—
342#1892—611no—
343#1809—611no—
344#1807—611no—
345#1763—611no—
346C2×C61—42no—
347C2×C61—81no—
348C2×C61#242711yescharacteristic
349C2×C61#305611yescharacteristic
350C2×C61#305711yescharacteristic
351C2×C61#331611yescharacteristic
352C2×C61#332111yescharacteristic
353C2×C61#334411yescharacteristic
354C2×C61#334811yescharacteristic
355C22×M16C2×D6111yescharacteristic
356C23×C8C2×D6111yescharacteristic
357C2×C4⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×C4×Q8—611no—
361C2×C4×D4C2×D6111yescharacteristic
362#475—1224no—
363#435—24no—
364C2×#107—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×#106—611no—
368C2×C4⋊C8C2×D6111yescharacteristic
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×C22⋊C8C2×D6111yescharacteristic
374C2×C22⋊C8C2×D6111yescharacteristic
375C2×#97C2×D6111yescharacteristic
376#398—1221no—
377#394—1221no—
378#393—1221no—
379#393—1221no—
380#390—1221no—
381#388—1221no—
382#388—1221no—
383#382—611no—
384#382—1221no—
385#380—611no—
386#380—611no—
387#380—611no—
388#380—611no—
389#380—611no—
390#380—611no—
391C4×C4⋊C4—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400C61#2535611yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8—1221no—
403C22×D4D24411yescharacteristic
404C23×C4C22×D6111yescharacteristic
405C23×C4C22×D6111yescharacteristic
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—41no—
409C22×C8—22no—
410C22×C8—22no—
411C22×C8—41no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—1222no—
417C4⋊Q8—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—2441no—
421C22⋊Q8—2441no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×D4—22no—
425C4×D4—22no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4D24411yescharacteristic
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4D24411yescharacteristic
447C2×C22⋊C4D24411yescharacteristic
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#107—2441no—
452#106—1222no—
453#106—1222no—
454#106—2441no—
455C4⋊C8C22⋊D6114yes—
456#103—1222no—
457#103—1222no—
458#103—1222no—
459#103—1222no—
460C22⋊C8—22no—
461C22⋊C8—22no—
462#97—22no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24C2×D24411yescharacteristic
473C24C2×D24411yescharacteristic
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—4881no—
481C2×Q8—4881no—
482C2×Q8—2442no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—41no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4C2×D24411yescharacteristic
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C23×D6111yescharacteristic
511M16—44no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—44no—
522C2×C8#1175822412yes—
523C2×C8C2×C22⋊D6112yes—
524C2×C8C2×C22⋊D6112yes—
525C4⋊C4—22no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—4881no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C42—2442no—
555C42—2442no—
556C42#1175822212yes—
557C42C2×C22⋊D6112yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23#1451981012yes—
572C23C22×D24411yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—4884no—
576Q8—2441no—
577Q8—2441no—
578D4—44no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—1222no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451975112yes—
622C2×C4#1451975112yes—
623C2×C4C22×C22⋊D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822411yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—22no—
630C8—22no—
631C8—22no—
632C8—41no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22C2×#1451975111yescharacteristic
648C22#1684040711yescharacteristic
649C22#1684041611yescharacteristic
650C22C2×#1451981011yescharacteristic
651C22#1684056211yescharacteristic
652C22#1684057411yescharacteristic
653C22#1684057411yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—4881no—
659C4—4881no—
660C4—4881no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—2442no—
666C4#1684029411yescharacteristic
667C4#1684037711yescharacteristic
668C4#1684040611yescharacteristic
669C4#1684041911yescharacteristic
670C2—42no—
671C2—81no—
672C2#2144939011yescharacteristic
673C2#2145544911yescharacteristic
674C2#2145545811yescharacteristic
675C2#2145793211yescharacteristic
676C2#2145800711yescharacteristic
677C2#2145827611yescharacteristic
678C2#2145833611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#435C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×#435D61
hypocenterC61#25356
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C2×#16840048C2semidirect
4. #21447252C2semidirect
5. #21447252C2semidirect
6. #21447023C2semidirect
7. #21447016C2semidirect
8. #21446792C2semidirect
9. #21446589C2semidirect
10. #21446252C2semidirect
11. #21446242C2semidirect
12. #21445852C2semidirect
13. #21445844C2semidirect
14. #21445650C2semidirect
400. C61#25356semidirect

Maximal subgroups

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

subgroupindex
C2×#168400482
#214509922
#214507282
#214472522
#214470232
#214470162
#214467922
#214465892
#214462522
#214462422
#214458522
#214458442
#214456502
C2×C61×#4352
#2535661