Finite Group Databasebeta

← #29718929

Subgroups — #29718929

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455767C211yesmaximal, characteristic, complemented
2C2×#16839913C211yesmaximal, characteristic
3#21454477C211yesmaximal, characteristic, complemented
4#21447179C212yesmaximal, complemented
5#21447081C212yesmaximal, complemented
6#21446725C212yesmaximal, complemented
7#21446484C212yesmaximal, complemented
8#21446287C211yesmaximal, characteristic, complemented
9#21446024C211yesmaximal, characteristic, complemented
10#21445740C211yesmaximal, characteristic
11C2×C61×#418C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C2×#14519756C2211yescharacteristic
14#16840103—21no—
15#16840103—21no—
16#16840097—21no—
17#16840092—21no—
18#16839913—21no—
19#16839913—21no—
20#16839913—22no—
21#16839823—22no—
22#16839819—21no—
23#16839815—21no—
24C2×#14519712C2212yescomplemented
25#16839805C2211yescharacteristic, complemented
26#16839805C2211yescharacteristic, complemented
27#16839789C2212yescomplemented
28#16839770C2211yescharacteristic, complemented
29#16839770C2211yescharacteristic, complemented
30C2×#14519698C2212yes—
31C2×#14519695C2212yescomplemented
32C2×#14519694C2212yes—
33C2×#14519694C2212yes—
34#16839726C2211yescharacteristic
35#16839726C2211yescharacteristic
36C2×C22⋊#92418C2211yescharacteristic
37#16839718C2212yescomplemented
38#16839530C2212yescomplemented
39#16839526C2212yescomplemented
40C23×D4×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#418—21no—
44C61×#418—21no—
45C61×#418C2212yes—
46C61×#418C2212yes—
47C2×#102×C61—21no—
48C2×#102×C61—21no—
49C22×Dic122C2311yescharacteristic
50C2×#11758200—24no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758200—42no—
54C2×#11758194C2312yes—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C22×#92418C2312yes—
58#14519756—21no—
59#14519756—21no—
60#14519756—21no—
61#14519756—21no—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—21no—
64C2×C22⋊Dic61—21no—
65#14519712—42no—
66#14519710—21no—
67#14519710—21no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—22no—
71C4⋊#92418—22no—
72#14519700—21no—
73#14519700—21no—
74#14519700—21no—
75#14519700—21no—
76#14519698—22no—
77#14519698—22no—
78#14519695—42no—
79#14519694—24no—
80#14519694—42no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85C22⋊#92418D411yescharacteristic
86C22⋊#92418D411yescharacteristic
87C22⋊#92418D412yes—
88#14519688—22no—
89C4×#92418—22no—
90C25×C61—21no—
91C22×D4×C61—24no—
92C22×D4×C61—21no—
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61—42no—
98C22×D4×C61C2312yes—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—21no—
101C2×M16×C61—42no—
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61D411yescharacteristic
107#117×C61D411yescharacteristic
108#117×C61C2311yescharacteristic
109#117×C61C2311yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#102×C61—22no—
115#102×C61—22no—
116#102×C61—22no—
117#102×C61—22no—
118C24×C61—22no—
119C24×C61—44no—
120C24×C61—41no—
121C24×C61—41no—
122C24×C61—82no—
123C24×C61—82no—
124C24×C61#4111yescharacteristic
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—44no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61—41no—
138C2×D4×C61—41no—
139C2×D4×C61#4114yes—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4111yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156M16×C61—22no—
157M16×C61—22no—
158M16×C61—42no—
159C2×C8×C61—22no—
160C2×C8×C61—42no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C2×D411yescharacteristic
164C4⋊C4×C61C2×D411yescharacteristic
165C4⋊C4×C61C2×D411yescharacteristic
166C4⋊C4×C61C2×D411yescharacteristic
167C22⋊C4×C61—21no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177#11758200—44no—
178#11758200—44no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758196—41no—
182#11758196—41no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—24no—
197C23×C61—41no—
198C23×C61—44no—
199C23×C61—44no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—82no—
203C23×C61—82no—
204C23×C61—82no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—816no—
208C23×C61—82no—
209C23×C61#12111yescharacteristic
210C23×C61#12111yescharacteristic
211C23×C61C2×#4111yescharacteristic
212D4×C61—24no—
213D4×C61—24no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—42no—
217D4×C61—42no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—42no—
230C2×C4×C61—81no—
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—81no—
241Dic122—41no—
242Dic122—41no—
243Dic122—42no—
244C8×C61—42no—
245#92418—22no—
246#92418—22no—
247#92418—41no—
248#92418—41no—
249#92418—42no—
250#24521—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—84no—
261C22×C61—84no—
262C22×C61—88no—
263C22×C61—88no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61#47112yes—
267C22×C61C2×#12111yescharacteristic
268C22×C61#55112yes—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61#52511yescharacteristic
276C4×C61#53511yescharacteristic
277C4×C61#54612yes—
278Dic61—81no—
279Dic61—81no—
280#3095—611no—
281C2×#450—611no—
282#2924—611no—
283C2×#418D6111yescharacteristic
284#2063—612no—
285#2046—612no—
286#1977—612no—
287#1931—612no—
288#1901—611no—
289#1855—611no—
290#1783—611no—
291C2×C61—22no—
292C2×C61—22no—
293C2×C61—41no—
294C2×C61—84no—
295C2×C61#241111yescharacteristic
296C2×C61C2×#47111yescharacteristic
297C2×C61#306811yescharacteristic
298C23×D4C2×D6111yescharacteristic
299C22×SD16—612no—
300C22×M16C2×D6111yescharacteristic
301C2×C22⋊Q8—611no—
302C2×#117C2×D6111yescharacteristic
303#485—1221no—
304#485—1221no—
305#484—1221no—
306#483—1221no—
307#450—1221no—
308#450—1221no—
309#450—1222no—
310#433—1222no—
311#432—1221no—
312#431—1221no—
313C2×#108—612no—
314#428—611no—
315#428—611no—
316#424—612no—
317#419—611no—
318#419—611no—
319#418—21no—
320#418—21no—
321#418C2×D6112yes—
322#418C2×D6112yes—
323C2×#102—21no—
324C2×#102—21no—
325C2×#102—612no—
326C2×#101—612no—
327C2×#100—612no—
328C2×#100—612no—
329#407—611no—
330#407—611no—
331C2×C22⋊C8—611no—
332#405—612no—
333#343—612no—
334#341—612no—
335C61#2452111yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8—611no—
338C22×D4—24no—
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—21no—
344C22×D4—42no—
345C22×D4C22×D6112yes—
346C23×C4C22×D6111yescharacteristic
347C2×SD16—1222no—
348C2×SD16—1222no—
349C2×SD16—1224no—
350C2×SD16—2442no—
351C2×M16—21no—
352C2×M16—42no—
353C2×M16—611no—
354C2×M16—612no—
355C2×M16—611no—
356C2×M16C22×D6112yes—
357C22×C8—21no—
358C22×C8—612no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8—1221no—
362C22⋊Q8—1221no—
363#117—21no—
364#117—21no—
365#117C22⋊D6111yescharacteristic
366#117C22⋊D6111yescharacteristic
367#117C22×D6111yescharacteristic
368#117C22×D6111yescharacteristic
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22⋊D6111yescharacteristic
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1221no—
375C2×C42—21no—
376#108—2442no—
377#107—1221no—
378#107—1221no—
379#106—1221no—
380#106—1221no—
381C4⋊C8—1222no—
382C4⋊C8—1222no—
383#103—1221no—
384#103—1221no—
385#103—1221no—
386#103—1221no—
387#102—22no—
388#102—22no—
389#102—22no—
390#102—22no—
391#102—1222no—
392#102—1222no—
393#101—2442no—
394#100—1224no—
395#100—2442no—
396C22⋊C8—611no—
397C22⋊C8—611no—
398C22⋊C8—612no—
399C22⋊C8—1221no—
400C22⋊C8—1221no—
401C22⋊C8—1221no—
402C22⋊C8—1221no—
403#97—1222no—
404C4×C8—1222no—
405C24—22no—
406C24—44no—
407C24—41no—
408C24—41no—
409C24—82no—
410C24—82no—
411C24#1175822411yescharacteristic
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4#1175822214yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4#1175822411yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443SD16—2444no—
444SD16—2444no—
445M16—22no—
446M16—22no—
447M16—42no—
448M16—1222no—
449M16—1222no—
450M16—1222no—
451M16—1222no—
452C2×C8—22no—
453C2×C8—42no—
454C2×C8—612no—
455C2×C8—612no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1221no—
460C2×C8—1221no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4C2×C22⋊D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C4⋊C4C2×C22⋊D6111yescharacteristic
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—41no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—816no—
495C23—82no—
496C23#1451974911yescharacteristic
497C23#1451974911yescharacteristic
498C23C2×#1175822411yescharacteristic
499Q8—2441no—
500Q8—2441no—
501Q8—2442no—
502D4—24no—
503D4—24no—
504D4—44no—
505D4—44no—
506D4—42no—
507D4—42no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—81no—
521C2×C4—4881no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4#1451974911yescharacteristic
526C2×C4#1451974911yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822212yes—
530C2×C4#1451982212yes—
531C8—42no—
532C8—1222no—
533C8—1222no—
534C8—2441no—
535C8—2441no—
536C8—2442no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—44no—
546C22—84no—
547C22—88no—
548C22—84no—
549C22—88no—
550C22—82no—
551C22—82no—
552C22#1684002312yes—
553C22C2×#1451974911yescharacteristic
554C22#1684043812yes—
555C4—21no—
556C4—21no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—4881no—
562C4—4881no—
563C4#1684027311yescharacteristic
564C4#1684032911yescharacteristic
565C4#1684040212yes—
566C2—22no—
567C2—22no—
568C2—41no—
569C2—84no—
570C2#2144920411yescharacteristic
571C2C2×#1684002311yescharacteristic
572C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC22×C61C2×#121
prime radicalC23×D4×C61C22
hypercenterC2×#418D61
hypocenterC61#24521
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
1. #21455767C2semidirect
3. #21454477C2semidirect
4. #21447179C2semidirect
5. #21447081C2semidirect
6. #21446725C2semidirect
7. #21446484C2semidirect
8. #21446287C2semidirect
9. #21446024C2semidirect
24. C2×#14519712C22semidirect
25. #16839805C22semidirect
26. #16839805C22semidirect
27. #16839789C22semidirect
28. #16839770C22semidirect
29. #16839770C22semidirect
31. C2×#14519695C22semidirect
37. #16839718C22semidirect
38. #16839530C22semidirect
39. #16839526C22semidirect
335. C61#24521semidirect

Maximal subgroups

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

subgroupindex
#214557672
C2×#168399132
#214544772
#214471792
#214470812
#214467252
#214464842
#214462872
#214460242
#214457402
C2×C61×#4182
#2452161