Finite Group Databasebeta

← #29670211

Subgroups — #29670211

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458740C211yesmaximal, characteristic
2Q8×#11758200C211yesmaximal, characteristic
3#21458495C211yesmaximal, characteristic, complemented
4#21445494C211yesmaximal, characteristic, complemented
5#21445323C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic
7#21445211C211yesmaximal, characteristic, complemented
8#21445186C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic
10#21445059C211yesmaximal, characteristic, complemented
11#21445038C211yesmaximal, characteristic, complemented
12#21444984C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic
15D4×Q16×C61C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840490C2211yescharacteristic
18#16840168—21no—
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840158—21no—
22#16840158C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840113C2211yescharacteristic
25#16840111—21no—
26#16840062—21no—
27#16840057C2211yescharacteristic
28#16840056—21no—
29#16840054—21no—
30#16840052—21no—
31#16840052C2211yescharacteristic
32#16840044—21no—
33#16840044C2211yescharacteristic
34#16840044C2211yescharacteristic
35#16840042C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839970—21no—
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839862—21no—
42#16839854—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C22×Q16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#467—21no—
61C61×#462C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C61×#451—21no—
65C4×Q16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200D411yescharacteristic
69#14519777—21no—
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519776—21no—
73#14519776—41no—
74#14519775—21no—
75#14519769—21no—
76#14519767—41no—
77Q8×Dic61—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80C4×Dic122C2311yescharacteristic
81#14519742—21no—
82#14519710—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—41no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—41no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×Q8×C61—21no—
116C22×Q8×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×Q16×C61—44no—
122C2×Q16×C61D411yescharacteristic
123C22×C8×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127C4⋊Q8×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131C22⋊Q8×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61—81no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162C2×Dic122C2×D411yescharacteristic
163Q16×C61—22no—
164Q16×C61—42no—
165Q16×C61—42no—
166Q16×C61—42no—
167Q16×C61—81no—
168C2×C8×C61—21no—
169C2×C8×C61—42no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189#11758202—42no—
190#11758200Q1614yescomplemented
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199#11758196—41no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—21no—
203C4×Dic61—41no—
204C4×Dic61—41no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C23×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—82no—
220Q8×C61—82no—
221Q8×C61AGL(1,ℤ/8)12yes—
222D4×C61—21no—
223D4×C61C2×Q1612yes—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242Dic122—21no—
243Dic122—41no—
244Dic122—41no—
245Dic122—81no—
246Dic122—161no—
247Dic122C2×Q1612yescomplemented
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418C2×Q1612yescomplemented
257#19812—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61D4211yescharacteristic
262C4×C61—22no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C22×Q1611yescharacteristic
271C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C61#57711yescharacteristic
273C4×C61#57811yescharacteristic
274Dic61—22no—
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278#3386—611no—
279Q8×SD16—611no—
280#3363—611no—
281D4×Q16D6111yescharacteristic
282#1723—611no—
283#1692—611no—
284#1683—611no—
285#1674—611no—
286#1670—611no—
287#1649—611no—
288#1648—611no—
289#1645—611no—
290#1637—611no—
291#1584—611no—
292SD16⋊C8—611no—
293C2×C61—42no—
294C2×C61#326511yescharacteristic
295C2×C61D4×Q1611yescharacteristic
296C2×C61#328911yescharacteristic
297C22×Q16—21no—
298Q82—611no—
299#557—611no—
300D4×Q8C2×D6111yescharacteristic
301D4×Q8C2×D6111yescharacteristic
302#501—1221no—
303#500—611no—
304#500—611no—
305#498—611no—
306#498—1221no—
307#494C2×D6111yescharacteristic
308#487—611no—
309#487—611no—
310#487—1221no—
311#478—1221no—
312#477—611no—
313#477—1221no—
314#477—1221no—
315#476—611no—
316#476—1221no—
317#475—611no—
318#475—611no—
319#475—1221no—
320#474—611no—
321#467—21no—
322#462—611no—
323#462—1221no—
324#462C2×D6111yescharacteristic
325#462C2×D6111yescharacteristic
326#451—21no—
327#451—21no—
328#446—611no—
329C8×Q8—611no—
330C8×Q8—611no—
331#440—1221no—
332C4×Q16—1221no—
333C4×Q16C2×D6111yescharacteristic
334C4×SD16—611no—
335C4×SD16—1221no—
336C8×D4C2×D6111yescharacteristic
337#335—611no—
338#335—611no—
339#333—611no—
340#332—611no—
341#332—611no—
342#332—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346D4⋊C8—611no—
347#322—611no—
348C61#1981211yescharacteristic, complemented, Sylow
349C22×Q8—21no—
350C22×Q8—21no—
351C2×Q16—21no—
352C2×Q16—21no—
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16—44no—
356C2×Q16—1221no—
357C2×Q16C22⋊D6111yescharacteristic
358C2×SD16—611no—
359C22×C8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8—2441no—
368C4⋊Q8C22×D6111yescharacteristic
369C4⋊Q8C22×D6111yescharacteristic
370#125—1221no—
371#125—2441no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—41no—
375C22⋊Q8—41no—
376C4×Q8—611no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×Q8C22×D6111yescharacteristic
382C4×Q8C22×D6111yescharacteristic
383C4×D4—21no—
384C4×D4—21no—
385C4×D4C22×D6111yescharacteristic
386#107—1221no—
387#107—1221no—
388#107—1221no—
389#107—1221no—
390#107—2441no—
391#107C22⋊D6111yescharacteristic
392#106—611no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106—1221no—
397#106—1221no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—21no—
407#103—611no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—2441no—
415#103C22⋊D6111yescharacteristic
416#103C22⋊D6111yescharacteristic
417#102—611no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—21no—
421#97—1221no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—1221no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—81no—
434C2×Q8—81no—
435C2×Q8—611no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—2441no—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4D4×D6111yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—41no—
445Q16—22no—
446Q16—42no—
447Q16—42no—
448Q16—42no—
449Q16—81no—
450Q16—2442no—
451SD16—614no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8C2×C22⋊D6111yescharacteristic
461C2×C8C2×C22⋊D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—611no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4D4×D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C42—21no—
488C42—21no—
489C42—1221no—
490C42—2441no—
491C42—2441no—
492C42C23×D6111yescharacteristic
493C23—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—82no—
503Q8—612no—
504Q8—1221no—
505Q8—9761no—
506Q8—4881no—
507Q8—2441no—
508Q8—2441no—
509Q8#1451980312yes—
510D4—21no—
511D4Q16×D6112yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4#1451982211yescharacteristic
530C8—21no—
531C8—21no—
532C8—41no—
533C8—612no—
534C8—1221no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22D4×C22⋊D6111yescharacteristic
543C4—22no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—1222no—
552C4—4881no—
553C4—4881no—
554C4—2441no—
555C4C2×Q16×D6111yescharacteristic
556C4C2×#1451980311yescharacteristic
557C4#1684057311yescharacteristic
558C4#1684057511yescharacteristic
559C2—42no—
560C2#2145731011yescharacteristic
561C2Q16×C22⋊D6111yescharacteristic
562C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×Q16×C61C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenterD4×Q16D61
hypocenterC61#19812
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
3. #21458495C2semidirect
4. #21445494C2semidirect
5. #21445323C2semidirect
7. #21445211C2semidirect
8. #21445186C2semidirect
10. #21445059C2semidirect
11. #21445038C2semidirect
13. #21444670C2semidirect
190. #11758200Q16semidirect
247. Dic122C2×Q16semidirect
256. #92418C2×Q16semidirect
348. C61#19812semidirect

Maximal subgroups

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

subgroupindex
#214587402
Q8×#117582002
#214584952
#214454942
#214453232
#214452702
#214452112
#214451862
#214450682
#214450592
#214450382
#214449842
#214446702
#214446312
D4×Q16×C612
#1981261