Finite Group Databasebeta

← #29670233

Subgroups — #29670233

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458725C211yesmaximal, characteristic
2Q8×#11758202C211yesmaximal, characteristic, complemented
3#21458678C211yesmaximal, characteristic
4#21457698C211yesmaximal, characteristic
5#21445498C211yesmaximal, characteristic, complemented
6#21445323C211yesmaximal, characteristic, complemented
7#21445275C211yesmaximal, characteristic, complemented
8#21445222C211yesmaximal, characteristic, complemented
9#21445217C211yesmaximal, characteristic
10#21444981C211yesmaximal, characteristic
11#21444961C211yesmaximal, characteristic, complemented
12#21444956C211yesmaximal, characteristic
13#21444672C211yesmaximal, characteristic, complemented
14#21444634C211yesmaximal, characteristic
15C61×#1645C211yesmaximal, characteristic, complemented
16Q8×Dic122C2211yescharacteristic
17#16840472C2211yescharacteristic
18#16840427C2211yescharacteristic
19#16840168—21no—
20#16840165—21no—
21#16840165C2211yescharacteristic
22#16840165C2211yescharacteristic
23#16840159—21no—
24#16840158C2211yescharacteristic
25#16840142C2211yescharacteristic
26#16840112C2211yescharacteristic
27#16840096—21no—
28#16840069—21no—
29#16840058C2211yescharacteristic
30#16840057—21no—
31#16840057C2211yescharacteristic
32#16840057C2211yescharacteristic
33#16840054—21no—
34#16840054C2211yescharacteristic
35#16840053C2211yescharacteristic
36#16840051—21no—
37#16840049C2211yescharacteristic
38#16840045—21no—
39#16840039—21no—
40#16840000—21no—
41#16839916—21no—
42#16839893C2211yescharacteristic
43Q8×#92418C2211yescharacteristic
44#16839886C2211yescharacteristic
45#16839860—21no—
46C4×#11758202—21no—
47C4×#11758202C2211yescharacteristic
48C4×#11758201C2211yescharacteristic
49#16839847—21no—
50#16839827C2211yescharacteristic
51#103×D61—21no—
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839510C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839508C2211yescharacteristic
57#16839494C2211yescharacteristic
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic
60Q82×C61C2211yescharacteristic
61C61×#487C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#462C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×Q8⋊C8C2211yescharacteristic
66C61×Q8⋊C8C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Dic244—41no—
69C2×#11758202D411yescharacteristic
70C2×#11758201D411yescharacteristic
71C2×C8×D61—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519776—21no—
77#14519776—41no—
78#14519769—21no—
79#14519755—41no—
80#14519755—41no—
81#14519753—21no—
82#14519753—21no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86C4×D244—21no—
87C4×D244C2311yescharacteristic
88#14519739—21no—
89C4⋊C4×D61—21no—
90#14519711—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710D411yescharacteristic
99#14519709—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708D411yescharacteristic
104C4⋊#92418—21no—
105C4⋊#92418—21no—
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519702—21no—
110#14519701—21no—
111#14519701—21no—
112#14519701—21no—
113#14519701—21no—
114#14519701D411yescharacteristic
115#14519701D411yescharacteristic
116#14519700—21no—
117#14519700D411yescharacteristic
118#14519699—21no—
119#14519697—21no—
120#14519697D411yescharacteristic
121#14519689—21no—
122#14519688—21no—
123C4×#92418—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C2×Q16×C61—21no—
128C4⋊Q8×C61—21no—
129C4⋊Q8×C61—21no—
130C4⋊Q8×C61—41no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×Q8×C61—21no—
134C4×Q8×C61—21no—
135C4×Q8×C61C2311yescharacteristic
136C4×Q8×C61C2311yescharacteristic
137#106×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142#103×C61—41no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149Q8×D61—42no—
150Q8×D61—81no—
151C2×D244C2×D411yescharacteristic
152C2×C4×D61—21no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—41no—
158C2×Dic122—41no—
159C2×Dic122C2×D411yescharacteristic
160Q16×C61—81no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—41no—
177C42×C61C2411yescharacteristic
178#11758202—22no—
179#11758201Q1614yescomplemented
180#11758198—21no—
181#11758198—41no—
182#11758198—41no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61C2×D411yescharacteristic
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—21no—
196C4×Dic61—41no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418C2×D411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203Dic244—82no—
204C8×D61—81no—
205C22×D61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—81no—
212Q8×C61—161no—
213Q8×C61C2×Q1612yes—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228D244—21no—
229D244C2×Q1612yes—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—82no—
233C4×D61—161no—
234Dic122—21no—
235Dic122—82no—
236Dic122—81no—
237Dic122—81no—
238Dic122—161no—
239Dic122#13712yes—
240C8×C61—41no—
241C8×C61—41no—
242C8×C61—41no—
243#92418—21no—
244#92418—21no—
245#92418—21no—
246#92418—41no—
247#92418—41no—
248#92418C2×Q1612yes—
249#19813—611nomaximal, Sylow
250C22×C61D4211yescharacteristic
251C2×D61—22no—
252C2×D61—41no—
253C2×D61—41no—
254C4×C61—22no—
255C4×C61—21no—
256C4×C61—41no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—81no—
260C4×C61—81no—
261C4×C61C22×Q1611yescharacteristic
262C4×C61C2×#13711yescharacteristic
263C4×C61#57711yescharacteristic
264C4×C61#57811yescharacteristic
265Dic61—22no—
266Dic61—41no—
267Dic61—41no—
268Dic61—81no—
269Dic61—81no—
270#3385—611no—
271Q8×Q16—611no—
272Q8×SD16—611no—
273#3299—611no—
274#1723—611no—
275#1692—611no—
276#1684—611no—
277#1676—611no—
278#1675—611no—
279#1645D6111yescharacteristic, complemented
280#1637—611no—
281#1633—611no—
282#1632—611no—
283#1584—611no—
284SD16⋊C8—611no—
285C2×C61#326611yescharacteristic
286C2×C61D4×Q1611yescharacteristic
287C2×C61#328811yescharacteristic
288D61—42no—
289Q82—611no—
290Q82C2×D6111yescharacteristic
291#554—611no—
292D4×Q8—611no—
293#501—1221no—
294#500—611no—
295#500—611no—
296#500—1221no—
297#498—611no—
298#498—1221no—
299#494—611no—
300#487—611no—
301#487C2×D6111yescharacteristic
302#484—1221no—
303#479—1221no—
304#477—611no—
305#477—611no—
306#477—611no—
307#477—611no—
308#477—1221no—
309#477—1221no—
310#476—611no—
311#476—611no—
312#476—1221no—
313#475—1221no—
314#475C2×D6111yescharacteristic
315#474—1221no—
316#467—1221no—
317#462C2×D6111yescharacteristic
318#451—1221no—
319#446—611no—
320C8×Q8—611no—
321C8×Q8—611no—
322#440—1221no—
323C4×Q16—611no—
324C4×Q16—1221no—
325C4×SD16—611no—
326C4×SD16—1221no—
327C8×D4—611no—
328C2×#103—1221no—
329#335—611no—
330#335—611no—
331#333—611no—
332#333—611no—
333#333C2×D6111yescharacteristic
334#332—611no—
335Q8⋊C8—611no—
336Q8⋊C8C2×D6111yescharacteristic
337Q8⋊C8C2×D6111yescharacteristic
338D4⋊C8—611no—
339#322—611no—
340C61#1981311yescharacteristic, complemented, Sylow
341C22×Q8—1221no—
342C2×Q16—21no—
343C2×Q16—611no—
344C2×Q16—2441no—
345C2×SD16—611no—
346C22×C8—1221no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—41no—
350C4⋊Q8—611no—
351C4⋊Q8—611no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—2441no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358#125—1221no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8—2441no—
362C22⋊Q8—2441no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8C22×D6111yescharacteristic
369C4×Q8C22×D6111yescharacteristic
370C4×D4—611no—
371C4×D4—1221no—
372C4×D4—1221no—
373C2×C4⋊C4—1221no—
374#107—611no—
375#107—1221no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107—1221no—
382#107—1221no—
383#106—21no—
384#106—611no—
385#106—1221no—
386#106—1221no—
387#106—1221no—
388#106—1221no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8C22×D6111yescharacteristic
395C4⋊C8C22×D6111yescharacteristic
396#103—21no—
397#103—21no—
398#103—41no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#102—611no—
409#102—1221no—
410#102—1221no—
411C22⋊C8—1221no—
412#97—1221no—
413C4×C8—611no—
414C4×C8—611no—
415C4×C8—1221no—
416C4×C8—1221no—
417C4×C8C22×D6111yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—41no—
421C2×Q8—611no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—4881no—
425C2×Q8—2441no—
426C2×Q8—2441no—
427C2×Q8—2442no—
428C2×Q8D4×D6111yescharacteristic
429C2×Q8D4×D6111yescharacteristic
430C2×D4—611no—
431C22×C4—1221no—
432C22×C4—2441no—
433C22×C4—2441no—
434Q16—81no—
435Q16—1222no—
436Q16—4882no—
437SD16—614no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—4881no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—611no—
457C4⋊C4—611no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4D4×D6111yescharacteristic
469C4⋊C4D4×D6111yescharacteristic
470C22⋊C4—1221no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C42—21no—
474C42—21no—
475C42—41no—
476C42—1221no—
477C42—2441no—
478C42C23×D6111yescharacteristic
479C23—1221no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—41no—
484Q8—41no—
485Q8—81no—
486Q8—161no—
487Q8—612no—
488Q8—1221no—
489Q8—9761no—
490Q8—4881no—
491Q8—4881no—
492Q8—4882no—
493Q8Q16×D6112yes—
494D4—612no—
495D4—1221no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—9761no—
506C2×C4—4882no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4#1451982211yescharacteristic
514C8—41no—
515C8—41no—
516C8—41no—
517C8—612no—
518C8—1221no—
519C8—1221no—
520C8—1221no—
521C8—2441no—
522C8—2441no—
523C22—1222no—
524C22—2441no—
525C22—2441no—
526C22#1684039411yescharacteristic
527C4—22no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—81no—
533C4—81no—
534C4—1222no—
535C4—4881no—
536C4—4881no—
537C4—2441no—
538C4—2441no—
539C4C2×Q16×D6111yescharacteristic
540C4#137×D6111yescharacteristic
541C4#1684056811yescharacteristic
542C4#1684057811yescharacteristic
543C2—2442no—
544C2#2145733211yescharacteristic
545C2#2145756411yescharacteristic
546C2#2145758811yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1645C2
socleC22×C61D42
prime radicalC2×D244C2×D4
hypercenter#1645D61
hypocenterC61#19813
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1645

Complements

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

NH ≅ G/Ntype
2. Q8×#11758202C2semidirect
5. #21445498C2semidirect
6. #21445323C2semidirect
7. #21445275C2semidirect
8. #21445222C2semidirect
11. #21444961C2semidirect
13. #21444672C2semidirect
15. C61×#1645C2semidirect
179. #11758201Q16semidirect
279. #1645D61semidirect
340. C61#19813semidirect

Maximal subgroups

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

subgroupindex
#214587252
Q8×#117582022
#214586782
#214576982
#214454982
#214453232
#214452752
#214452222
#214452172
#214449812
#214449612
#214449562
#214446722
#214446342
C61×#16452
#1981361