Finite Group Databasebeta

← #29670256

Subgroups — #29670256

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458824C211yesmaximal, characteristic, complemented
2#21458740C211yesmaximal, characteristic
3#21458694C211yesmaximal, characteristic
4#21445498C211yesmaximal, characteristic, complemented
5#21445327C211yesmaximal, characteristic, complemented
6#21445274C211yesmaximal, characteristic, complemented
7#21445211C211yesmaximal, characteristic, complemented
8#21445184C211yesmaximal, characteristic
9#21445068C211yesmaximal, characteristic
10#21445059C211yesmaximal, characteristic, complemented
11#21445033C211yesmaximal, characteristic
12#21444975C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic
15C61×#3299C211yesmaximal, characteristic
16#16840490C2211yescharacteristic
17#16840490C2211yescharacteristic
18#16840168—21no—
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840161—21no—
22#16840158C2211yescharacteristic
23#16840142C2211yescharacteristic
24#16840140—21no—
25#16840112C2211yescharacteristic
26#16840066—21no—
27#16840062—21no—
28#16840062—21no—
29#16840054—21no—
30#16840054C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16840042C2211yescharacteristic
34#16839972C2211yescharacteristic
35#16839972C2211yescharacteristic
36#16839971—21no—
37#16839971—21no—
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839862—21no—
42#16839856—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#484—21no—
60C61×#477C2211yescharacteristic
61C61×#467—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×Dic244—21no—
68C2×Dic244—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200D411yescharacteristic
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519775—21no—
75#14519775—21no—
76#14519769—21no—
77#14519769—21no—
78#14519767—41no—
79#14519767—41no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519742—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—41no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519701—41no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
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—
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C22×C8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122C2×D411yescharacteristic
158Q16×C61—41no—
159Q16×C61—81no—
160C2×C8×C61—21no—
161C2×C8×C61—81no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
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×C61—41no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61C2411yescharacteristic
182#11758202—81no—
183#11758202—81no—
184#11758200Q1614yescomplemented
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206Dic244—42no—
207Dic244—81no—
208C23×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216Q8×C61—82no—
217D4×C61—21no—
218D4×C61C2×Q1612yes—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61—161no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237Dic122—21no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—41no—
242Dic122C2×Q1612yes—
243C8×C61—21no—
244C8×C61—21no—
245C8×C61—41no—
246#92418—21no—
247#92418—41no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418C2×Q1612yes—
252#19815—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61D4211yescharacteristic
257C4×C61—21no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C22×Q1611yescharacteristic
266C4×C61#57511yescharacteristic
267C4×C61#57611yescharacteristic
268C4×C61#57811yescharacteristic
269Dic61—22no—
270Dic61—41no—
271Dic61—81no—
272Dic61—81no—
273#3395—611no—
274#3386—611no—
275#3381—611no—
276#3299D6111yescharacteristic
277#1723—611no—
278#1692—611no—
279#1684—611no—
280#1674—611no—
281#1669—611no—
282#1649—611no—
283#1648—611no—
284#1644—611no—
285#1636—611no—
286#1584—611no—
287SD16⋊C8—611no—
288C2×C61—42no—
289C2×C61#326811yescharacteristic
290C2×C61D4×Q1611yescharacteristic
291C2×C61#328911yescharacteristic
292#557—611no—
293#557—611no—
294#554C2×D6111yescharacteristic
295D4×Q8C2×D6111yescharacteristic
296#501—1221no—
297#500—611no—
298#500—611no—
299#499—1221no—
300#498—611no—
301#494—611no—
302#494—1221no—
303#487—611no—
304#487C2×D6111yescharacteristic
305#484—21no—
306#479—1221no—
307#478—1221no—
308#478—1221no—
309#477—611no—
310#477—1221no—
311#477C2×D6111yescharacteristic
312#475—611no—
313#474—611no—
314#474—611no—
315#467—21no—
316#462—611no—
317#462—611no—
318#462—1221no—
319#462—1221no—
320#462C2×D6111yescharacteristic
321#451—21no—
322#446—611no—
323C8×Q8—611no—
324C8×Q8—611no—
325#440—1221no—
326C4×Q16—1221no—
327C4×Q16C2×D6111yescharacteristic
328C4×SD16—611no—
329C4×SD16—1221no—
330C8×D4C2×D6111yescharacteristic
331C2×#103—21no—
332#335—611no—
333#335—611no—
334#333—611no—
335#332—611no—
336#332—611no—
337#332—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341D4⋊C8—611no—
342#322—611no—
343C61#1981511yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C2×Q16—21no—
346C2×Q16—1221no—
347C2×Q16—1221no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×Q16C22⋊D6111yescharacteristic
351C2×SD16—611no—
352C22×C8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#125—21no—
361#125—1221no—
362#125—1221no—
363#125—2441no—
364#125—2441no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—41no—
368C22⋊Q8—41no—
369C4×Q8—611no—
370C4×Q8—1221no—
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×Q8—1221no—
374C4×Q8C22×D6111yescharacteristic
375C4×Q8C22×D6111yescharacteristic
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C22×D6111yescharacteristic
379C2×C4⋊C4—21no—
380#107—21no—
381#107—1221no—
382#107—1221no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—2441no—
387#107C22⋊D6111yescharacteristic
388#106—611no—
389#106—1221no—
390#106—1221no—
391#106—1221no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6111yescharacteristic
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103—611no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—2441no—
409#103C22⋊D6111yescharacteristic
410#103C22⋊D6111yescharacteristic
411#102—611no—
412#102—1221no—
413#102—1221no—
414C22⋊C8—21no—
415#97—1221no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—1221no—
419C4×C8—1221no—
420C4×C8C22×D6111yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—41no—
424C2×Q8—42no—
425C2×Q8—81no—
426C2×Q8—611no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8C2×C22⋊D6111yescharacteristic
430C2×Q8C2×C22⋊D6111yescharacteristic
431C2×D4D4×D6111yescharacteristic
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—41no—
435Q16—41no—
436Q16—81no—
437Q16—4881no—
438Q16—4881no—
439Q16—4881no—
440Q16—2442no—
441SD16—614no—
442C2×C8—21no—
443C2×C8—81no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8C2×C22⋊D6111yescharacteristic
451C2×C8C2×C22⋊D6111yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—611no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4D4×D6111yescharacteristic
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—21no—
480C42—21no—
481C42—1221no—
482C42—2441no—
483C42—2441no—
484C42C23×D6111yescharacteristic
485C23—21no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—82no—
494Q8—612no—
495Q8—1221no—
496Q8—2441no—
497Q8—2441no—
498Q8—2441no—
499Q8—2441no—
500D4—21no—
501D4Q16×D6112yes—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—161no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4#1451982211yescharacteristic
520C8—21no—
521C8—21no—
522C8—41no—
523C8—612no—
524C8—1221no—
525C8—2441no—
526C8—2441no—
527C8—2441no—
528C8—2441no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22D4×C22⋊D6111yescharacteristic
533C4—21no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—81no—
540C4—81no—
541C4—1222no—
542C4—4881no—
543C4—4881no—
544C4—2441no—
545C4C2×Q16×D6111yescharacteristic
546C4#1684056011yescharacteristic
547C4#1684056511yescharacteristic
548C4#1684057511yescharacteristic
549C2—42no—
550C2#2145735511yescharacteristic
551C2Q16×C22⋊D6111yescharacteristic
552C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3299C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3299D61
hypocenterC61#19815
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3299

Complements

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

NH ≅ G/Ntype
1. #21458824C2semidirect
4. #21445498C2semidirect
5. #21445327C2semidirect
6. #21445274C2semidirect
7. #21445211C2semidirect
10. #21445059C2semidirect
12. #21444975C2semidirect
13. #21444670C2semidirect
184. #11758200Q16semidirect
343. C61#19815semidirect

Maximal subgroups

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

subgroupindex
#214588242
#214587402
#214586942
#214454982
#214453272
#214452742
#214452112
#214451842
#214450682
#214450592
#214450332
#214449752
#214446702
#214446312
C61×#32992
#1981561