Finite Group Databasebeta

← #29687847

Subgroups — #29687847

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455607C211yesmaximal, characteristic
2#21454749C211yesmaximal, characteristic, complemented
3C4×#14519784C211yesmaximal, characteristic
4#21446762C211yesmaximal, characteristic, complemented
5#21446592C211yesmaximal, characteristic, complemented
6#21446416C212yesmaximal, complemented
7#21446092C212yesmaximal
8#21445702C212yesmaximal, complemented
9#21445688C211yesmaximal, characteristic, complemented
10#21445664C212yesmaximal
11C61×#1806C211yesmaximal, characteristic
12C2×#14519784C2211yescharacteristic
13C2×#14519784C2211yescharacteristic
14#16840319C2211yescharacteristic
15#16840069C412yescomplemented
16#16840062C412yescomplemented
17#16840057C411yescharacteristic, complemented
18#16840052C411yescharacteristic, complemented
19#16840047C411yescharacteristic, complemented
20#16840042C411yescharacteristic, complemented
21#16839889—21no—
22#16839889—21no—
23#16839886—21no—
24#16839886—21no—
25C2×#14519712C2212yes—
26#16839798C2212yes—
27#16839790C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29#16839758C2212yes—
30#16839720C2212yes—
31#16839718—21no—
32#16839718—21no—
33#16839718C2211yescharacteristic
34#16839718C2211yescharacteristic
35C4×#11758194—21no—
36C4×#11758194C2211yescharacteristic
37C2×#14519688C2212yes—
38C2×C4×#92418—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839526C2212yes—
41#16839522C2212yes—
42#16839522C2212yes—
43#16839517C2212yes—
44C4×#41×C61C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×#104×C61C2212yes—
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#339C2212yes—
49#14519784—22no—
50#14519784—22no—
51#14519784—22no—
52#14519784Q814yescomplemented
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C22×#92418—21no—
58C22×#92418—21no—
59C22×#92418C2312yes—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519769C2×C412yes—
65#14519756—21no—
66#14519756—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69#14519736C2312yes—
70#14519712—24no—
71#14519710C2×C412yes—
72#14519710C2×C412yes—
73#14519708C2×C412yes—
74#14519708C2×C412yes—
75C4⋊#92418C2×C411yescharacteristic
76C4⋊#92418C2×C411yescharacteristic
77C4⋊#92418C2×C412yes—
78#14519706—22no—
79#14519706—22no—
80#14519701C2×C412yes—
81#14519701C2×C412yes—
82#14519698C2×C412yes—
83#14519698C2×C412yes—
84#14519688—21no—
85#14519688—21no—
86#14519688—21no—
87#14519688—24no—
88#14519687—22no—
89C4×#92418—21no—
90C4×#92418—21no—
91C4×#92418—21no—
92C4×#92418D411yescharacteristic
93C4×#92418D411yescharacteristic
94C4×#92418D411yescharacteristic
95C4×#92418D411yescharacteristic
96C8×Dic61—22no—
97C2×#41×C61C2311yescharacteristic
98C2×M16×C61C2312yes—
99C4×Q8×C61C2×C411yescharacteristic
100C4×Q8×C61C2×C411yescharacteristic
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61C2×C411yescharacteristic
104C4×D4×C61C2×C411yescharacteristic
105#117×C61—21no—
106#117×C61C2311yescharacteristic
107C2×C4⋊C4×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2312yes—
110C2×C42×C61C2311yescharacteristic
111C4⋊C8×C61—21no—
112C4⋊C8×C61—21no—
113#104×C61—24no—
114#95×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61C2×Q812yes—
118#41×C61#4112yes—
119C2×Q8×C61C22×C411yescharacteristic
120C2×D4×C61—21no—
121C2×D4×C61C22×C411yescharacteristic
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×Dic122—21no—
130C2×Dic122—21no—
131M16×C61—22no—
132M16×C61—22no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—42no—
137C4⋊C4×C61C22×C411yescharacteristic
138C4⋊C4×C61C22×C411yescharacteristic
139C22⋊C4×C61—21no—
140C22⋊C4×C61—21no—
141C22⋊C4×C61—21no—
142C42×C61—21no—
143C42×C61—21no—
144C42×C61—21no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61C22×C411yescharacteristic
148C42×C61C22×C411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C22⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61C22×C412yes—
155C4⋊Dic61C22×C412yes—
156#11758196—22no—
157#11758196—22no—
158C4×Dic61—22no—
159#11758194—24no—
160#11758194—21no—
161#11758194—21no—
162#11758194—24no—
163#11758194C2×Q812yes—
164#11758194#4112yes—
165C2×#92418—24no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—24no—
173C2×#92418C22×C412yes—
174C2×#92418C22×C412yes—
175C2×#92418C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177C2×#92418C2×D411yescharacteristic
178C2×#92418C2×D411yescharacteristic
179C2×#92418C2×Q812yes—
180C2×#92418#4112yes—
181C23×C61—21no—
182C23×C61C2×#4111yescharacteristic
183Q8×C61—21no—
184Q8×C61C4×Q812yes—
185D4×C61—22no—
186D4×C61—22no—
187D4×C61—21no—
188D4×C61C4×Q812yes—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61C4×D411yescharacteristic
203C2×C4×C61C4×D411yescharacteristic
204C2×C4×C61C4×D411yescharacteristic
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×Q811yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×Dic61—22no—
212C2×Dic61—22no—
213Dic122—42no—
214Dic122—42no—
215C8×C61—42no—
216#92418—22no—
217#92418—22no—
218#92418—22no—
219#92418—21no—
220#92418—22no—
221#92418—21no—
222#92418C22⋊Q812yes—
223#92418C22⋊Q812yes—
224#21628—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—22no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61C2×C4×D411yescharacteristic
231C22×C61#52911yescharacteristic
232C22×C61#53311yescharacteristic
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—22no—
237C4×C61—41no—
238C4×C61—41no—
239C4×C61—82no—
240C4×C61#44412yes—
241C4×C61C2×C4×Q811yescharacteristic
242C4×C61C4×#4111yescharacteristic
243C4×C61C2×C22⋊Q811yescharacteristic
244C4×C61#52511yescharacteristic
245Dic61—42no—
246Dic61—42no—
247#3076—611no—
248#2960—611no—
249C4×#131—611no—
250#1986—611no—
251#1950—611no—
252#1922—612no—
253#1866—612no—
254#1806D6111yescharacteristic
255#1773—612no—
256#1770—611no—
257#1765—612no—
258C2×C61—21no—
259C2×C61—21no—
260C2×C61—42no—
261C2×C61C4×C22⋊Q811yescharacteristic
262C2×C61C2×#44411yescharacteristic
263C2×C61#295511yescharacteristic
264C2×#131—611no—
265C2×#131—611no—
266#533—611no—
267C4×#41C2×D6111yescharacteristic
268#479—612no—
269#478—612no—
270#477—611no—
271#476—611no—
272#475—611no—
273#474—611no—
274#446—1221no—
275#446—1221no—
276C8×Q8—1221no—
277C8×Q8—1221no—
278C2×#108—612no—
279#427—612no—
280#424—612no—
281#423C2×D6111yescharacteristic
282C2×C4⋊C8—611no—
283C2×#104C2×D6112yes—
284#416—612no—
285#405—612no—
286#405—611no—
287#405—611no—
288#405—1221no—
289#405—1221no—
290C4×M16—611no—
291C4×M16—1221no—
292C2×#97—612no—
293C2×C4×C8—611no—
294C2×C4×C8—1221no—
295C4×C4⋊C4C2×D6111yescharacteristic
296#341—612no—
297#339—612no—
298#339—612no—
299#339C2×D6112yes—
300#336—612no—
301C61#2162811yescharacteristic, complemented, Sylow
302C2×#41C22×D6111yescharacteristic
303#131—614no—
304#131—1222no—
305#131—1222no—
306#131—1222no—
307C2×M16—611no—
308C2×M16—611no—
309C2×M16—1221no—
310C2×M16—1221no—
311C2×M16C22×D6112yes—
312C22×C8—611no—
313C22×C8—611no—
314C22×C8—612no—
315C22×C8—1221no—
316C22×C8—1221no—
317C4⋊Q8—611no—
318C4⋊Q8—611no—
319#125—612no—
320C22⋊Q8—1221no—
321C22⋊Q8—1221no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×Q8C4×D6111yescharacteristic
325C4×Q8C4×D6111yescharacteristic
326C4×D4—21no—
327C4×D4—21no—
328C4×D4C4×D6111yescharacteristic
329C4×D4C4×D6111yescharacteristic
330#117—21no—
331#117—612no—
332#117C22×D6111yescharacteristic
333C2×C4⋊C4—21no—
334C2×C42—21no—
335C2×C42C22×D6112yes—
336C2×C42C22×D6111yescharacteristic
337#108—1224no—
338#107—612no—
339#107—612no—
340#106—612no—
341#106—612no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—611no—
345C4⋊C8—611no—
346C4⋊C8—612no—
347C4⋊C8—1222no—
348C4⋊C8—1222no—
349#104—24no—
350#103—612no—
351#103—612no—
352#102—612no—
353#102—612no—
354#97—1224no—
355#97—1221no—
356#97—1221no—
357#97—1222no—
358#97—1221no—
359C4×C8—611no—
360C4×C8—611no—
361C4×C8—611no—
362C4×C8—611no—
363C4×C8—1221no—
364C4×C8—1221no—
365C4×C8—1222no—
366C4×C8—1221no—
367#95—21no—
368#41—21no—
369#41—21no—
370#41#1175822412yes—
371#41Q8×D6112yes—
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8C2×C4×D6111yescharacteristic
375C2×D4—21no—
376C2×D4C2×C4×D6111yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4#1175822411yescharacteristic
382C22×C4#1175822411yescharacteristic
383C22×C4C23×D6111yescharacteristic
384M16—22no—
385M16—22no—
386M16—612no—
387M16—612no—
388M16—1224no—
389M16—1224no—
390M16—1221no—
391M16—1221no—
392C2×C8—22no—
393C2×C8—611no—
394C2×C8—611no—
395C2×C8—612no—
396C2×C8—612no—
397C2×C8—612no—
398C2×C8—612no—
399C2×C8—611no—
400C2×C8—611no—
401C2×C8—1224no—
402C2×C8—1224no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1221no—
408C2×C8—1221no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—612no—
413C4⋊C4—612no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4C2×C4×D6111yescharacteristic
419C4⋊C4C2×C4×D6111yescharacteristic
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—1222no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—22no—
428C42—22no—
429C42—1222no—
430C42C2×C4×D6111yescharacteristic
431C42C2×C4×D6111yescharacteristic
432C42D4×D6111yescharacteristic
433C42D4×D6111yescharacteristic
434C23—21no—
435C23C2×#1175822411yescharacteristic
436Q8—21no—
437Q8—2442no—
438Q8—2442no—
439Q8#1451974212yes—
440D4—22no—
441D4—22no—
442D4—21no—
443D4#1451974212yes—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4#1451973711yescharacteristic
459C2×C4#1451973711yescharacteristic
460C2×C4#1451973711yescharacteristic
461C2×C4#1451973711yescharacteristic
462C2×C4C22×C4×D6111yescharacteristic
463C2×C4C2×D4×D6111yescharacteristic
464C2×C4C2×Q8×D6111yescharacteristic
465C2×C4#41×D6111yescharacteristic
466C2×C4#41×D6111yescharacteristic
467C2×C4C2×#1175822411yescharacteristic
468C8—42no—
469C8—612no—
470C8—612no—
471C8—1222no—
472C8—1222no—
473C8—1222no—
474C8—1221no—
475C8—1221no—
476C8—1222no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22C2×#1451973711yescharacteristic
483C22#1684029311yescharacteristic
484C22#1684031411yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—2442no—
493C4—2442no—
494C4#1683988112yes—
495C4C2×#1451974211yescharacteristic
496C4#1684023011yescharacteristic
497C4C22⋊Q8×D6111yescharacteristic
498C4#1684027211yescharacteristic
499C2—21no—
500C2—21no—
501C2—42no—
502C2#2144834711yescharacteristic
503C2C2×#1683988111yescharacteristic
504C2#2145471811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1806C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#1806D61
hypocenterC61#21628
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21454749C2semidirect
4. #21446762C2semidirect
5. #21446592C2semidirect
6. #21446416C2semidirect
8. #21445702C2semidirect
9. #21445688C2semidirect
15. #16840069C4semidirect
16. #16840062C4semidirect
17. #16840057C4semidirect
18. #16840052C4semidirect
19. #16840047C4semidirect
20. #16840042C4semidirect
52. #14519784Q8semidirect
301. C61#21628semidirect

Maximal subgroups

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

subgroupindex
#214556072
#214547492
C4×#145197842
#214467622
#214465922
#214464162
#214460922
#214457022
#214456882
#214456642
C61×#18062
#2162861