Finite Group Databasebeta

← #29662915

Subgroups — #29662915

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456195C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic
3Q8×#11758194C211yesmaximal, characteristic, complemented
4#21445389C212yesmaximal, complemented
5#21445361C212yesmaximal, complemented
6#21445265C212yesmaximal, complemented
7#21445237C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444486C211yesmaximal, characteristic
11C61×#3104C211yesmaximal, characteristic, complemented
12C22×#11758201—21no—
13C2×#14519774C2211yescharacteristic
14#16840128—22no—
15#16840110C2212yes—
16#16840110C2212yes—
17#16839962—21no—
18#16839962—21no—
19#16839962C2212yes—
20#16839962C2212yes—
21#16839893—22no—
22#16839893C2212yescomplemented
23Q8×#92418C2212yescomplemented
24#16839786—21no—
25#16839786C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839766—21no—
28#16839766—21no—
29C2×#14519697—21no—
30C4×#11758194—21no—
31C4×#11758194C2211yescharacteristic
32#16839514C2212yescomplemented
33#16839512C2212yescomplemented
34#16839511C2212yescomplemented
35#16839508C2212yescomplemented
36#16839505C2212yes—
37#16839505C2212yes—
38C61×#533C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C61×#484—21no—
41C61×#477C2211yescharacteristic, complemented
42C61×#477C2211yescharacteristic, complemented
43C61×#462C2211yescharacteristic, complemented
44C61×#462C2211yescharacteristic, complemented
45C61×#451—21no—
46C4×Q16×C61—21no—
47C4×Q16×C61—21no—
48C61×#431C2211yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—44no—
55C2×#11758194—21no—
56C2×#11758194D411yescharacteristic
57C2×#11758194D411yescharacteristic
58C22×#92418—21no—
59#14519774—21no—
60#14519774—21no—
61#14519774—21no—
62#14519774C2312yes—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65C4⋊#92418—22no—
66C4⋊#92418—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519697—22no—
70#14519697—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75#14519688—22no—
76#14519688D412yes—
77C4×#92418—22no—
78C4×#92418C2312yes—
79C22×Q8×C61D411yescharacteristic
80C2×Q16×C61—21no—
81C2×Q16×C61—21no—
82C4⋊Q8×C61C2311yescharacteristic
83C4⋊Q8×C61C2311yescharacteristic
84#125×C61—21no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61D411yescharacteristic
92C4×Q8×C61D411yescharacteristic
93C4×Q8×C61C2311yescharacteristic
94C4×Q8×C61C2311yescharacteristic
95#117×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61C2311yescharacteristic
100#107×C61—21no—
101#107×C61—21no—
102C4⋊C8×C61C2311yescharacteristic
103C4⋊C8×C61C2311yescharacteristic
104#103×C61—21no—
105#103×C61—21no—
106#103×C61—21no—
107#103×C61—21no—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C23×D61—41no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—41no—
116C2×Q8×C61C2×D411yescharacteristic
117C2×Q8×C61C2×D411yescharacteristic
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—42no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—41no—
133C2×D244—82no—
134Q16×C61—42no—
135Q16×C61—42no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61C2×D411yescharacteristic
150C4⋊C4×C61C2×D411yescharacteristic
151C22⋊C4×C61—41no—
152C42×C61—21no—
153C42×C61—21no—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61—41no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61#4111yescharacteristic
160C42×C61C2411yescharacteristic
161#11758201—44no—
162#11758201—44no—
163#11758194—22no—
164#11758194—22no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—42no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171C23×C61#12111yescharacteristic
172C22×D61—42no—
173C22×D61—82no—
174C22×D61—82no—
175C22×D61—82no—
176Q8×C61—21no—
177Q8×C61—21no—
178Q8×C61—21no—
179Q8×C61—42no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—81no—
183Q8×C61AGL(1,ℤ/8)12yes—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—81no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#13512yes—
199C2×C4×C61#13712yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203D244—42no—
204D244—42no—
205D244—42no—
206D244—42no—
207D244—82no—
208D244—82no—
209D244—82no—
210D244—82no—
211C8×C61—41no—
212C8×C61—41no—
213#92418—22no—
214#92418—22no—
215#92418—42no—
216#92418—42no—
217#19174—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61#46412yes—
221C22×C61C2×#12111yescharacteristic
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—164no—
226C4×C61—21no—
227C4×C61—22no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61#46412yes—
236C4×C61C2×#13511yescharacteristic
237C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C61C2×#13711yescharacteristic
239C4×C61#57511yescharacteristic
240#3138—611no—
241#3104D6111yescharacteristic, complemented
242C2×#460—611no—
243Q8×M16—611no—
244#1702—612no—
245#1698—612no—
246#1682—612no—
247#1678—612no—
248#1565—611no—
249#1550—611no—
250#1545—611no—
251C2×C61—22no—
252C2×C61C2×#46411yescharacteristic
253C2×C61#305511yescharacteristic
254C2×C61#306811yescharacteristic
255D61—162no—
256C22×SD16—1221no—
257#533C2×D6111yescharacteristic
258C2×#127—611no—
259C2×C4×Q8C2×D6111yescharacteristic
260#490—1222no—
261#486—612no—
262#486—612no—
263#484—21no—
264#477C2×D6111yescharacteristic, complemented
265#477C2×D6111yescharacteristic, complemented
266#462C2×D6111yescharacteristic, complemented
267#462C2×D6111yescharacteristic, complemented
268#460—612no—
269#460—612no—
270#460—1221no—
271#460—1221no—
272#451—21no—
273#446—612no—
274#446—1222no—
275C8×Q8—612no—
276C4×Q16—21no—
277C4×Q16—21no—
278#431C2×D6111yescharacteristic
279#423—611no—
280#423—1221no—
281C2×C4⋊C8—611no—
282#418—1221no—
283#418—1221no—
284C2×#102—1221no—
285C4×M16—611no—
286C4×M16—1221no—
287#335—612no—
288#334—612no—
289#333—612no—
290#332—612no—
291#331—612no—
292#331—612no—
293C61#1917411yescharacteristic, complemented, Sylow
294C22×Q8C22⋊D6111yescharacteristic
295C22×D4—1221no—
296C22×D4—1221no—
297C22×D4—1221no—
298C2×Q16—21no—
299C2×Q16—21no—
300C2×SD16—1222no—
301C2×SD16—1222no—
302C2×SD16—2444no—
303C2×M16—611no—
304C2×M16—611no—
305C2×M16—1221no—
306C22×C8—1221no—
307C4⋊Q8C22×D6111yescharacteristic
308C4⋊Q8C22×D6111yescharacteristic
309#127—612no—
310#127—1221no—
311#127—1221no—
312#127—1221no—
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D6111yescharacteristic
321C4×Q8C22⋊D6111yescharacteristic
322C4×Q8C22×D6111yescharacteristic
323C4×Q8C22×D6111yescharacteristic
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D6111yescharacteristic
327C2×C42—21no—
328C2×C42C22×D6111yescharacteristic
329#107—21no—
330#107—21no—
331C4⋊C8—612no—
332C4⋊C8—612no—
333C4⋊C8—1222no—
334C4⋊C8—1222no—
335C4⋊C8—1221no—
336C4⋊C8—1221no—
337C4⋊C8C22×D6111yescharacteristic
338C4⋊C8C22×D6111yescharacteristic
339#103—21no—
340#103—21no—
341#103—21no—
342#103—21no—
343#102—1222no—
344#102—1222no—
345#102—1222no—
346#102—1222no—
347#102—1222no—
348#102—1222no—
349C22⋊C8—21no—
350#97—612no—
351#97—1222no—
352C4×C8—21no—
353C4×C8—612no—
354C4×C8—1222no—
355C24—2441no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8C2×C22⋊D6111yescharacteristic
362C2×Q8C2×C22⋊D6111yescharacteristic
363C2×D4—1222no—
364C2×D4—1222no—
365C2×D4—1222no—
366C2×D4—4882no—
367C2×D4—2442no—
368C2×D4—2442no—
369C2×D4—2442no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2442no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D6111yescharacteristic
377C22×C4#1175822411yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—42no—
380Q16—42no—
381SD16—2444no—
382SD16—2444no—
383M16—1222no—
384M16—1222no—
385M16—2442no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—612no—
389C2×C8—612no—
390C2×C8—1222no—
391C2×C8—1222no—
392C2×C8—2442no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C2×C22⋊D6111yescharacteristic
405C4⋊C4C2×C22⋊D6111yescharacteristic
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42D4×D6111yescharacteristic
413C42#1175822411yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C23×D6111yescharacteristic
416C23—4882no—
417C23—4882no—
418C23—4882no—
419C23—2442no—
420C23#1451974911yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—41no—
426Q8—41no—
427Q8—81no—
428Q8#1451980212yes—
429D4—4882no—
430D4—4882no—
431D4—4882no—
432D4—4882no—
433D4—2442no—
434D4—2442no—
435D4—2442no—
436D4—2442no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4#1451974911yescharacteristic
449C2×C4#1451974911yescharacteristic
450C2×C4#1451974911yescharacteristic
451C2×C4#1451980012yes—
452C2×C4#1451980912yes—
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C22×C22⋊D6111yescharacteristic
455C2×C4C2×#1175822411yescharacteristic
456C8—41no—
457C8—41no—
458C8—1222no—
459C8—1222no—
460C8—2442no—
461C8—2442no—
462C22—21no—
463C22—21no—
464C22—9764no—
465C22—4882no—
466C22—4882no—
467C22—4882no—
468C22#1683998612yes—
469C22C2×#1451974911yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4#1683998612yes—
480C4C2×#1451980011yescharacteristic
481C4C2×#1451980211yescharacteristic
482C4C2×#1451980911yescharacteristic
483C4#1684056011yescharacteristic
484C2—22no—
485C2—9762no—
486C2C2×#1683998611yescharacteristic
487C2#2145544011yescharacteristic
488C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3104C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3104D61
hypocenterC61#19174
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. Q8×#11758194C2semidirect
4. #21445389C2semidirect
5. #21445361C2semidirect
6. #21445265C2semidirect
7. #21445237C2semidirect
8. #21444572C2semidirect
9. #21444509C2semidirect
11. C61×#3104C2semidirect
22. #16839893C22semidirect
23. Q8×#92418C22semidirect
32. #16839514C22semidirect
33. #16839512C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
41. C61×#477C22semidirect
42. C61×#477C22semidirect
43. C61×#462C22semidirect
44. C61×#462C22semidirect
241. #3104D61semidirect
264. #477C2×D61semidirect
265. #477C2×D61semidirect
266. #462C2×D61semidirect
267. #462C2×D61semidirect
293. C61#19174semidirect

Maximal subgroups

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

subgroupindex
#214561952
C2×#168399622
Q8×#117581942
#214453892
#214453612
#214452652
#214452372
#214445722
#214445092
#214444862
C61×#31042
#1917461