Finite Group Databasebeta

← #27153236

Subgroups — #27153236

Order 12416.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690838C211yesmaximal, characteristic, complemented
2C22×#13095500C211yesmaximal, characteristic, complemented
3#20261394C211yesmaximal, characteristic, complemented
4D4×D388C211yesmaximal, characteristic, complemented
5#20261124C211yesmaximal, characteristic, complemented
6#20260981C211yesmaximal, characteristic, complemented
7#20260971C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260936C211yesmaximal, characteristic, complemented
10#20260888C211yesmaximal, characteristic, complemented
11#20260882C211yesmaximal, characteristic, complemented
12#20260876C211yesmaximal, characteristic, complemented
13#20260871C211yesmaximal, characteristic, complemented
14#20260837C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic, complemented
17C22×Dic194C2211yescharacteristic, complemented
18C22×D388C2211yescharacteristic
19C22×D388C2211yescharacteristic
20C2×D4×D97—21no—
21#15690838—24no—
22C2×#13095500—24no—
23C2×#13095500C2211yescharacteristic
24C2×#13095500C2211yescharacteristic
25C2×#13095500C2211yescharacteristic, complemented
26C2×#13095500C2211yescharacteristic, complemented
27C2×#13095500C2211yescharacteristic, complemented
28C2×#13095500C2211yescharacteristic, complemented
29C2×D776C2211yescharacteristic
30C2×D776C2211yescharacteristic, complemented
31C2×D776C2211yescharacteristic, complemented
32#15690807C2211yescharacteristic
33#15690798C2211yescharacteristic, complemented
34#15690796—21no—
35#15690789C2211yescharacteristic, complemented
36#15690788—21no—
37#15690785—21no—
38#15690784C2211yescharacteristic
39#15690784C2211yescharacteristic
40#15690784C2211yescharacteristic, complemented
41#15690782—21no—
42#15690780—21no—
43#15690780—21no—
44C4×Dic194C2211yescharacteristic, complemented
45C4×D388C2211yescharacteristic
46#15690770—21no—
47#15690744C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690735C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic, complemented
52#15690732C2211yescharacteristic, complemented
53#15690732C2211yescharacteristic, complemented
54C2×M16×C97C2211yescharacteristic, complemented
55C22×C8×C97C2211yescharacteristic, complemented
56C4×D4×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic, complemented
59C22⋊C8×C97C2211yescharacteristic, complemented
60#97×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095511—22no—
64#13095511—41no—
65C2×Dic194—22no—
66C2×Dic194—21no—
67C2×Dic194—41no—
68C2×Dic194C2311yescharacteristic
69C2×Dic194C2311yescharacteristic
70C2×D388—22no—
71C2×D388—22no—
72C2×D388—21no—
73C2×D388—21no—
74C2×D388—21no—
75C2×D388—21no—
76C2×D388—41no—
77C2×D388—41no—
78C2×D388C2311yescharacteristic
79C2×D388C2311yescharacteristic
80C2×D388C2311yescharacteristic
81C2×D388C2311yescharacteristic
82C2×C22⋊D97—21no—
83C2×C22⋊D97—21no—
84C2×C22⋊D97—21no—
85C2×C22⋊D97—21no—
86D4×D97—44no—
87C22×Dic97—21no—
88C2×C4×D97—21no—
89C2×C4×D97—21no—
90#13095500—22no—
91#13095500—22no—
92#13095500—22no—
93#13095500—22no—
94#13095500—41no—
95#13095500D414yescomplemented
96D776—22no—
97D776—22no—
98D776—41no—
99C4⋊Dic97C2311yescharacteristic
100C4⋊Dic97C2311yescharacteristic
101#13095490—21no—
102#13095490—21no—
103C22⋊Dic97—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488—21no—
109#13095488—21no—
110C4×Dic97—21no—
111C2×D4×C97D411yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C22×C4×C97C2311yescharacteristic
114M16×C97—22no—
115C2×C8×C97—22no—
116C2×C8×C97C2311yescharacteristic
117C2×C8×C97C2311yescharacteristic
118C2×C8×C97C2311yescharacteristic
119C2×C8×C97C2311yescharacteristic
120C4⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C22⋊C4×C97D411yescharacteristic
123C42×C97C2311yescharacteristic
124C23×C97C2×D411yescharacteristic
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—81no—
135C22×D97—81no—
136D4×C97—22no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97C2×D411yescharacteristic
142C2×C4×C97C2×D411yescharacteristic
143C2×C4×C97C2×D411yescharacteristic
144C2×C4×C97C2×D411yescharacteristic
145C2×C4×C97C2411yescharacteristic
146C22⋊D97—42no—
147C22⋊D97—42no—
148C22⋊D97—42no—
149C22⋊D97—42no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—42no—
155D388—21no—
156D388—21no—
157D388—21no—
158D388—21no—
159D388—21no—
160D388—21no—
161D388—21no—
162D388—42no—
163D388—42no—
164D388—42no—
165D388—81no—
166D388C2×D412yes—
167C4×D97—42no—
168C4×D97—42no—
169Dic194—21no—
170Dic194—21no—
171Dic194—21no—
172Dic194—42no—
173Dic194C2×D412yescomplemented
174C8×C97—21no—
175C8×C97—21no—
176C8×C97—21no—
177C8×C97C2×D412yescomplemented
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—21no—
182C22×C97—21no—
183C22×C97AGL(1,ℤ/8)12yescomplemented
184C22×C97C22×D411yescharacteristic
185C2×D97—22no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—41no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—84no—
198C2×D97—84no—
199C4×C97—21no—
200C4×C97—21no—
201C4×C97—21no—
202C4×C97—21no—
203C4×C97C22×D411yescharacteristic
204C4×C97H(5,𝔽2)11yescharacteristic
205Dic97—22no—
206Dic97—41no—
207Dic97—41no—
208Dic97—41no—
209C2×C97—22no—
210C2×C97—41no—
211C2×C97D4211yescharacteristic
212C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C97#57711yescharacteristic
214D97—42no—
215D97—81no—
216D97—81no—
217D97—81no—
218#3265—971nomaximal, Sylow
219C97#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C22×SD16—971no—
222#548—971no—
223D42—971no—
224#496—971no—
225#468—971no—
226#466—971no—
227#463—971no—
228#460—971no—
229#450—971no—
230#449—971no—
231#448—971no—
232#447—971no—
233#440—971no—
234#435D9711yescharacteristic, complemented
235C2×#41—971no—
236C22×Q8—971no—
237C22×D4—971no—
238C22×D4—971no—
239C22×D4—1941no—
240AGL(1,ℤ/8)—1944no—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×SD16—971no—
244C2×SD16—971no—
245C2×SD16—971no—
246C2×SD16—971no—
247C2×SD16—1944no—
248C2×D8—971no—
249C2×D8—971no—
250C2×D8—971no—
251C2×M16C2×D9711yescharacteristic, complemented
252C22×C8C2×D9711yescharacteristic, complemented
253#127—971no—
254#124—971no—
255#124—1941no—
256C22⋊Q8—971no—
257C22⋊Q8—1941no—
258#121—971no—
259#121—971no—
260#121—971no—
261#121—1941no—
262#121—1941no—
263#120—1941no—
264#120—1941no—
265C4×Q8—971no—
266C4×D4—971no—
267C4×D4—1941no—
268C4×D4C2×D9711yescharacteristic
269#107—971no—
270C4⋊C8C2×D9711yescharacteristic, complemented
271#103—971no—
272#103—971no—
273#102—971no—
274#102—971no—
275#102—971no—
276#102—971no—
277C22⋊C8C2×D9711yescharacteristic, complemented
278C22⋊C8C2×D9711yescharacteristic, complemented
279#97C2×D9711yescharacteristic, complemented
280C24—1941no—
281C24—1941no—
282#41—3881no—
283#41—1942no—
284C2×Q8—971no—
285C2×Q8—971no—
286C2×Q8—3881no—
287C2×Q8—1942no—
288C2×Q8—1941no—
289C2×D4—971no—
290C2×D4—971no—
291C2×D4—971no—
292C2×D4—971no—
293C2×D4—3881no—
294C2×D4—3881no—
295C2×D4—3884no—
296C2×D4—1942no—
297C2×D4—1942no—
298C2×D4—1941no—
299C2×D4—1941no—
300C2×D4—1941no—
301C2×D4—1941no—
302C2×D4—1941no—
303C2×D4—1941no—
304C2×D4—1941no—
305C2×D4—1941no—
306C2×D4D38811yescharacteristic
307C22×C4—1941no—
308C22×C4—1941no—
309C22×C4—1941no—
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312SD16—974no—
313SD16—3881no—
314SD16—1942no—
315SD16—1942no—
316SD16—1942no—
317SD16—1942no—
318D8—3881no—
319D8—1942no—
320D8—1942no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D9711yescharacteristic
324C2×C8C22×D9711yescharacteristic
325C2×C8C22×D9711yescharacteristic
326C2×C8C22×D9711yescharacteristic
327C4⋊C4—971no—
328C4⋊C4—971no—
329C4⋊C4—1941no—
330C4⋊C4—1941no—
331C4⋊C4D38811yescharacteristic
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4—1941no—
335C22⋊C4—1941no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C22⋊C4—1941no—
339C22⋊C4D38811yescharacteristic
340C22⋊C4D38811yescharacteristic
341C42—1941no—
342C42C22×D9711yescharacteristic
343C23—7761no—
344C23—7761no—
345C23—3882no—
346C23—3882no—
347C23—3882no—
348C23—3882no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23C2×D38811yescharacteristic
354C23C2×D38811yescharacteristic
355Q8—972no—
356Q8—3882no—
357Q8—1941no—
358Q8—1941no—
359Q8—1941no—
360D4—22no—
361D4—7761no—
362D4—972no—
363D4—3882no—
364D4—3882no—
365D4—3882no—
366D4—3882no—
367D4—3882no—
368D4—3882no—
369D4—3882no—
370D4—1941no—
371D4—1941no—
372D4—1941no—
373D4—1941no—
374D4—1941no—
375D4—1941no—
376D4—1941no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3882no—
382C2×C4—3882no—
383C2×C4—3882no—
384C2×C4—1941no—
385C2×C4—1941no—
386C2×C4—1941no—
387C2×C4—1941no—
388C2×C4C2×D38811yescharacteristic
389C2×C4C2×D38811yescharacteristic
390C2×C4C2×D38811yescharacteristic
391C2×C4C2×D38811yescharacteristic
392C2×C4C23×D9711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D9712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—7764no—
403C22—7764no—
404C22—3881no—
405C22—3881no—
406C22—3881no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—3881no—
415C22—1942no—
416C22#1569083812yescomplemented
417C22C22×D38811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—3881no—
423C4—3881no—
424C4—3881no—
425C4—1942no—
426C4C2×D4×D9711yescharacteristic
427C4#1569085611yescharacteristic
428C2—22no—
429C2—41no—
430C2—7761no—
431C2—7761no—
432C2—7761no—
433C2—3882no—
434C2D4×D38811yescharacteristic
435C2C2×#1569083811yescharacteristic
436C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#435×C97C2
socleC22×C97C22×D4
hypercenter#435D97
hypocenterC97#3265
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. C22×#13095500C2semidirect
3. #20261394C2semidirect
4. D4×D388C2semidirect
5. #20261124C2semidirect
6. #20260981C2semidirect
7. #20260971C2semidirect
8. #20260951C2semidirect
9. #20260936C2semidirect
10. #20260888C2semidirect
11. #20260882C2semidirect
12. #20260876C2semidirect
13. #20260871C2semidirect
14. #20260837C2semidirect
15. #435×C97C2semidirect
16. C2×#13095511C22semidirect
17. C22×Dic194C22semidirect
25. C2×#13095500C22semidirect
26. C2×#13095500C22semidirect
27. C2×#13095500C22semidirect
28. C2×#13095500C22semidirect
30. C2×D776C22semidirect
31. C2×D776C22semidirect
33. #15690798C22semidirect
35. #15690789C22semidirect
40. #15690784C22semidirect
44. C4×Dic194C22semidirect
47. #15690744C22semidirect
48. #15690735C22semidirect
49. #15690735C22semidirect
51. #15690732C22semidirect
52. #15690732C22semidirect
53. #15690732C22semidirect
54. C2×M16×C97C22semidirect
55. C22×C8×C97C22semidirect
57. C4⋊C8×C97C22semidirect
58. C22⋊C8×C97C22semidirect
59. C22⋊C8×C97C22semidirect
60. #97×C97C22semidirect
95. #13095500D4semidirect
173. Dic194C2×D4semidirect
177. C8×C97C2×D4semidirect
183. C22×C97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
234. #435D97semidirect
251. C2×M16C2×D97semidirect
252. C22×C8C2×D97semidirect
270. C4⋊C8C2×D97semidirect
277. C22⋊C8C2×D97semidirect
278. C22⋊C8C2×D97semidirect
279. #97C2×D97semidirect
396. C8D4×D97semidirect
416. C22#15690838semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C22×#130955002
#202613942
D4×D3882
#202611242
#202609812
#202609712
#202609512
#202609362
#202608882
#202608822
#202608762
#202608712
#202608372
#435×C972
#326597