Finite Group Databasebeta

← #27153555

Subgroups — #27153555

Order 12416.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261443C211yesmaximal, characteristic, complemented
2#20261369C211yesmaximal, characteristic
3#20261080C211yesmaximal, characteristic, complemented
4#20261055C211yesmaximal, characteristic, complemented
5#20261051C211yesmaximal, characteristic, complemented
6#20261024C211yesmaximal, characteristic, complemented
7#20260967C211yesmaximal, characteristic, complemented
8#20260933C211yesmaximal, characteristic, complemented
9#20260885C211yesmaximal, characteristic, complemented
10#20260884C211yesmaximal, characteristic, complemented
11#20260820C211yesmaximal, characteristic, complemented
12C8×C22⋊D97C211yesmaximal, characteristic, complemented
13C2×#15690732C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#465×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C2×D4×D97—21no—
18C2×D4×D97C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C2×#13095498C2211yescharacteristic, complemented
21C2×C8×D97C2211yescharacteristic, complemented
22#15690809C2211yescharacteristic, complemented
23#15690806C2211yescharacteristic
24#15690800—21no—
25#15690790—21no—
26#15690787—21no—
27#15690786C2211yescharacteristic, complemented
28#15690785C2211yescharacteristic
29#15690783—21no—
30#15690780—21no—
31#15690778—21no—
32Q8×Dic97C2211yescharacteristic, complemented
33#15690772C2211yescharacteristic
34C4×C22⋊D97C2211yescharacteristic
35#15690770—21no—
36C2×C4⋊Dic97C2211yescharacteristic, complemented
37C4⋊C4×D97C2211yescharacteristic, complemented
38#15690741C2211yescharacteristic, complemented
39#15690741C2211yescharacteristic, complemented
40#15690739C2211yescharacteristic, complemented
41#15690732C2211yescharacteristic
42#15690732C2211yescharacteristic
43#15690732C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic, complemented
45#15690731C2211yescharacteristic, complemented
46#15690731C2211yescharacteristic, complemented
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic, complemented
49C22⋊#11715895C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×SD16×C97C2211yescharacteristic, complemented
53C22×C8×C97C2211yescharacteristic, complemented
54C22⋊Q8×C97C2211yescharacteristic, complemented
55#121×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C23×D97—21no—
61C2×Dic194—21no—
62C2×D388—22no—
63C2×D388—21no—
64C2×D388—21no—
65C2×D388—41no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97D411yescharacteristic
72D4×D97—22no—
73D4×D97—41no—
74D4×D97—44no—
75C22×Dic97—21no—
76C2×C4×D97—21no—
77C2×C4×D97—21no—
78C2×C4×D97C2311yescharacteristic
79#13095498—22no—
80#13095498—41no—
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C8×D97—41no—
84C4⋊Dic97—21no—
85C4⋊Dic97—21no—
86C4⋊Dic97C2311yescharacteristic
87C4⋊Dic97C2311yescharacteristic
88#13095490—21no—
89#13095490—21no—
90#13095490—21no—
91#13095490D411yescharacteristic
92C22⋊Dic97D411yescharacteristic
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488D411yescharacteristic
97C4×Dic97—21no—
98C4×Dic97C2311yescharacteristic
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97—21no—
101C2×D4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103SD16×C97—41no—
104C2×C8×C97—41no—
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97—21no—
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C22⋊C4×C97—21no—
112C23×C97—21no—
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115C22×D97—21no—
116C22×D97—42no—
117C22×D97—42no—
118C22×D97—42no—
119C22×D97—42no—
120C22×D97—81no—
121C22×D97—81no—
122C22×D97C2×D411yescharacteristic
123Q8×C97—21no—
124Q8×C97—21no—
125D4×C97—21no—
126D4×C97—21no—
127D4×C97—42no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97C2×D411yescharacteristic
134C2×C4×C97C2411yescharacteristic
135C22⋊D97—42no—
136C22⋊D97—42no—
137C22⋊D97—81no—
138C22⋊D97SD1614yescomplemented
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—81no—
142C2×Dic97C2×D411yescharacteristic
143C2×Dic97C2×D411yescharacteristic
144C2×Dic97C2×D411yescharacteristic
145D388—21no—
146D388—21no—
147D388—21no—
148D388—42no—
149D388—42no—
150D388#4112yes—
151C4×D97—21no—
152C4×D97—21no—
153C4×D97—42no—
154C4×D97—81no—
155Dic194—81no—
156C8×C97—21no—
157C8×C97—21no—
158#11715895—21no—
159#11715895—21no—
160C22×C97—21no—
161C22×C97—21no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97C2×SD1612yes—
166C22×C97C22×D411yescharacteristic
167C2×D97—21no—
168C2×D97—21no—
169C2×D97—22no—
170C2×D97—41no—
171C2×D97—41no—
172C2×D97—41no—
173C2×D97—41no—
174C2×D97—41no—
175C2×D97—84no—
176C2×D97—84no—
177C2×D97C2×SD1612yes—
178C4×C97—21no—
179C4×C97—41no—
180C4×C97—41no—
181C4×C97—41no—
182C4×C97C2×#4111yescharacteristic
183C4×C97H(5,𝔽2)11yescharacteristic
184Dic97—21no—
185Dic97—21no—
186Dic97—41no—
187Dic97—41no—
188Dic97C2×SD1612yes—
189C2×C97—22no—
190C2×C97—81no—
191C2×C97#54611yescharacteristic
192C2×C97C22×SD1611yescharacteristic
193C2×C97#57611yescharacteristic
194D97—22no—
195D97—42no—
196D97—81no—
197#3295—971nomaximal, Sylow
198C97#329511yescharacteristic, complemented, Sylow
199#554—971no—
200D42—971no—
201#486—971no—
202#481—971no—
203#481—971no—
204#476—971no—
205#465—971no—
206#465D9711yescharacteristic, complemented
207#460—971no—
208#450—971no—
209#450—971no—
210C4×SD16—971no—
211C8×D4—971no—
212C2×#102—971no—
213C2×#102—971no—
214C22×D4—971no—
215C22×D4—971no—
216C22×D4—1941no—
217C2×SD16—971no—
218C2×SD16—971no—
219C2×SD16C2×D9711yescharacteristic, complemented
220C22×C8—971no—
221C22×C8C2×D9711yescharacteristic, complemented
222C4⋊Q8—971no—
223#127—971no—
224#125—1941no—
225C22⋊Q8—971no—
226C22⋊Q8—1941no—
227C22⋊Q8—1941no—
228C22⋊Q8C2×D9711yescharacteristic, complemented
229#121—971no—
230#121—1941no—
231#121C2×D9711yescharacteristic
232#120—1941no—
233#120—1941no—
234C4×Q8—971no—
235C4×D4—971no—
236C4×D4—971no—
237C4×D4—1941no—
238C2×C4⋊C4—971no—
239C2×C4⋊C4—971no—
240#106—971no—
241#106—971no—
242#106C2×D9711yescharacteristic, complemented
243C4⋊C8—971no—
244#103C2×D9711yescharacteristic, complemented
245#102—971no—
246#102—971no—
247#102—971no—
248#102—971no—
249#102—971no—
250#102—971no—
251#102—971no—
252#102—971no—
253#102C2×D9711yescharacteristic, complemented
254C22⋊C8—971no—
255C22⋊C8—971no—
256C4×C8—971no—
257C24—1941no—
258C24—1941no—
259C2×Q8—1941no—
260C2×Q8C22×D9711yescharacteristic
261C2×D4—21no—
262C2×D4—971no—
263C2×D4—971no—
264C2×D4—971no—
265C2×D4—3881no—
266C2×D4—3884no—
267C2×D4—3881no—
268C2×D4—1942no—
269C2×D4—1941no—
270C2×D4—1941no—
271C2×D4—1941no—
272C2×D4—1941no—
273C2×D4—1942no—
274C2×D4—1941no—
275C2×D4C22×D9711yescharacteristic
276C22×C4—971no—
277C22×C4—1941no—
278C22×C4—1941no—
279C22×C4—1941no—
280C22×C4C22×D9711yescharacteristic
281SD16—41no—
282SD16—3881no—
283SD16—1942no—
284C2×C8—41no—
285C2×C8—971no—
286C2×C8—971no—
287C2×C8—3881no—
288C2×C8C22×D9711yescharacteristic
289C2×C8C22×D9711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—971no—
292C4⋊C4—971no—
293C4⋊C4—971no—
294C4⋊C4—1941no—
295C4⋊C4—1941no—
296C4⋊C4—1941no—
297C4⋊C4—1941no—
298C4⋊C4—1941no—
299C4⋊C4C22×D9711yescharacteristic
300C4⋊C4C22×D9711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—971no—
304C22⋊C4—971no—
305C22⋊C4—1941no—
306C22⋊C4—1941no—
307C22⋊C4—1941no—
308C42—971no—
309C42—1941no—
310C23—21no—
311C23—7761no—
312C23—7761no—
313C23—971no—
314C23—3882no—
315C23—3882no—
316C23—3882no—
317C23—3882no—
318C23—1941no—
319C23—1941no—
320C23D4×D9711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—7761no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—7761no—
328D4—974no—
329D4—972no—
330D4—3882no—
331D4—3882no—
332D4—3882no—
333D4—3882no—
334D4—1941no—
335D4—1941no—
336D4—1941no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—7761no—
343C2×C4—7761no—
344C2×C4—971no—
345C2×C4—971no—
346C2×C4—971no—
347C2×C4—3882no—
348C2×C4—1941no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4D4×D9711yescharacteristic
353C2×C4C23×D9711yescharacteristic
354C8—21no—
355C8—21no—
356C8—1941no—
357C8—1941no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—7764no—
364C22—7764no—
365C22—972no—
366C22—3881no—
367C22—3881no—
368C22—3881no—
369C22—3881no—
370C22—3881no—
371C22—1942no—
372C22—1941no—
373C22—1941no—
374C22SD16×D9712yes—
375C22C2×D4×D9711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—972no—
381C4—3881no—
382C4—3881no—
383C4—1941no—
384C4—1941no—
385C4#41×D9711yescharacteristic
386C4#1569085511yescharacteristic
387C2—22no—
388C2—81no—
389C2—7761no—
390C2—3882no—
391C2—1942no—
392C2#2026137411yescharacteristic
393C2C2×SD16×D9711yescharacteristic
394C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#465×C97C2
socleC22×C97C22×D4
prime radical#20261369C2
hypercenter#465D97
hypocenterC97#3295
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20261443C2semidirect
3. #20261080C2semidirect
4. #20261055C2semidirect
5. #20261051C2semidirect
6. #20261024C2semidirect
7. #20260967C2semidirect
8. #20260933C2semidirect
9. #20260885C2semidirect
10. #20260884C2semidirect
11. #20260820C2semidirect
12. C8×C22⋊D97C2semidirect
13. C2×#15690732C2semidirect
14. #102×D97C2semidirect
15. #465×C97C2semidirect
20. C2×#13095498C22semidirect
21. C2×C8×D97C22semidirect
22. #15690809C22semidirect
27. #15690786C22semidirect
32. Q8×Dic97C22semidirect
36. C2×C4⋊Dic97C22semidirect
37. C4⋊C4×D97C22semidirect
38. #15690741C22semidirect
39. #15690741C22semidirect
40. #15690739C22semidirect
43. #15690732C22semidirect
44. #15690732C22semidirect
45. #15690731C22semidirect
46. #15690731C22semidirect
48. #15690730C22semidirect
49. C22⋊#11715895C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
52. C2×SD16×C97C22semidirect
53. C22×C8×C97C22semidirect
54. C22⋊Q8×C97C22semidirect
56. #106×C97C22semidirect
57. #103×C97C22semidirect
58. #102×C97C22semidirect
138. C22⋊D97SD16semidirect
198. C97#3295semidirect
206. #465D97semidirect
219. C2×SD16C2×D97semidirect
221. C22×C8C2×D97semidirect
228. C22⋊Q8C2×D97semidirect
242. #106C2×D97semidirect
244. #103C2×D97semidirect
253. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614432
#202613692
#202610802
#202610552
#202610512
#202610242
#202609672
#202609332
#202608852
#202608842
#202608202
C8×C22⋊D972
C2×#156907322
#102×D972
#465×C972
#329597