Finite Group Databasebeta

← #27152220

Subgroups — #27152220

Order 12416.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C2×#15690834C211yesmaximal, characteristic, complemented
3C2×#15690831C211yesmaximal, characteristic, complemented
4C2×D8×D97C211yesmaximal, characteristic, complemented
5#20261303C211yesmaximal, characteristic, complemented
6#20261298C211yesmaximal, characteristic, complemented
7#20261124C211yesmaximal, characteristic, complemented
8#20261123C211yesmaximal, characteristic, complemented
9#20261121C211yesmaximal, characteristic, complemented
10#20261117C211yesmaximal, characteristic, complemented
11#20261116C211yesmaximal, characteristic, complemented
12#20261111C211yesmaximal, characteristic, complemented
13#20261110C211yesmaximal, characteristic, complemented
14#20260692C211yesmaximal, characteristic, complemented
15#496×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095510C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic
20#15690837—24no—
21#15690834—24no—
22#15690831—24no—
23C2×#13095501C2211yescharacteristic, complemented
24C2×#13095500C2211yescharacteristic, complemented
25C2×#13095500C2211yescharacteristic, complemented
26C2×#13095499C2211yescharacteristic, complemented
27C2×#13095498C2211yescharacteristic, complemented
28C2×D776C2211yescharacteristic
29C2×D776C2211yescharacteristic, complemented
30C2×#13095496C2211yescharacteristic
31C2×#13095496C2211yescharacteristic
32C2×#13095496C2211yescharacteristic, complemented
33C2×#13095496C2211yescharacteristic, complemented
34C2×#13095496C2211yescharacteristic, complemented
35D8×D97—24no—
36C2×#13095494C2211yescharacteristic
37C2×C8×D97C2211yescharacteristic
38#15690807C2211yescharacteristic, complemented
39#15690806C2211yescharacteristic, complemented
40#15690806C2211yescharacteristic, complemented
41#15690799C2211yescharacteristic, complemented
42#15690798C2211yescharacteristic, complemented
43#15690797C2211yescharacteristic, complemented
44#15690783—22no—
45#15690782—22no—
46#15690780—22no—
47#15690778—22no—
48#15690767C2211yescharacteristic
49#15690720C2211yescharacteristic, complemented
50C4×#11715895C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×SD16×C97C2211yescharacteristic, complemented
53C2×SD16×C97C2211yescharacteristic, complemented
54C2×D8×C97C2211yescharacteristic
55C2×D8×C97C2211yescharacteristic, complemented
56#127×C97C2211yescharacteristic, complemented
57#124×C97C2211yescharacteristic, complemented
58#97×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C23×D97—21no—
61#13095512—22no—
62#13095512—22no—
63#13095510—22no—
64#13095510—22no—
65C2×Dic194C2311yescharacteristic
66C2×D388—21no—
67C2×D388—22no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×D388C2311yescharacteristic
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—21no—
74C2×C22⋊D97—22no—
75C2×C22⋊D97—22no—
76D4×D97—22no—
77D4×D97—22no—
78D4×D97—22no—
79D4×D97—22no—
80C22×Dic97—21no—
81C2×C4×D97—21no—
82C2×C4×D97C2311yescharacteristic
83#13095501—22no—
84#13095500—22no—
85#13095500—22no—
86#13095499—22no—
87#13095498—22no—
88D776—22no—
89D776—22no—
90#13095496—22no—
91#13095496—22no—
92#13095496—22no—
93#13095496—22no—
94#13095496—22no—
95#13095494D414yescomplemented
96C2×#11715895C2311yescharacteristic
97C2×#11715895C2311yescharacteristic
98C8×D97D414yescomplemented
99#13095490D412yes—
100C22⋊Dic97—22no—
101#13095488—22no—
102#13095488—22no—
103#13095488D412yes—
104C4×Dic97C2311yescharacteristic
105C2×Q8×C97C2311yescharacteristic
106C2×D4×C97—22no—
107C2×D4×C97C2311yescharacteristic
108C2×D4×C97C2311yescharacteristic
109C2×D4×C97C2311yescharacteristic
110SD16×C97—22no—
111SD16×C97—22no—
112D8×C97—22no—
113D8×C97—22no—
114C2×C8×C97C2311yescharacteristic
115C2×C8×C97C2311yescharacteristic
116C22⋊C4×C97—22no—
117C42×C97C2311yescharacteristic
118C23×C97—21no—
119C23×C97—21no—
120C23×C97—21no—
121C22×D97—21no—
122C22×D97—21no—
123C22×D97—21no—
124C22×D97—42no—
125C22×D97—42no—
126C22×D97—82no—
127C22×D97—82no—
128C22×D97C2×D411yescharacteristic
129Q8×C97—21no—
130Q8×C97—21no—
131D4×C97—21no—
132D4×C97—21no—
133D4×C97—21no—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—21no—
137D4×C97—82no—
138C2×C4×C97—21no—
139C2×C4×C97C2×D412yes—
140C2×C4×C97C2411yescharacteristic
141C22⋊D97—42no—
142C22⋊D97—42no—
143C22⋊D97—42no—
144C22⋊D97—82no—
145C22⋊D97—82no—
146C2×Dic97—21no—
147C2×Dic97—42no—
148C2×Dic97C2×D412yes—
149C2×Dic97C2×D411yescharacteristic
150D388—21no—
151D388—21no—
152D388—21no—
153D388—21no—
154D388—21no—
155D388—21no—
156D388—42no—
157D388—82no—
158C4×D97—42no—
159C4×D97C2×D412yes—
160C4×D97C2×D412yes—
161Dic194—21no—
162Dic194—21no—
163C8×C97C2×D412yes—
164C8×C97C2×D412yes—
165#11715895C2×D412yes—
166#11715895C2×D412yes—
167C22×C97—41no—
168C22×C97—41no—
169C22×C97—41no—
170C22×C97—42no—
171C22×C97—42no—
172C22×C97—42no—
173C22×C97C22×D411yescharacteristic
174C2×D97—22no—
175C2×D97—41no—
176C2×D97—41no—
177C2×D97—41no—
178C2×D97—42no—
179C2×D97—42no—
180C2×D97—42no—
181C2×D97—84no—
182C2×D97—84no—
183C2×D97#12712yes—
184C4×C97—22no—
185C4×C97—41no—
186C4×C97C22×D411yescharacteristic
187C4×C97C22×D411yescharacteristic
188Dic97—22no—
189Dic97—41no—
190Dic97#12712yes—
191C2×C97—81no—
192C2×C97—81no—
193C2×C97—81no—
194C2×C97C2×#12711yescharacteristic
195C2×C97#57612yes—
196D97—22no—
197D97—81no—
198D97—81no—
199D97—81no—
200#3155—971nomaximal, Sylow
201C97#315511yescharacteristic, complemented, Sylow
202C2×AGL(1,ℤ/8)—971no—
203C2×AGL(1,ℤ/8)—971no—
204C2×#135—971no—
205C22×D8—971no—
206#535—971no—
207#535—971no—
208#496—971no—
209#496—971no—
210#496—971no—
211#496D9711yescharacteristic, complemented
212#495—971no—
213#495—971no—
214#493—971no—
215#493—971no—
216#405—971no—
217C2×#41—971no—
218C2×#41—971no—
219C22×D4—971no—
220C22×D4—971no—
221AGL(1,ℤ/8)—1944no—
222AGL(1,ℤ/8)—1944no—
223#135—1944no—
224C2×Q16—971no—
225C2×SD16—971no—
226C2×SD16—971no—
227C2×SD16—971no—
228C2×SD16—971no—
229C2×SD16C2×D9711yescharacteristic, complemented
230C2×SD16C2×D9711yescharacteristic, complemented
231C2×D8—971no—
232C2×D8—971no—
233C2×D8—971no—
234C2×D8—971no—
235C2×D8—971no—
236C2×D8—971no—
237C2×D8—971no—
238C2×D8—1944no—
239C2×D8C2×D9711yescharacteristic
240C2×D8C2×D9711yescharacteristic, complemented
241C2×M16—971no—
242C22×C8—971no—
243#127—971no—
244#127—971no—
245#127—971no—
246#127C2×D9711yescharacteristic, complemented
247#124—971no—
248#124—971no—
249#124—971no—
250#124C2×D9711yescharacteristic, complemented
251#121—1942no—
252#121—1942no—
253#120—1942no—
254#120—1942no—
255#117—971no—
256#97—971no—
257#97C2×D9711yescharacteristic, complemented
258C4×C8—971no—
259C4×C8—971no—
260C24—1941no—
261C24—1941no—
262#41—1942no—
263#41—1942no—
264#41—1942no—
265#41—1942no—
266C2×Q8—971no—
267C2×Q8C22×D9711yescharacteristic
268C2×D4—22no—
269C2×D4—971no—
270C2×D4—971no—
271C2×D4—971no—
272C2×D4—1941no—
273C2×D4—1942no—
274C2×D4—1942no—
275C2×D4—1942no—
276C2×D4—1942no—
277C2×D4—1942no—
278C2×D4—1942no—
279C2×D4—1941no—
280C2×D4—1941no—
281C2×D4—1941no—
282C2×D4—1942no—
283C2×D4C22×D9711yescharacteristic
284C2×D4C22×D9711yescharacteristic
285C2×D4C22×D9711yescharacteristic
286C22×C4—971no—
287C22×C4—1941no—
288C22×C4—1941no—
289Q16—1942no—
290SD16—22no—
291SD16—22no—
292SD16—1942no—
293SD16—1942no—
294SD16—1942no—
295SD16—1942no—
296D8—22no—
297D8—22no—
298D8—1942no—
299D8—1942no—
300D8—1942no—
301D8—1942no—
302D8—1942no—
303D8—1942no—
304D8—1942no—
305M16—974no—
306C2×C8—971no—
307C2×C8—971no—
308C2×C8—974no—
309C2×C8C22×D9711yescharacteristic
310C2×C8C22×D9711yescharacteristic
311C4⋊C4—972no—
312C22⋊C4—22no—
313C22⋊C4—972no—
314C22⋊C4—1942no—
315C22⋊C4—1942no—
316C22⋊C4—1942no—
317C42—971no—
318C42C22×D9711yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—7762no—
323C23—7762no—
324C23—971no—
325C23—3882no—
326C23—3882no—
327C23—1941no—
328C23—1941no—
329C23—1941no—
330Q8—21no—
331Q8—21no—
332Q8—1941no—
333Q8—1941no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—7762no—
342D4—7762no—
343D4—7762no—
344D4—3882no—
345D4—3882no—
346D4—3882no—
347D4—3882no—
348D4—1941no—
349D4—1941no—
350D4—1941no—
351D4—1941no—
352D4—1941no—
353D4—1941no—
354C2×C4—21no—
355C2×C4—971no—
356C2×C4—972no—
357C2×C4—972no—
358C2×C4—972no—
359C2×C4—3882no—
360C2×C4—3882no—
361C2×C4—1941no—
362C2×C4D4×D9712yes—
363C2×C4C23×D9711yescharacteristic
364C8—972no—
365C8—972no—
366C8D4×D9712yes—
367C8D4×D9712yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—7764no—
375C22—7764no—
376C22—972no—
377C22—3881no—
378C22—3881no—
379C22—3881no—
380C22—3882no—
381C22—3882no—
382C22—3882no—
383C22—1942no—
384C22C2×D4×D9711yescharacteristic
385C4—22no—
386C4—41no—
387C4—972no—
388C4—3881no—
389C4—1942no—
390C4C2×D4×D9711yescharacteristic
391C4C2×D4×D9711yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—7761no—
396C2—7761no—
397C2—7761no—
398C2—1942no—
399C2#127×D9711yescharacteristic
400C2#2026153912yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#496×C97C2
socleC22×C97C22×D4
hypercenter#496D97
hypocenterC97#3155
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690834C2semidirect
3. C2×#15690831C2semidirect
4. C2×D8×D97C2semidirect
5. #20261303C2semidirect
6. #20261298C2semidirect
7. #20261124C2semidirect
8. #20261123C2semidirect
9. #20261121C2semidirect
10. #20261117C2semidirect
11. #20261116C2semidirect
12. #20261111C2semidirect
13. #20261110C2semidirect
14. #20260692C2semidirect
15. #496×C97C2semidirect
23. C2×#13095501C22semidirect
24. C2×#13095500C22semidirect
25. C2×#13095500C22semidirect
26. C2×#13095499C22semidirect
27. C2×#13095498C22semidirect
29. C2×D776C22semidirect
32. C2×#13095496C22semidirect
33. C2×#13095496C22semidirect
34. C2×#13095496C22semidirect
38. #15690807C22semidirect
39. #15690806C22semidirect
40. #15690806C22semidirect
41. #15690799C22semidirect
42. #15690798C22semidirect
43. #15690797C22semidirect
49. #15690720C22semidirect
50. C4×#11715895C22semidirect
51. C8×Dic97C22semidirect
52. C2×SD16×C97C22semidirect
53. C2×SD16×C97C22semidirect
55. C2×D8×C97C22semidirect
56. #127×C97C22semidirect
57. #124×C97C22semidirect
58. #97×C97C22semidirect
95. #13095494D4semidirect
98. C8×D97D4semidirect
201. C97#3155semidirect
211. #496D97semidirect
229. C2×SD16C2×D97semidirect
230. C2×SD16C2×D97semidirect
240. C2×D8C2×D97semidirect
246. #127C2×D97semidirect
250. #124C2×D97semidirect
257. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908342
C2×#156908312
C2×D8×D972
#202613032
#202612982
#202611242
#202611232
#202611212
#202611172
#202611162
#202611112
#202611102
#202606922
#496×C972
#315597