Finite Group Databasebeta

← #27151312

Subgroups — #27151312

Order 12416.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690831C211yesmaximal, characteristic, complemented
2C22×C8×D97C211yesmaximal, characteristic, complemented
3#20261248C211yesmaximal, characteristic, complemented
4#20261245C211yesmaximal, characteristic, complemented
5#20260975C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260970C211yesmaximal, characteristic, complemented
8#20260969C211yesmaximal, characteristic, complemented
9#20260965C211yesmaximal, characteristic, complemented
10#20260964C211yesmaximal, characteristic, complemented
11#20260961C211yesmaximal, characteristic, complemented
12#20260770C211yesmaximal, characteristic, complemented
13#20260730C211yesmaximal, characteristic, complemented
14#20260728C211yesmaximal, characteristic, complemented
15#465×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095510C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19#15690831—24no—
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095500C2211yescharacteristic, complemented
22C2×#13095496C2211yescharacteristic, complemented
23C22×#11715895C2211yescharacteristic
24C2×C8×D97—24no—
25C2×C8×D97C2211yescharacteristic
26C2×C8×D97C2211yescharacteristic
27C2×C8×D97C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic
29#15690795—21no—
30#15690792—21no—
31#15690790C2211yescharacteristic
32#15690789C2211yescharacteristic
33#15690786C2211yescharacteristic
34#15690785C2211yescharacteristic
35#15690784C2211yescharacteristic
36#15690781C2211yescharacteristic
37#15690780—21no—
38#15690778—21no—
39D4×Dic97—21no—
40#15690772—21no—
41#15690770—21no—
42#15690770—21no—
43#15690768C2211yescharacteristic
44#15690768C2211yescharacteristic
45#15690743C2211yescharacteristic, complemented
46#15690743C2211yescharacteristic, complemented
47#15690742C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690734C2211yescharacteristic, complemented
50#15690733C2211yescharacteristic, complemented
51#15690732C2211yescharacteristic, complemented
52#15690731C2211yescharacteristic, complemented
53#15690730C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic, complemented
55C22×C8×C97C2211yescharacteristic
56C22⋊Q8×C97C2211yescharacteristic
57#121×C97C2211yescharacteristic
58#106×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97D411yescharacteristic
62#13095512—22no—
63#13095512—22no—
64#13095510—22no—
65#13095510—22no—
66C2×Dic194C2311yescharacteristic
67C2×D388—21no—
68C2×D388—21no—
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C22×Dic97—21no—
74C22×Dic97D411yescharacteristic
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97—21no—
78C2×C4×D97D411yescharacteristic
79C2×C4×D97D411yescharacteristic
80C2×C4×D97C2311yescharacteristic
81C2×C4×D97C2311yescharacteristic
82#13095501—22no—
83#13095500—22no—
84#13095496—22no—
85C2×#11715895—22no—
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C8×D97—22no—
89C8×D97—22no—
90C8×D97—22no—
91C8×D97D414yescomplemented
92C4⋊Dic97—21no—
93C4⋊Dic97C2311yescharacteristic
94C4⋊Dic97C2311yescharacteristic
95#13095490—21no—
96#13095490—21no—
97C22⋊Dic97—21no—
98C22⋊Dic97—21no—
99#13095488—21no—
100#13095488—21no—
101#13095488—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105C4×Dic97—21no—
106C4×Dic97—21no—
107C2×Q8×C97C2311yescharacteristic
108C2×D4×C97—21no—
109C2×D4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111SD16×C97—22no—
112C2×C8×C97—22no—
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C4⋊C4×C97—21no—
116C4⋊C4×C97C2311yescharacteristic
117C4⋊C4×C97C2311yescharacteristic
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C23×C97—21no—
121C23×C97C2×D411yescharacteristic
122C22×D97—22no—
123C22×D97—22no—
124C22×D97—22no—
125C22×D97—21no—
126C22×D97C2×D411yescharacteristic
127C22×D97C2×D411yescharacteristic
128Q8×C97—21no—
129Q8×C97—21no—
130D4×C97—21no—
131D4×C97—21no—
132D4×C97—42no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2411yescharacteristic
140C22⋊D97—42no—
141C22⋊D97—42no—
142C22⋊D97—42no—
143C2×Dic97—22no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—42no—
148C2×Dic97C2×D411yescharacteristic
149C2×Dic97C2×D411yescharacteristic
150D388—21no—
151D388—21no—
152D388—42no—
153D388—42no—
154C4×D97—22no—
155C4×D97—21no—
156C4×D97—21no—
157C4×D97—22no—
158C4×D97—42no—
159C4×D97C2×D412yes—
160C4×D97#4112yes—
161Dic194—21no—
162Dic194—21no—
163C8×C97—21no—
164C8×C97C2×D412yes—
165#11715895—21no—
166#11715895C2×D412yes—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—41no—
170C22×C97—41no—
171C22×C97—41no—
172C22×C97#13512yescomplemented
173C22×C97C22×D411yescharacteristic
174C2×D97—28no—
175C2×D97—21no—
176C2×D97—21no—
177C2×D97—21no—
178C2×D97—21no—
179C2×D97—41no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97#12112yes—
183C2×D97#13512yescomplemented
184C4×C97—21no—
185C4×C97—41no—
186C4×C97—41no—
187C4×C97—41no—
188C4×C97C22×D411yescharacteristic
189C4×C97C2×#4111yescharacteristic
190Dic97—21no—
191Dic97—41no—
192Dic97—41no—
193Dic97—41no—
194Dic97#12112yes—
195C2×C97—22no—
196C2×C97—81no—
197C2×C97C2×#12111yescharacteristic
198C2×C97C2×#13511yescharacteristic
199C2×C97C2×#13511yescharacteristic
200D97—22no—
201D97—22no—
202D97—81no—
203#3050—971nomaximal, Sylow
204C97#305011yescharacteristic, complemented, Sylow
205C2×#135—971no—
206C23×C8—971no—
207#525—971no—
208#525—971no—
209#467—971no—
210#467—971no—
211#466—971no—
212#466—971no—
213#465—971no—
214#465—971no—
215#465—971no—
216#465D9711yescharacteristic, complemented
217#427—971no—
218#416—971no—
219#416—971no—
220C2×#41—971no—
221C2×#41—971no—
222C23×C4—971no—
223#135—1944no—
224C2×Q16—971no—
225C2×SD16—971no—
226C2×SD16C2×D9711yescharacteristic, complemented
227C2×D8—971no—
228C22×C8—971no—
229C22×C8—971no—
230C22×C8—971no—
231C22×C8—971no—
232C22×C8—971no—
233C22×C8—1944no—
234C22×C8C2×D9711yescharacteristic
235#123—1941no—
236#123—1941no—
237C22⋊Q8—971no—
238C22⋊Q8—971no—
239C22⋊Q8—971no—
240C22⋊Q8C2×D9711yescharacteristic
241#121—971no—
242#121—971no—
243#121—971no—
244#121C2×D9711yescharacteristic
245#120—1941no—
246#120—1941no—
247C4×D4—1941no—
248C4×D4—1941no—
249C4×D4—1941no—
250C4×D4—1941no—
251#117—971no—
252#117—971no—
253#107—971no—
254#107—971no—
255#106—971no—
256#106C2×D9711yescharacteristic, complemented
257#103—971no—
258#103—971no—
259#103—971no—
260#103C2×D9711yescharacteristic, complemented
261#102—971no—
262#102—971no—
263#102—971no—
264#102C2×D9711yescharacteristic, complemented
265C24—971no—
266#41—1942no—
267#41—1942no—
268#41—1942no—
269#41—1942no—
270C2×Q8—971no—
271C2×Q8C22×D9711yescharacteristic
272C2×D4—21no—
273C2×D4—971no—
274C2×D4—1941no—
275C2×D4—1941no—
276C2×D4—1941no—
277C2×D4—1941no—
278C2×D4—1941no—
279C2×D4C22×D9711yescharacteristic
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—971no—
283C22×C4—971no—
284C22×C4—971no—
285C22×C4—1941no—
286C22×C4—1942no—
287C22×C4—1942no—
288C22×C4—1941no—
289C22×C4C22×D9711yescharacteristic
290Q16—1942no—
291SD16—22no—
292SD16—1942no—
293D8—1942no—
294C2×C8—22no—
295C2×C8—971no—
296C2×C8—974no—
297C2×C8—971no—
298C2×C8—1942no—
299C2×C8—1942no—
300C2×C8—1942no—
301C2×C8—1942no—
302C2×C8C22×D9711yescharacteristic
303C2×C8C22×D9711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—971no—
306C4⋊C4—971no—
307C4⋊C4—1941no—
308C4⋊C4—1941no—
309C4⋊C4—1941no—
310C4⋊C4C22×D9711yescharacteristic
311C4⋊C4C22×D9711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—1941no—
315C22⋊C4—1941no—
316C22⋊C4—1941no—
317C22⋊C4—1941no—
318C22⋊C4—1941no—
319C22⋊C4—1941no—
320C22⋊C4—1941no—
321C22⋊C4—1941no—
322C42—1941no—
323C42—1941no—
324C23—21no—
325C23—971no—
326C23—971no—
327C23—1942no—
328C23—1942no—
329C23—1941no—
330C23—1942no—
331C23D4×D9711yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—1941no—
335Q8—1941no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—3882no—
340D4—3882no—
341D4—3882no—
342D4—3882no—
343D4—3882no—
344D4—1941no—
345D4—1941no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—971no—
352C2×C4—972no—
353C2×C4—972no—
354C2×C4—971no—
355C2×C4—3882no—
356C2×C4—3882no—
357C2×C4—1942no—
358C2×C4—1942no—
359C2×C4—1942no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4D4×D9711yescharacteristic
366C2×C4C23×D9711yescharacteristic
367C8—21no—
368C8—972no—
369C8—1941no—
370C8D4×D9712yes—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—972no—
377C22—972no—
378C22—3881no—
379C22—3881no—
380C22—3881no—
381C22—1948no—
382C22—1941no—
383C22—1941no—
384C22—1941no—
385C22—1941no—
386C22#1569083112yescomplemented
387C22C2×D4×D9711yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—972no—
393C4—3881no—
394C4—3881no—
395C4—3881no—
396C4—1941no—
397C4C2×D4×D9711yescharacteristic
398C4#41×D9711yescharacteristic
399C2—22no—
400C2—81no—
401C2—7761no—
402C2—1942no—
403C2—1942no—
404C2#121×D9711yescharacteristic
405C2#135×D9711yescharacteristic
406C2C2×#1569083111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#465×C97C2
socleC22×C97C22×D4
hypercenter#465D97
hypocenterC97#3050
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. C2×#15690831C2semidirect
2. C22×C8×D97C2semidirect
3. #20261248C2semidirect
4. #20261245C2semidirect
5. #20260975C2semidirect
6. #20260974C2semidirect
7. #20260970C2semidirect
8. #20260969C2semidirect
9. #20260965C2semidirect
10. #20260964C2semidirect
11. #20260961C2semidirect
12. #20260770C2semidirect
13. #20260730C2semidirect
14. #20260728C2semidirect
15. #465×C97C2semidirect
20. C2×#13095501C22semidirect
21. C2×#13095500C22semidirect
22. C2×#13095496C22semidirect
45. #15690743C22semidirect
46. #15690743C22semidirect
47. #15690742C22semidirect
48. #15690735C22semidirect
49. #15690734C22semidirect
50. #15690733C22semidirect
51. #15690732C22semidirect
52. #15690731C22semidirect
53. #15690730C22semidirect
54. C2×SD16×C97C22semidirect
58. #106×C97C22semidirect
59. #103×C97C22semidirect
60. #102×C97C22semidirect
91. C8×D97D4semidirect
172. C22×C97#135semidirect
183. C2×D97#135semidirect
204. C97#3050semidirect
216. #465D97semidirect
226. C2×SD16C2×D97semidirect
256. #106C2×D97semidirect
260. #103C2×D97semidirect
264. #102C2×D97semidirect
386. C22#15690831semidirect

Maximal subgroups

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

subgroupindex
C2×#156908312
C22×C8×D972
#202612482
#202612452
#202609752
#202609742
#202609702
#202609692
#202609652
#202609642
#202609612
#202607702
#202607302
#202607282
#465×C972
#305097