Finite Group Databasebeta

← #27151359

Subgroups — #27151359

Order 12416.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690839C211yesmaximal, characteristic, complemented
2C22×#13095494C211yesmaximal, characteristic, complemented
3#20261245C211yesmaximal, characteristic
4C22⋊Q8×D97C211yesmaximal, characteristic, complemented
5#20260994C211yesmaximal, characteristic, complemented
6#20260992C211yesmaximal, characteristic, complemented
7#20260980C211yesmaximal, characteristic, complemented
8#20260978C211yesmaximal, characteristic, complemented
9#20260976C211yesmaximal, characteristic, complemented
10#20260974C211yesmaximal, characteristic, complemented
11#20260962C211yesmaximal, characteristic, complemented
12#20260774C211yesmaximal, characteristic, complemented
13#20260742C211yesmaximal, characteristic, complemented
14#20260733C211yesmaximal, characteristic, complemented
15#465×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19#15690839—24no—
20C2×Dic388C2211yescharacteristic, complemented
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095494—24no—
24C2×#13095494C2211yescharacteristic
25C2×#13095494C2211yescharacteristic
26C2×#13095494C2211yescharacteristic
27C2×#13095494C2211yescharacteristic
28C22×#11715895C2211yescharacteristic
29#15690792—21no—
30#15690790C2211yescharacteristic
31#15690790C2211yescharacteristic
32#15690789C2211yescharacteristic
33#15690789C2211yescharacteristic
34#15690788—21no—
35#15690787—21no—
36#15690786C2211yescharacteristic
37#15690781C2211yescharacteristic
38#15690778—21no—
39D4×Dic97—21no—
40#15690770—21no—
41#15690768C2211yescharacteristic
42C4⋊C4×D97—21no—
43C4⋊C4×D97C2211yescharacteristic
44C22⋊C4×D97—21no—
45#15690744C2211yescharacteristic, complemented
46#15690743C2211yescharacteristic, complemented
47#15690741C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690735C2211yescharacteristic, complemented
50#15690734C2211yescharacteristic, complemented
51#15690733C2211yescharacteristic, complemented
52#15690733C2211yescharacteristic, complemented
53#15690731C2211yescharacteristic, 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#13095510—22no—
63#13095510—22no—
64C2×Dic194—21no—
65C2×Dic194C2311yescharacteristic
66C2×Dic194C2311yescharacteristic
67Q8×D97—22no—
68Q8×D97—22no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C22×Dic97—21no—
72C22×Dic97D411yescharacteristic
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97—21no—
76C2×C4×D97—21no—
77C2×C4×D97D411yescharacteristic
78C2×C4×D97D411yescharacteristic
79C2×C4×D97C2311yescharacteristic
80C2×C4×D97C2311yescharacteristic
81Dic388—22no—
82#13095501—22no—
83#13095499—22no—
84#13095494—22no—
85#13095494—22no—
86#13095494—22no—
87#13095494D414yescomplemented
88C2×#11715895—22no—
89C2×#11715895C2311yescharacteristic
90C2×#11715895C2311yescharacteristic
91C4⋊Dic97—21no—
92C4⋊Dic97—21no—
93C4⋊Dic97C2311yescharacteristic
94C4⋊Dic97C2311yescharacteristic
95#13095490—21no—
96#13095490—21no—
97#13095490—21no—
98#13095490—21no—
99C22⋊Dic97—21no—
100C22⋊Dic97—21no—
101C22⋊Dic97—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—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—41no—
125C22×D97C2×D411yescharacteristic
126C22×D97C2×D411yescharacteristic
127Q8×C97—21no—
128Q8×C97—21no—
129D4×C97—21no—
130D4×C97—21no—
131D4×C97—42no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2411yescharacteristic
139C22⋊D97—42no—
140C22⋊D97—42no—
141C2×Dic97—22no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—42no—
147C2×Dic97C2×D411yescharacteristic
148C2×Dic97C2×D411yescharacteristic
149C4×D97—22no—
150C4×D97—21no—
151C4×D97—21no—
152C4×D97—22no—
153C4×D97—42no—
154C4×D97—42no—
155C4×D97C2×D412yes—
156C4×D97#4112yes—
157Dic194—21no—
158Dic194—21no—
159Dic194—21no—
160Dic194—21no—
161Dic194—42no—
162C8×C97—21no—
163C8×C97C2×D412yes—
164#11715895—21no—
165#11715895C2×D412yes—
166C22×C97—21no—
167C22×C97—21no—
168C22×C97—41no—
169C22×C97—41no—
170C22×C97—41no—
171C22×C97#13712yescomplemented
172C22×C97C22×D411yescharacteristic
173C2×D97—21no—
174C2×D97—21no—
175C2×D97—21no—
176C2×D97—21no—
177C2×D97—21no—
178C2×D97—44no—
179C2×D97#12112yes—
180C4×C97—21no—
181C4×C97—41no—
182C4×C97—41no—
183C4×C97—41no—
184C4×C97C22×D411yescharacteristic
185C4×C97C2×#4111yescharacteristic
186Dic97—21no—
187Dic97—41no—
188Dic97—41no—
189Dic97—41no—
190Dic97—41no—
191Dic97#12112yes—
192C2×C97—22no—
193C2×C97—81no—
194C2×C97C2×#12111yescharacteristic
195C2×C97C2×#13711yescharacteristic
196C2×C97#57511yescharacteristic
197D97—22no—
198D97—41no—
199#3056—971nomaximal, Sylow
200C97#305611yescharacteristic, complemented, Sylow
201C2×#137—971no—
202C22×M16—971no—
203#525—971no—
204C2×C22⋊Q8—971no—
205#470—971no—
206#470—971no—
207#468—971no—
208#468—971no—
209#467—971no—
210#467—971no—
211#465—971no—
212#465D9711yescharacteristic, complemented
213#428—971no—
214#419—971no—
215#417—971no—
216C2×#41—971no—
217C22×Q8—971no—
218C23×C4—971no—
219#137—1944no—
220C2×Q16—971no—
221C2×Q16—971no—
222C2×SD16—971no—
223C2×SD16C2×D9711yescharacteristic, complemented
224C2×M16—971no—
225C2×M16—971no—
226C2×M16—971no—
227C2×M16—971no—
228C2×M16—1944no—
229C22×C8—971no—
230C22×C8C2×D9711yescharacteristic
231#123—1941no—
232C22⋊Q8—971no—
233C22⋊Q8—971no—
234C22⋊Q8—971no—
235C22⋊Q8—971no—
236C22⋊Q8—971no—
237C22⋊Q8—1941no—
238C22⋊Q8—1941no—
239C22⋊Q8C2×D9711yescharacteristic
240#121—971no—
241#121C2×D9711yescharacteristic
242#120—1941no—
243C4×D4—1941no—
244C4×D4—1941no—
245#117—971no—
246C2×C4⋊C4—971no—
247C2×C4⋊C4—1941no—
248C2×C22⋊C4—1941no—
249#107—971no—
250#107—971no—
251#106—971no—
252#106C2×D9711yescharacteristic, complemented
253#103—971no—
254#103—971no—
255#103—971no—
256#103—971no—
257#103—971no—
258#103C2×D9711yescharacteristic, complemented
259#102—971no—
260#102C2×D9711yescharacteristic, complemented
261C24—971no—
262#41—1942no—
263#41—1942no—
264C2×Q8—971no—
265C2×Q8—971no—
266C2×Q8—1941no—
267C2×Q8—1942no—
268C2×Q8—1942no—
269C2×Q8C22×D9711yescharacteristic
270C2×D4—21no—
271C2×D4—1941no—
272C2×D4—1941no—
273C2×D4C22×D9711yescharacteristic
274C22×C4—971no—
275C22×C4—971no—
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—1942no—
280C22×C4—1942no—
281C22×C4—1941no—
282C22×C4—1941no—
283C22×C4—1941no—
284C22×C4C22×D9711yescharacteristic
285Q16—1942no—
286Q16—1942no—
287SD16—22no—
288SD16—1942no—
289M16—974no—
290M16—1942no—
291M16—1942no—
292M16—1942no—
293C2×C8—22no—
294C2×C8—971no—
295C2×C8—971no—
296C2×C8—1942no—
297C2×C8C22×D9711yescharacteristic
298C2×C8C22×D9711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—971no—
301C4⋊C4—971no—
302C4⋊C4—1941no—
303C4⋊C4—1941no—
304C4⋊C4—1941no—
305C4⋊C4—1941no—
306C4⋊C4—1941no—
307C4⋊C4—1941no—
308C4⋊C4C22×D9711yescharacteristic
309C4⋊C4C22×D9711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1941no—
313C22⋊C4—1941no—
314C22⋊C4—1941no—
315C22⋊C4—1941no—
316C22⋊C4—1941no—
317C22⋊C4—1941no—
318C22⋊C4—1941no—
319C42—1941no—
320C23—21no—
321C23—971no—
322C23—971no—
323C23—3881no—
324C23—1942no—
325C23—1942no—
326C23D4×D9711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—3882no—
330Q8—1941no—
331Q8—1941no—
332Q8—1941no—
333Q8—1941no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—3882no—
338D4—3882no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—971no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—971no—
348C2×C4—3882no—
349C2×C4—3882no—
350C2×C4—3882no—
351C2×C4—1942no—
352C2×C4—1942no—
353C2×C4—1942no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4—1941no—
358C2×C4—1941no—
359C2×C4—1941no—
360C2×C4D4×D9711yescharacteristic
361C2×C4C23×D9711yescharacteristic
362C8—21no—
363C8—972no—
364C8—1941no—
365C8D4×D9712yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—972no—
372C22—3884no—
373C22—1941no—
374C22—1941no—
375C22—1941no—
376C22—1941no—
377C22—1941no—
378C22#1569083912yescomplemented
379C22C2×D4×D9711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—972no—
385C4—3881no—
386C4—3881no—
387C4—3881no—
388C4—3881no—
389C4—1941no—
390C4C2×D4×D9711yescharacteristic
391C4#41×D9711yescharacteristic
392C2—22no—
393C2—81no—
394C2—3881no—
395C2—1942no—
396C2#121×D9711yescharacteristic
397C2C2×#1569083911yescharacteristic
398C2#2026153211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#465×C97C2
socleC22×C97C22×D4
prime radical#20261245C2
hypercenter#465D97
hypocenterC97#3056
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×#15690839C2semidirect
2. C22×#13095494C2semidirect
4. C22⋊Q8×D97C2semidirect
5. #20260994C2semidirect
6. #20260992C2semidirect
7. #20260980C2semidirect
8. #20260978C2semidirect
9. #20260976C2semidirect
10. #20260974C2semidirect
11. #20260962C2semidirect
12. #20260774C2semidirect
13. #20260742C2semidirect
14. #20260733C2semidirect
15. #465×C97C2semidirect
20. C2×Dic388C22semidirect
21. C2×#13095501C22semidirect
22. C2×#13095499C22semidirect
45. #15690744C22semidirect
46. #15690743C22semidirect
47. #15690741C22semidirect
48. #15690735C22semidirect
49. #15690735C22semidirect
50. #15690734C22semidirect
51. #15690733C22semidirect
52. #15690733C22semidirect
53. #15690731C22semidirect
54. C2×SD16×C97C22semidirect
58. #106×C97C22semidirect
59. #103×C97C22semidirect
60. #102×C97C22semidirect
87. #13095494D4semidirect
171. C22×C97#137semidirect
200. C97#3056semidirect
212. #465D97semidirect
223. C2×SD16C2×D97semidirect
252. #106C2×D97semidirect
258. #103C2×D97semidirect
260. #102C2×D97semidirect
378. C22#15690839semidirect

Maximal subgroups

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

subgroupindex
C2×#156908392
C22×#130954942
#202612452
C22⋊Q8×D972
#202609942
#202609922
#202609802
#202609782
#202609762
#202609742
#202609622
#202607742
#202607422
#202607332
#465×C972
#305697