Finite Group Databasebeta

← #27153241

Subgroups — #27153241

Order 12416.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690842C211yesmaximal, characteristic, complemented
2C2×SD16×D97C211yesmaximal, characteristic, complemented
3Q8×D388C211yesmaximal, characteristic, complemented
4#20261378C211yesmaximal, characteristic
5#20261130C211yesmaximal, characteristic, complemented
6#20260980C211yesmaximal, characteristic, complemented
7#20260975C211yesmaximal, characteristic, complemented
8#20260941C211yesmaximal, characteristic, complemented
9#20260900C211yesmaximal, characteristic, complemented
10#20260889C211yesmaximal, characteristic, complemented
11#20260887C211yesmaximal, characteristic, complemented
12#20260872C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260805C211yesmaximal, characteristic, complemented
15#464×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18C2×Q8×D97C2211yescharacteristic, complemented
19C2×D4×D97C2211yescharacteristic
20#15690842—24no—
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095501C2211yescharacteristic, complemented
23C2×#13095500C2211yescharacteristic
24C2×#13095500C2211yescharacteristic
25C2×#13095499C2211yescharacteristic
26C2×#13095499C2211yescharacteristic
27C2×#13095498C2211yescharacteristic
28SD16×D97—24no—
29C2×#13095494C2211yescharacteristic, complemented
30C2×C8×D97C2211yescharacteristic
31#15690810—21no—
32#15690810C2211yescharacteristic, complemented
33#15690792—21no—
34#15690790—21no—
35#15690790—21no—
36#15690790C2211yescharacteristic
37#15690790C2211yescharacteristic, complemented
38#15690786C2211yescharacteristic
39#15690783—21no—
40#15690781C2211yescharacteristic
41#15690780—21no—
42C4×D388—21no—
43C4×D388C2211yescharacteristic
44C4×D388C2211yescharacteristic
45C22⋊C4×D97—21no—
46#15690743C2211yescharacteristic, complemented
47#15690735C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690734C2211yescharacteristic, complemented
50#15690730C2211yescharacteristic
51#15690722C2211yescharacteristic
52#15690722C2211yescharacteristic, complemented
53#15690721C2211yescharacteristic, complemented
54C2×Q16×C97C2211yescharacteristic, complemented
55C2×SD16×C97C2211yescharacteristic
56#124×C97C2211yescharacteristic
57C4×Q8×C97C2211yescharacteristic, complemented
58C4⋊C8×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic
61C23×D97—21no—
62#13095512—22no—
63#13095512—41no—
64C2×Dic194—21no—
65C2×Dic194—21no—
66C2×Dic194C2311yescharacteristic
67C2×Dic194C2311yescharacteristic
68Q8×D97—22no—
69Q8×D97—22no—
70Q8×D97—41no—
71Q8×D97—41no—
72C2×D388—21no—
73C2×D388D411yescharacteristic
74C2×D388C2311yescharacteristic
75C2×C22⋊D97—21no—
76C2×C22⋊D97—21no—
77D4×D97—22no—
78D4×D97—41no—
79C2×C4×D97—21no—
80C2×C4×D97—21no—
81C2×C4×D97—21no—
82C2×C4×D97—21no—
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85#13095501—22no—
86#13095501—41no—
87#13095500—22no—
88#13095500—22no—
89#13095499—22no—
90#13095499—41no—
91#13095498D414yescomplemented
92#13095494—22no—
93C2×#11715895C2311yescharacteristic
94C2×#11715895C2311yescharacteristic
95C8×D97—22no—
96C4⋊Dic97—21no—
97C4⋊Dic97—21no—
98C4⋊Dic97—21no—
99C4⋊Dic97—21no—
100C4⋊Dic97D411yescharacteristic
101C4⋊Dic97C2311yescharacteristic
102#13095490—21no—
103C22⋊Dic97—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488D411yescharacteristic
109#13095488D411yescharacteristic
110C2×Q8×C97C2311yescharacteristic
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97C2311yescharacteristic
113Q16×C97—22no—
114SD16×C97—22no—
115C2×C8×C97C2311yescharacteristic
116C2×C8×C97C2311yescharacteristic
117C4⋊C4×C97—21no—
118C4⋊C4×C97C2311yescharacteristic
119C22⋊C4×C97—21no—
120C22⋊C4×C97—21no—
121C42×C97—21no—
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C22×D97—21no—
125C22×D97—42no—
126C22×D97—42no—
127C22×D97—81no—
128C22×D97C2×D411yescharacteristic
129C22×D97C2×D411yescharacteristic
130Q8×C97—21no—
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97C2×D412yes—
134D4×C97—21no—
135D4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2×D411yescharacteristic
141C2×C4×C97C2411yescharacteristic
142C22⋊D97—42no—
143C22⋊D97—81no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97C2×D411yescharacteristic
148C2×Dic97C2×D411yescharacteristic
149D388—22no—
150D388—21no—
151D388—42no—
152D388C2×D412yes—
153C4×D97—21no—
154C4×D97—21no—
155C4×D97—21no—
156C4×D97—21no—
157C4×D97—42no—
158C4×D97—42no—
159C4×D97—42no—
160C4×D97—81no—
161Dic194—21no—
162Dic194—21no—
163Dic194—21no—
164Dic194—21no—
165Dic194—42no—
166Dic194—42no—
167C8×C97—21no—
168C8×C97—21no—
169#11715895—21no—
170#11715895C2×D412yes—
171C22×C97—41no—
172C22×C97—41no—
173C22×C97—41no—
174C22×C97C22×D411yescharacteristic
175C2×D97—21no—
176C2×D97—21no—
177C2×D97—21no—
178C2×D97—22no—
179C2×D97—21no—
180C2×D97—21no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—84no—
184C2×D97#13712yescomplemented
185C4×C97—22no—
186C4×C97—21no—
187C4×C97—21no—
188C4×C97—41no—
189C4×C97—41no—
190C4×C97C22×D411yescharacteristic
191C4×C97H(5,𝔽2)11yescharacteristic
192Dic97—21no—
193Dic97—21no—
194Dic97—41no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—81no—
198C2×C97D4211yescharacteristic
199C2×C97C2×#13711yescharacteristic
200C2×C97#57711yescharacteristic
201D97—22no—
202D97—41no—
203D97—42no—
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137—971no—
207C22×SD16—971no—
208D4×Q8—971no—
209#546—971no—
210#497—971no—
211#468—971no—
212#467—971no—
213#464D9711yescharacteristic, complemented
214#461—971no—
215#452—971no—
216#451—971no—
217#450—971no—
218#448—971no—
219#440—971no—
220#435—971no—
221C2×#41—971no—
222C22×Q8—971no—
223C22×Q8—971no—
224C22×D4—971no—
225#137—1944no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×Q16C2×D9711yescharacteristic, complemented
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×SD16—1944no—
235C2×SD16C2×D9711yescharacteristic
236C2×M16—971no—
237C22×C8—971no—
238C4⋊Q8—971no—
239C4⋊Q8—1941no—
240#124C2×D9711yescharacteristic
241#123—1941no—
242C22⋊Q8—971no—
243C22⋊Q8—971no—
244C22⋊Q8—971no—
245C22⋊Q8—1941no—
246C22⋊Q8—1941no—
247#121—971no—
248#121—1941no—
249#120—1941no—
250C4×Q8C2×D9711yescharacteristic, complemented
251C4×D4—971no—
252C4×D4—971no—
253C4×D4—1941no—
254C2×C22⋊C4—1941no—
255#107—971no—
256C4⋊C8C2×D9711yescharacteristic, complemented
257#103—971no—
258#103—971no—
259#103—971no—
260#103C2×D9711yescharacteristic, complemented
261#102—971no—
262#102C2×D9711yescharacteristic
263C22⋊C8—971no—
264C22⋊C8—971no—
265#97—971no—
266C24—1941no—
267#41—3881no—
268#41—1942no—
269C2×Q8—971no—
270C2×Q8—971no—
271C2×Q8—3881no—
272C2×Q8—3881no—
273C2×Q8—1941no—
274C2×Q8—1941no—
275C2×Q8—1942no—
276C2×Q8—1942no—
277C2×Q8C22×D9711yescharacteristic
278C2×Q8C22×D9711yescharacteristic
279C2×D4—971no—
280C2×D4—971no—
281C2×D4—3881no—
282C2×D4—1941no—
283C2×D4—1941no—
284C2×D4—1942no—
285C2×D4—1941no—
286C2×D4C22×D9711yescharacteristic
287C22×C4—971no—
288C22×C4—971no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4—1941no—
292C22×C4—1941no—
293Q16—22no—
294Q16—3881no—
295Q16—1942no—
296SD16—22no—
297SD16—974no—
298SD16—3881no—
299SD16—1942no—
300SD16—1942no—
301SD16—1942no—
302M16—1942no—
303C2×C8—971no—
304C2×C8—971no—
305C2×C8—1942no—
306C2×C8C22×D9711yescharacteristic
307C2×C8C22×D9711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—971no—
310C4⋊C4—971no—
311C4⋊C4—1941no—
312C4⋊C4—1941no—
313C4⋊C4—1941no—
314C4⋊C4—1941no—
315C4⋊C4—1941no—
316C4⋊C4C22×D9711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—971no—
320C22⋊C4—971no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C42—21no—
327C42C22×D9711yescharacteristic
328C23—21no—
329C23—7761no—
330C23—971no—
331C23—971no—
332C23—3882no—
333C23—3882no—
334C23—1941no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—3882no—
339Q8—3882no—
340Q8—1941no—
341Q8—1941no—
342Q8—1941no—
343Q8—1941no—
344Q8D4×D9712yes—
345D4—21no—
346D4—21no—
347D4—7761no—
348D4—972no—
349D4—3882no—
350D4—3882no—
351D4—1942no—
352D4—1941no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—7761no—
357C2×C4—971no—
358C2×C4—971no—
359C2×C4—3882no—
360C2×C4—3882no—
361C2×C4—3882no—
362C2×C4—1941no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4—1941no—
368C2×C4—1941no—
369C2×C4D4×D9711yescharacteristic
370C2×C4D4×D9711yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—21no—
373C8—21no—
374C8—972no—
375C8—1941no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—7764no—
380C22—972no—
381C22—3881no—
382C22—3881no—
383C22—1941no—
384C22—1941no—
385C22—1941no—
386C22—1942no—
387C22—1941no—
388C22—1941no—
389C22C2×D4×D9711yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—3881no—
396C4—3881no—
397C4—3881no—
398C4—1941no—
399C4—1941no—
400C4C2×D4×D9711yescharacteristic
401C4#1569085611yescharacteristic
402C2—81no—
403C2—3882no—
404C2—3881no—
405C2—1942no—
406C2#2026136911yescharacteristic
407C2#137×D9711yescharacteristic
408C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#464×C97C2
socleC22×C97C22×D4
prime radical#20261378C2
hypercenter#464D97
hypocenterC97#3266
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#15690842C2semidirect
2. C2×SD16×D97C2semidirect
3. Q8×D388C2semidirect
5. #20261130C2semidirect
6. #20260980C2semidirect
7. #20260975C2semidirect
8. #20260941C2semidirect
9. #20260900C2semidirect
10. #20260889C2semidirect
11. #20260887C2semidirect
12. #20260872C2semidirect
13. #20260833C2semidirect
14. #20260805C2semidirect
15. #464×C97C2semidirect
18. C2×Q8×D97C22semidirect
21. C2×#13095501C22semidirect
22. C2×#13095501C22semidirect
29. C2×#13095494C22semidirect
32. #15690810C22semidirect
37. #15690790C22semidirect
46. #15690743C22semidirect
47. #15690735C22semidirect
48. #15690735C22semidirect
49. #15690734C22semidirect
52. #15690722C22semidirect
53. #15690721C22semidirect
54. C2×Q16×C97C22semidirect
57. C4×Q8×C97C22semidirect
58. C4⋊C8×C97C22semidirect
59. #103×C97C22semidirect
91. #13095498D4semidirect
184. C2×D97#137semidirect
205. C97#3266semidirect
213. #464D97semidirect
228. C2×Q16C2×D97semidirect
250. C4×Q8C2×D97semidirect
256. C4⋊C8C2×D97semidirect
260. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
C2×SD16×D972
Q8×D3882
#202613782
#202611302
#202609802
#202609752
#202609412
#202609002
#202608892
#202608872
#202608722
#202608332
#202608052
#464×C972
#326697