Finite Group Databasebeta

← #27151349

Subgroups — #27151349

Order 12416.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D97C211yesmaximal, characteristic, complemented
2#20261242C211yesmaximal, characteristic, complemented
3C2×#15690781C211yesmaximal, characteristic, complemented
4#20260985C212yesmaximal, complemented
5#20260978C212yesmaximal, complemented
6#20260968C212yesmaximal, complemented
7#20260961C212yesmaximal, complemented
8#20260777C211yesmaximal, characteristic, complemented
9#20260740C211yesmaximal, characteristic, complemented
10#20260736C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97—21no—
13C22×C4×D97C2211yescharacteristic
14C2×#13095495C2211yescharacteristic
15C2×#13095494C2212yescomplemented
16M16×D97—24no—
17C2×C8×D97C2212yescomplemented
18#15690791—21no—
19#15690786C2211yescharacteristic, complemented
20#15690786C2211yescharacteristic, complemented
21#15690781—21no—
22#15690781—21no—
23#15690781C2212yes—
24#15690781C2212yes—
25#15690781C2211yescharacteristic, complemented
26#15690781C2211yescharacteristic, complemented
27#15690778—21no—
28C4×C22⋊D97—21no—
29C4×C22⋊D97—21no—
30#15690769C2211yescharacteristic
31C2×C4⋊Dic97C2211yescharacteristic
32C2×C22⋊Dic97—21no—
33#15690744C2212yescomplemented
34#15690742C2212yescomplemented
35#15690734C2212yescomplemented
36#15690731C2212yescomplemented
37#15690731C2212yescomplemented
38#15690731C2212yescomplemented
39C2×#41×C97C2211yescharacteristic
40C22×D4×C97C2211yescharacteristic
41AGL(1,ℤ/8)×C97—24no—
42C2×SD16×C97C2212yescomplemented
43C2×D8×C97C2212yescomplemented
44C2×M16×C97C2211yescharacteristic
45C23×D97D411yescharacteristic
46C2×C22⋊D97—21no—
47C2×C22⋊D97—21no—
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—44no—
51C22×Dic97—21no—
52C22×Dic97D411yescharacteristic
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55C2×C4×D97D411yescharacteristic
56C2×C4×D97D411yescharacteristic
57C2×C4×D97C2311yescharacteristic
58C2×C4×D97C2311yescharacteristic
59#13095495—22no—
60#13095494—24no—
61C2×#11715895C2312yes—
62C8×D97—24no—
63C4⋊Dic97—21no—
64C4⋊Dic97C2312yes—
65C4⋊Dic97C2311yescharacteristic
66C4⋊Dic97C2311yescharacteristic
67#13095490—21no—
68#13095490—21no—
69C22⋊Dic97—21no—
70C22⋊Dic97—22no—
71C22⋊Dic97—22no—
72C22⋊Dic97—21no—
73C22⋊Dic97—21no—
74#13095488—21no—
75#13095488—21no—
76C4×Dic97—21no—
77C24×C97—21no—
78#41×C97—22no—
79#41×C97—22no—
80C2×Q8×C97C2311yescharacteristic
81C2×D4×C97—21no—
82C2×D4×C97—22no—
83C2×D4×C97—21no—
84C2×D4×C97—22no—
85C2×D4×C97C2312yes—
86C2×D4×C97C2311yescharacteristic
87C22×C4×C97—21no—
88C22×C4×C97C2311yescharacteristic
89SD16×C97—24no—
90D8×C97—24no—
91M16×C97D414yescomplemented
92C2×C8×C97C2312yes—
93C23×C97—21no—
94C23×C97—22no—
95C23×C97—42no—
96C23×C97—42no—
97C23×C97—42no—
98C23×C97C2×D411yescharacteristic
99C22×D97—22no—
100C22×D97—22no—
101C22×D97—41no—
102C22×D97C2×D411yescharacteristic
103C22×D97C2×D411yescharacteristic
104Q8×C97—21no—
105Q8×C97—21no—
106D4×C97—21no—
107D4×C97—22no—
108D4×C97—22no—
109D4×C97—21no—
110D4×C97—42no—
111D4×C97—42no—
112C2×C4×C97—21no—
113C2×C4×C97—42no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D412yes—
116C2×C4×C97#4112yes—
117C2×C4×C97C2411yescharacteristic
118C22⋊D97—42no—
119C22⋊D97—44no—
120C22⋊D97—42no—
121C22⋊D97—44no—
122C2×Dic97—22no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—22no—
126C2×Dic97—42no—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129C4×D97—22no—
130C4×D97—22no—
131C4×D97—21no—
132C4×D97—21no—
133C4×D97—21no—
134C4×D97—21no—
135C8×C97C2×D414yescomplemented
136#11715895—22no—
137C22×C97—21no—
138C22×C97—21no—
139C22×C97—41no—
140C22×C97—42no—
141C22×C97—42no—
142C22×C97—41no—
143C22×C97—41no—
144C22×C97—42no—
145C22×C97—84no—
146C22×C97#12112yes—
147C22×C97C22×D411yescharacteristic
148C2×D97—21no—
149C2×D97—21no—
150C2×D97—21no—
151C2×D97—21no—
152C2×D97—21no—
153C2×D97—44no—
154C2×D97AGL(1,ℤ/8)12yescomplemented
155C4×C97—41no—
156C4×C97#12112yes—
157C4×C97C22×D411yescharacteristic
158C4×C97C2×#4111yescharacteristic
159Dic97—21no—
160Dic97—21no—
161Dic97—42no—
162Dic97—41no—
163Dic97—41no—
164C2×C97—22no—
165C2×C97—81no—
166C2×C97—82no—
167C2×C97C2×#12111yescharacteristic
168C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C97#57511yescharacteristic
170D97—22no—
171D97—41no—
172#3055—971nomaximal, Sylow
173C97#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
175C22×M16—971no—
176#525—971no—
177C2×#121—971no—
178#469—972no—
179#468—972no—
180#466—972no—
181#465—972no—
182#428—971no—
183#418—971no—
184#417—971no—
185C2×#41C2×D9711yescharacteristic
186C22×D4—1941no—
187C22×D4C2×D9711yescharacteristic
188C23×C4—971no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D9712yescomplemented
191C2×D8C2×D9712yescomplemented
192C2×M16—971no—
193C2×M16—972no—
194C2×M16—1944no—
195C2×M16C2×D9711yescharacteristic
196C22×C8—972no—
197#123—1941no—
198C22⋊Q8—971no—
199C22⋊Q8—971no—
200#121—972no—
201#121—972no—
202#121—971no—
203#121—971no—
204#121—1941no—
205#121—1941no—
206#120—1941no—
207C4×D4—1941no—
208C4×D4—1941no—
209#117—971no—
210C2×C4⋊C4—971no—
211C2×C22⋊C4—1941no—
212#107—972no—
213#106—972no—
214#103—972no—
215#102—972no—
216#102—972no—
217#102—972no—
218C24—21no—
219C24—971no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D9711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—3884no—
228C2×D4—1941no—
229C2×D4—1941no—
230C2×D4—1942no—
231C2×D4—1942no—
232C2×D4C22×D9711yescharacteristic
233C2×D4C22×D9712yes—
234C22×C4—21no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—1941no—
241C22×C4—1942no—
242C22×C4—1942no—
243C22×C4C22×D9711yescharacteristic
244SD16—24no—
245D8—24no—
246M16—1942no—
247M16—1944no—
248M16C22⋊D9714yescomplemented
249C2×C8—972no—
250C2×C8—1944no—
251C2×C8C22×D9712yes—
252C4⋊C4—971no—
253C4⋊C4—971no—
254C4⋊C4—972no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C22⋊C4—1941no—
259C22⋊C4—1941no—
260C22⋊C4—1942no—
261C22⋊C4—1942no—
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C42—1941no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—971no—
272C23—971no—
273C23—3881no—
274C23—1942no—
275C23—1942no—
276C23D4×D9711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—3884no—
286D4—3882no—
287D4—3884no—
288D4—3882no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—971no—
292C2×C4—971no—
293C2×C4—3882no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1942no—
297C2×C4—1942no—
298C2×C4—1942no—
299C2×C4—1942no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4D4×D9711yescharacteristic
305C2×C4C2×C22⋊D9712yes—
306C2×C4#1309551012yes—
307C2×C4C23×D9711yescharacteristic
308C8—1942no—
309C8C2×C22⋊D9714yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—972no—
320C22—3884no—
321C22—1941no—
322C22—1941no—
323C22—1941no—
324C22—1941no—
325C22—1941no—
326C22#1569078112yes—
327C22C2×D4×D9711yescharacteristic
328C4—41no—
329C4—3882no—
330C4—3881no—
331C4—3881no—
332C4—1941no—
333C4—1941no—
334C4#1569078112yes—
335C4C22×C22⋊D9711yescharacteristic
336C4C2×#1309551011yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—3881no—
341C2—1942no—
342C2C2×#1569078111yescharacteristic
343C2AGL(1,ℤ/8)×D9711yescharacteristic
344C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3055
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D97C2semidirect
2. #20261242C2semidirect
3. C2×#15690781C2semidirect
4. #20260985C2semidirect
5. #20260978C2semidirect
6. #20260968C2semidirect
7. #20260961C2semidirect
8. #20260777C2semidirect
9. #20260740C2semidirect
10. #20260736C2semidirect
11. C2×AGL(1,ℤ/8)×C97C2semidirect
15. C2×#13095494C22semidirect
17. C2×C8×D97C22semidirect
19. #15690786C22semidirect
20. #15690786C22semidirect
25. #15690781C22semidirect
26. #15690781C22semidirect
33. #15690744C22semidirect
34. #15690742C22semidirect
35. #15690734C22semidirect
36. #15690731C22semidirect
37. #15690731C22semidirect
38. #15690731C22semidirect
42. C2×SD16×C97C22semidirect
43. C2×D8×C97C22semidirect
91. M16×C97D4semidirect
135. C8×C97C2×D4semidirect
154. C2×D97AGL(1,ℤ/8)semidirect
173. C97#3055semidirect
174. C2×AGL(1,ℤ/8)D97semidirect
190. C2×SD16C2×D97semidirect
191. C2×D8C2×D97semidirect
248. M16C22⋊D97semidirect
309. C8C2×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D972
#202612422
C2×#156907812
#202609852
#202609782
#202609682
#202609612
#202607772
#202607402
#202607362
C2×AGL(1,ℤ/8)×C972
#305597