Finite Group Databasebeta

← #27151319

Subgroups — #27151319

Order 12416.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690833C211yesmaximal, characteristic, complemented
2C22×C8×D97C211yesmaximal, characteristic, complemented
3#20261248C212yesmaximal, complemented
4#20260973C211yesmaximal, characteristic, complemented
5#20260972C211yesmaximal, characteristic, complemented
6#20260971C211yesmaximal, characteristic, complemented
7#20260967C212yesmaximal, complemented
8#20260966C212yesmaximal, complemented
9#20260772C211yesmaximal, characteristic, complemented
10#20260730C212yesmaximal, complemented
11#467×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C22×C4×D97C2211yescharacteristic
14#15690833—24no—
15C2×#13095498C2212yescomplemented
16C2×D776C2211yescharacteristic, complemented
17C22×#11715895C2211yescharacteristic
18C2×C8×D97—24no—
19C2×C8×D97C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690795—22no—
24#15690786C2212yes—
25#15690785C2212yes—
26#15690784C2212yes—
27#15690780—22no—
28#15690772—22no—
29#15690770—22no—
30#15690768C2212yes—
31#15690744C2211yescharacteristic, complemented
32#15690741C2212yescomplemented
33#15690734C2212yescomplemented
34#15690732C2212yescomplemented
35#15690730C2212yescomplemented
36C2×Q16×C97C2211yescharacteristic, complemented
37C22×C8×C97C2211yescharacteristic
38C22⋊Q8×C97C2212yes—
39#107×C97C2211yescharacteristic, complemented
40#103×C97C2212yescomplemented
41C23×D97D411yescharacteristic
42#13095512—24no—
43#13095512—24no—
44C2×D388—22no—
45C2×D388—22no—
46C2×D388C2312yes—
47C2×C22⋊D97—22no—
48C22×Dic97D411yescharacteristic
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97D411yescharacteristic
53C2×C4×D97D411yescharacteristic
54C2×C4×D97C2311yescharacteristic
55C2×C4×D97C2311yescharacteristic
56#13095498—24no—
57D776—22no—
58C2×#11715895—22no—
59C2×#11715895C2311yescharacteristic
60C2×#11715895C2311yescharacteristic
61C8×D97—22no—
62C8×D97—22no—
63C8×D97—22no—
64C8×D97D414yescomplemented
65C4⋊Dic97C2312yes—
66#13095490—22no—
67#13095488—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71C4×Dic97—22no—
72C2×Q8×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74Q16×C97—22no—
75C2×C8×C97—22no—
76C2×C8×C97C2311yescharacteristic
77C2×C8×C97C2311yescharacteristic
78C4⋊C4×C97—22no—
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—22no—
81C23×C97C2×D411yescharacteristic
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97C2×D411yescharacteristic
87C22×D97C2×D411yescharacteristic
88Q8×C97—22no—
89Q8×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—44no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97C2×D411yescharacteristic
100C2×Dic97C2×D411yescharacteristic
101D388—22no—
102D388—22no—
103D388—44no—
104D388—44no—
105C4×D97—22no—
106C4×D97—21no—
107C4×D97—21no—
108C4×D97—22no—
109C4×D97—44no—
110C4×D97C2×D412yes—
111C4×D97#4112yes—
112C8×C97—21no—
113C8×C97C2×D412yes—
114#11715895—21no—
115#11715895C2×D412yes—
116C22×C97—21no—
117C22×C97—21no—
118C22×C97#13512yescomplemented
119C22×C97C22×D411yescharacteristic
120C2×D97—28no—
121C2×D97—21no—
122C2×D97—21no—
123C2×D97—21no—
124C2×D97—21no—
125C2×D97—42no—
126C2×D97—42no—
127C2×D97—42no—
128C2×D97#12112yes—
129C2×D97#13512yescomplemented
130C4×C97—21no—
131C4×C97—42no—
132C4×C97—42no—
133C4×C97C22×D411yescharacteristic
134C4×C97C2×#4111yescharacteristic
135Dic97—21no—
136Dic97—42no—
137Dic97#12112yes—
138C2×C97—22no—
139C2×C97C2×#12111yescharacteristic
140C2×C97C2×#13511yescharacteristic
141C2×C97C2×#13511yescharacteristic
142D97—22no—
143D97—22no—
144D97—82no—
145#3050—971nomaximal, Sylow
146C97#305011yescharacteristic, complemented, Sylow
147C2×#135—971no—
148C23×C8—971no—
149#525—972no—
150#467—971no—
151#467D9711yescharacteristic, complemented
152#466—971no—
153#466—971no—
154#465—972no—
155#465—972no—
156#427—971no—
157#416—972no—
158C2×#41—972no—
159C23×C4—971no—
160#135—1944no—
161C2×Q16C2×D9711yescharacteristic, complemented
162C2×SD16—972no—
163C2×D8—971no—
164C22×C8—971no—
165C22×C8—971no—
166C22×C8—971no—
167C22×C8—971no—
168C22×C8—971no—
169C22×C8—1944no—
170C22×C8C2×D9711yescharacteristic
171#123—1942no—
172C22⋊Q8—972no—
173C22⋊Q8C2×D9712yes—
174#121—972no—
175#121—972no—
176#120—1942no—
177C4×D4—1942no—
178C4×D4—1942no—
179#117—972no—
180#107—971no—
181#107C2×D9711yescharacteristic, complemented
182#106—972no—
183#103—972no—
184#103C2×D9712yescomplemented
185#102—972no—
186#102—972no—
187C24—971no—
188#41—1944no—
189#41—1944no—
190C2×Q8C22×D9712yes—
191C2×D4—972no—
192C2×D4—1942no—
193C2×D4—1942no—
194C2×D4—1942no—
195C22×C4—971no—
196C22×C4—971no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—1942no—
201C22×C4—1942no—
202C22×C4—1942no—
203C22×C4C22×D9711yescharacteristic
204Q16—22no—
205SD16—1944no—
206D8—1942no—
207C2×C8—22no—
208C2×C8—971no—
209C2×C8—974no—
210C2×C8—971no—
211C2×C8—1942no—
212C2×C8—1942no—
213C2×C8—1942no—
214C2×C8—1942no—
215C2×C8C22×D9711yescharacteristic
216C2×C8C22×D9711yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—972no—
219C4⋊C4—1942no—
220C4⋊C4C22×D9712yes—
221C22⋊C4—22no—
222C22⋊C4—1942no—
223C22⋊C4—1942no—
224C22⋊C4—1942no—
225C22⋊C4—1942no—
226C42—1942no—
227C23—971no—
228C23—971no—
229C23—1942no—
230C23—1942no—
231C23—1942no—
232C23—1942no—
233C23D4×D9711yescharacteristic
234Q8—22no—
235Q8—22no—
236D4—3884no—
237D4—3884no—
238D4—3884no—
239D4—1942no—
240D4—1942no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—971no—
246C2×C4—972no—
247C2×C4—972no—
248C2×C4—971no—
249C2×C4—3884no—
250C2×C4—1942no—
251C2×C4—1942no—
252C2×C4—1942no—
253C2×C4—1942no—
254C2×C4—1941no—
255C2×C4—1941no—
256C2×C4D4×D9711yescharacteristic
257C2×C4C23×D9711yescharacteristic
258C8—21no—
259C8—972no—
260C8—1941no—
261C8D4×D9712yes—
262C22—21no—
263C22—21no—
264C22—972no—
265C22—972no—
266C22—3882no—
267C22—3882no—
268C22—3882no—
269C22—1948no—
270C22—1941no—
271C22—1941no—
272C22—1941no—
273C22—1941no—
274C22#1569083312yescomplemented
275C22C2×D4×D9711yescharacteristic
276C4—21no—
277C4—42no—
278C4—42no—
279C4—972no—
280C4—3882no—
281C4—1941no—
282C4C2×D4×D9711yescharacteristic
283C4#41×D9711yescharacteristic
284C2—22no—
285C2—7762no—
286C2—1942no—
287C2—1942no—
288C2#121×D9711yescharacteristic
289C2#135×D9711yescharacteristic
290C2C2×#1569083311yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#467×C97C2
socleC22×C97C22×D4
hypercenter#467D97
hypocenterC97#3050
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. C22×C8×D97C2semidirect
3. #20261248C2semidirect
4. #20260973C2semidirect
5. #20260972C2semidirect
6. #20260971C2semidirect
7. #20260967C2semidirect
8. #20260966C2semidirect
9. #20260772C2semidirect
10. #20260730C2semidirect
11. #467×C97C2semidirect
15. C2×#13095498C22semidirect
16. C2×D776C22semidirect
31. #15690744C22semidirect
32. #15690741C22semidirect
33. #15690734C22semidirect
34. #15690732C22semidirect
35. #15690730C22semidirect
36. C2×Q16×C97C22semidirect
39. #107×C97C22semidirect
40. #103×C97C22semidirect
64. C8×D97D4semidirect
118. C22×C97#135semidirect
129. C2×D97#135semidirect
146. C97#3050semidirect
151. #467D97semidirect
161. C2×Q16C2×D97semidirect
181. #107C2×D97semidirect
184. #103C2×D97semidirect
274. C22#15690833semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
C22×C8×D972
#202612482
#202609732
#202609722
#202609712
#202609672
#202609662
#202607722
#202607302
#467×C972
#305097