Finite Group Databasebeta

← #27150977

Subgroups — #27150977

Order 12416.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#13095512C211yesmaximal, characteristic, complemented
2C2×#15690835C211yesmaximal, characteristic, complemented
3#20261242C211yesmaximal, characteristic, complemented
4#20260896C214yesmaximal, complemented
5#20260879C214yesmaximal, complemented
6#20260744C211yesmaximal, characteristic, complemented
7#20260741C211yesmaximal, characteristic, complemented
8#20260698C211yesmaximal, characteristic, complemented
9C2×#137×C97C211yesmaximal, characteristic, complemented
10C2×#13095512—24no—
11C2×#13095512—22no—
12C2×#13095512—24no—
13C2×#13095512C2214yes—
14C22×D388—21no—
15C22×D388C2211yescharacteristic
16C22×C4×D97—21no—
17C22×C4×D97C2211yescharacteristic
18#15690835—24no—
19C2×#13095498C2212yescomplemented
20C2×#13095496C2212yescomplemented
21C2×#13095495C2211yescharacteristic
22#15690791—21no—
23#15690786C2212yescomplemented
24#15690781C2212yescomplemented
25#15690778—21no—
26C4×C22⋊D97—22no—
27#15690769C2211yescharacteristic
28#15690734C2214yescomplemented
29#15690732C2214yescomplemented
30#15690722C2214yescomplemented
31C2×#41×C97C2211yescharacteristic
32C22×Q8×C97C2211yescharacteristic
33#137×C97—24no—
34C2×Q16×C97C2212yescomplemented
35C2×SD16×C97C2212yescomplemented
36C2×M16×C97C2211yescharacteristic
37C23×D97—21no—
38C23×D97D411yescharacteristic
39#13095512—216no—
40#13095512—28no—
41#13095512—28no—
42C2×D388—28no—
43C2×D388—24no—
44C2×D388—21no—
45C2×D388—22no—
46C2×D388D414yescomplemented
47C2×D388C2312yes—
48C2×C22⋊D97—22no—
49C22×Dic97D411yescharacteristic
50C2×C4×D97—28no—
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97—24no—
54C2×C4×D97D412yes—
55C2×C4×D97C2312yes—
56#13095498—24no—
57#13095496—24no—
58#13095495—22no—
59C2×#11715895C2312yes—
60C4⋊Dic97C2312yes—
61#13095490—22no—
62C22⋊Dic97—21no—
63C22⋊Dic97—22no—
64#13095488—22no—
65C4×Dic97—21no—
66#41×C97—22no—
67#41×C97—22no—
68C2×Q8×C97—22no—
69C2×Q8×C97—21no—
70C2×Q8×C97D414yescomplemented
71C2×Q8×C97C2311yescharacteristic
72C2×Q8×C97C2312yes—
73C2×D4×C97—21no—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97—21no—
77C22×C4×C97C2311yescharacteristic
78Q16×C97—24no—
79SD16×C97—24no—
80M16×C97—22no—
81C2×C8×C97C2312yes—
82C23×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—22no—
85C22×D97—24no—
86C22×D97—22no—
87C22×D97—42no—
88C22×D97—44no—
89C22×D97C2×D412yes—
90Q8×C97—24no—
91Q8×C97—22no—
92Q8×C97—21no—
93Q8×C97—21no—
94Q8×C97C2×D414yes—
95D4×C97—21no—
96D4×C97—21no—
97D4×C97—42no—
98C2×C4×C97—24no—
99C2×C4×C97—22no—
100C2×C4×C97—21no—
101C2×C4×C97—42no—
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97C2×D412yes—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2411yescharacteristic
106C22⋊D97—44no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97C2×D412yes—
110D388—216no—
111D388—24no—
112D388—22no—
113D388C2×D414yes—
114C4×D97—216no—
115C4×D97—24no—
116C4×D97—22no—
117C4×D97—22no—
118C8×C97—22no—
119#11715895—22no—
120C22×C97—22no—
121C22×C97—41no—
122C22×C97—42no—
123C22×C97#12012yes—
124C22×C97C22×D411yescharacteristic
125C2×D97—22no—
126C2×D97—24no—
127C2×D97—22no—
128C2×D97—22no—
129C2×D97—44no—
130C2×D97—48no—
131C2×D97—44no—
132C4×C97—24no—
133C4×C97—41no—
134C4×C97#12012yes—
135C4×C97C22×D411yescharacteristic
136C4×C97C22×D411yescharacteristic
137Dic97—22no—
138Dic97—42no—
139C2×C97—22no—
140C2×C97—81no—
141C2×C97C2×#12011yescharacteristic
142C2×C97#57512yes—
143D97—42no—
144D97—44no—
145#3005—971nomaximal, Sylow
146C97#300511yescharacteristic, complemented, Sylow
147C22×#41—971no—
148C2×#137D9711yescharacteristic, complemented
149C2×AGL(1,ℤ/8)—971no—
150#525—971no—
151#452—974no—
152#449—974no—
153#419—971no—
154#418—971no—
155#407—971no—
156C2×#41—974no—
157C2×#41—1942no—
158C2×#41—1944no—
159C2×#41—1944no—
160C2×#41C2×D9711yescharacteristic
161C22×Q8C2×D9711yescharacteristic
162C22×D4—971no—
163C22×D4—1941no—
164C23×C4—971no—
165C23×C4—1941no—
166#137—24no—
167AGL(1,ℤ/8)—1944no—
168C2×Q16C2×D9712yescomplemented
169C2×SD16—972no—
170C2×SD16C2×D9712yescomplemented
171C2×D8—972no—
172C2×M16—971no—
173C2×M16C2×D9711yescharacteristic
174#123—1941no—
175C22⋊Q8—972no—
176#121—972no—
177#120—1941no—
178C4×D4—1942no—
179#117—971no—
180#103—974no—
181#102—974no—
182C22⋊C8—974no—
183C24—971no—
184C24—1941no—
185#41—22no—
186#41—22no—
187#41—1948no—
188#41—19416no—
189#41—1948no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8C22⋊D9714yescomplemented
193C2×Q8C22×D9711yescharacteristic
194C2×Q8C22×D9712yes—
195C2×D4—21no—
196C2×D4—974no—
197C2×D4—972no—
198C2×D4—1944no—
199C2×D4—1948no—
200C2×D4—1942no—
201C2×D4—1941no—
202C2×D4—1942no—
203C2×D4C22×D9711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—971no—
207C22×C4—972no—
208C22×C4—972no—
209C22×C4—1944no—
210C22×C4—1948no—
211C22×C4—1942no—
212C22×C4—1942no—
213C22×C4C22×D9711yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—1944no—
217D8—1944no—
218M16—22no—
219M16—1942no—
220C2×C8—972no—
221C2×C8C22×D9712yes—
222C4⋊C4—972no—
223C4⋊C4—1942no—
224C22⋊C4—1942no—
225C22⋊C4—1942no—
226C22⋊C4—1941no—
227C42—1941no—
228C23—21no—
229C23—972no—
230C23—3884no—
231C23—3882no—
232C23—1944no—
233C23—1942no—
234C23—1942no—
235C23D4×D9711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D9714yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—974no—
245D4—3884no—
246D4—19416no—
247D4—1944no—
248D4—1942no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—972no—
254C2×C4—19416no—
255C2×C4—1942no—
256C2×C4—1942no—
257C2×C4—1944no—
258C2×C4—1942no—
259C2×C4—1942no—
260C2×C4D4×D9711yescharacteristic
261C2×C4D4×D9712yes—
262C2×C4C2×C22⋊D9712yes—
263C2×C4C23×D9711yescharacteristic
264C8—22no—
265C8—1942no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—3888no—
270C22—3884no—
271C22—3884no—
272C22—1944no—
273C22—1942no—
274C22—1942no—
275C22—1942no—
276C22#1569077812yes—
277C22C2×D4×D9711yescharacteristic
278C4—24no—
279C4—41no—
280C4—3882no—
281C4—1942no—
282C4#1569077812yes—
283C4C2×D4×D9711yescharacteristic
284C4C22×C22⋊D9711yescharacteristic
285C2—22no—
286C2—81no—
287C2—3884no—
288C2—3882no—
289C2C2×#1569077811yescharacteristic
290C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
hypercenterC2×#137D97
hypocenterC97#3005
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. C2×#15690835C2semidirect
3. #20261242C2semidirect
4. #20260896C2semidirect
5. #20260879C2semidirect
6. #20260744C2semidirect
7. #20260741C2semidirect
8. #20260698C2semidirect
9. C2×#137×C97C2semidirect
19. C2×#13095498C22semidirect
20. C2×#13095496C22semidirect
23. #15690786C22semidirect
24. #15690781C22semidirect
28. #15690734C22semidirect
29. #15690732C22semidirect
30. #15690722C22semidirect
34. C2×Q16×C97C22semidirect
35. C2×SD16×C97C22semidirect
46. C2×D388D4semidirect
70. C2×Q8×C97D4semidirect
146. C97#3005semidirect
148. C2×#137D97semidirect
168. C2×Q16C2×D97semidirect
170. C2×SD16C2×D97semidirect
192. C2×Q8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955122
C2×#156908352
#202612422
#202608962
#202608792
#202607442
#202607412
#202606982
C2×#137×C972
#300597