Finite Group Databasebeta

← #27151382

Subgroups — #27151382

Order 12416.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D97C211yesmaximal, characteristic, complemented
2C2×#15690818C211yesmaximal, characteristic, complemented
3#20261301C212yesmaximal, complemented
4#20260989C211yesmaximal, characteristic, complemented
5#20260988C211yesmaximal, characteristic, complemented
6#20260985C211yesmaximal, characteristic, complemented
7#20260970C212yesmaximal, complemented
8#20260969C212yesmaximal, complemented
9#20260770C211yesmaximal, characteristic, complemented
10#20260738C212yesmaximal, complemented
11#469×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×D4×D97C2212yes—
14C2×D776C2211yescharacteristic
15C2×#13095496C2212yes—
16D8×D97—24no—
17#15690818—24no—
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic, complemented
20C22×#11715895C2211yescharacteristic, complemented
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic, complemented
23#15690806—22no—
24#15690796—22no—
25#15690785C2212yescomplemented
26#15690784C2212yescomplemented
27#15690781C2212yescomplemented
28#15690780—22no—
29#15690778—22no—
30#15690768C2212yes—
31#15690743C2212yescomplemented
32#15690742C2211yescharacteristic, complemented
33#15690732C2212yescomplemented
34#15690731C2212yescomplemented
35#15690730C2212yescomplemented
36C2×D8×C97C2211yescharacteristic
37C2×M16×C97C2211yescharacteristic, complemented
38#121×C97C2212yescomplemented
39#106×C97C2211yescharacteristic, complemented
40#102×C97C2212yescomplemented
41C23×D97—22no—
42#13095511—22no—
43#13095511—22no—
44C2×Dic194D411yescharacteristic
45C2×D388—22no—
46C2×D388D411yescharacteristic
47C2×D388C2312yes—
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51C2×C22⋊D97D411yescharacteristic
52C2×C22⋊D97D411yescharacteristic
53D4×D97—24no—
54D4×D97—24no—
55C2×C4×D97C2311yescharacteristic
56C2×C4×D97C2311yescharacteristic
57D776—22no—
58#13095496—24no—
59#13095494—22no—
60#13095494—22no—
61C2×#11715895—22no—
62C2×#11715895C2311yescharacteristic
63C2×#11715895C2311yescharacteristic
64C8×D97—22no—
65C8×D97D414yescomplemented
66C4⋊Dic97C2312yes—
67C22⋊Dic97—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71C4×Dic97—22no—
72C2×D4×C97—22no—
73C2×D4×C97C2312yes—
74C22×C4×C97C2311yescharacteristic
75D8×C97—22no—
76M16×C97—22no—
77C2×C8×C97C2311yescharacteristic
78C2×C8×C97C2311yescharacteristic
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—22no—
81C23×C97—22no—
82C23×C97C2×D411yescharacteristic
83C22×D97—22no—
84C22×D97—44no—
85C22×D97—82no—
86C22×D97—82no—
87C22×D97C2×D411yescharacteristic
88C22×D97C2×D411yescharacteristic
89D4×C97—22no—
90D4×C97—22no—
91D4×C97—82no—
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—22no—
98C22⋊D97—22no—
99C22⋊D97—44no—
100C22⋊D97—82no—
101C22⋊D97—82no—
102C2×Dic97—22no—
103C2×Dic97C2×D411yescharacteristic
104C2×Dic97C2×D411yescharacteristic
105D388—22no—
106D388—22no—
107D388—22no—
108D388—82no—
109C4×D97—21no—
110C4×D97—21no—
111C4×D97C2×D412yes—
112C4×D97#4112yes—
113Dic194—22no—
114C8×C97—21no—
115C8×C97C2×D412yes—
116#11715895—21no—
117#11715895C2×D412yes—
118C22×C97—21no—
119C22×C97—22no—
120C22×C97—42no—
121C22×C97—42no—
122C22×C97—42no—
123C22×C97C22×D411yescharacteristic
124C2×D97—22no—
125C2×D97—21no—
126C2×D97—22no—
127C2×D97—42no—
128C2×D97—42no—
129C2×D97—42no—
130C2×D97—88no—
131C2×D97#12112yes—
132C4×C97—21no—
133C4×C97—42no—
134C4×C97C22×D411yescharacteristic
135C4×C97C2×#4111yescharacteristic
136Dic97—21no—
137Dic97—42no—
138Dic97#12112yes—
139C2×C97—41no—
140C2×C97—82no—
141C2×C97C2×#12111yescharacteristic
142C2×C97#57612yes—
143D97—22no—
144D97—41no—
145D97—82no—
146#3058—971nomaximal, Sylow
147C97#305811yescharacteristic, complemented, Sylow
148C22×D8—971no—
149C2×#131—971no—
150#535—972no—
151#469—971no—
152#469—971no—
153#469—971no—
154#469D9711yescharacteristic, complemented
155#466—972no—
156#466—972no—
157#427—971no—
158#418—972no—
159C2×#41—971no—
160C22×D4—972no—
161C2×D8—972no—
162C2×D8—971no—
163C2×D8—1944no—
164C2×D8C2×D9711yescharacteristic
165#131—1944no—
166C2×M16—971no—
167C2×M16—971no—
168C2×M16C2×D9711yescharacteristic, complemented
169C22×C8—971no—
170C22×C8—971no—
171C22×C8—971no—
172#127—1942no—
173#124—1942no—
174#121—972no—
175#121—972no—
176#121—972no—
177#121C2×D9712yescomplemented
178#120—1942no—
179#120—1942no—
180#117—972no—
181#107—972no—
182#106—971no—
183#106C2×D9711yescharacteristic, complemented
184#102—972no—
185#102—972no—
186#102—972no—
187#102C2×D9712yescomplemented
188C24—1942no—
189#41—1942no—
190#41—1942no—
191C2×Q8—971no—
192C2×D4—22no—
193C2×D4—971no—
194C2×D4—971no—
195C2×D4—971no—
196C2×D4—972no—
197C2×D4—1942no—
198C2×D4—1944no—
199C2×D4—1944no—
200C2×D4—1942no—
201C2×D4—1942no—
202C2×D4—1942no—
203C2×D4C22×D9712yes—
204C22×C4—971no—
205C22×C4—971no—
206C22×C4C22×D9711yescharacteristic
207D8—22no—
208D8—1944no—
209D8—1942no—
210M16—22no—
211M16—1942no—
212M16—1942no—
213C2×C8—971no—
214C2×C8—974no—
215C2×C8—971no—
216C2×C8—1942no—
217C2×C8—1942no—
218C2×C8C22×D9711yescharacteristic
219C2×C8C22×D9711yescharacteristic
220C4⋊C4—972no—
221C4⋊C4C22×D9712yes—
222C22⋊C4—22no—
223C22⋊C4—1942no—
224C22⋊C4—1942no—
225C22⋊C4—1942no—
226C22⋊C4—1942no—
227C42—1942no—
228C23—22no—
229C23—7762no—
230C23—7762no—
231C23—971no—
232C23—971no—
233C23—3884no—
234C23—1942no—
235C23D4×D9711yescharacteristic
236Q8—1942no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—7762no—
241D4—7762no—
242D4—7762no—
243D4—3884no—
244D4—1942no—
245D4—1942no—
246D4—1942no—
247D4—1942no—
248D4—1942no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—971no—
253C2×C4—972no—
254C2×C4—972no—
255C2×C4—971no—
256C2×C4—1942no—
257C2×C4—1941no—
258C2×C4—1941no—
259C2×C4D4×D9711yescharacteristic
260C2×C4C23×D9711yescharacteristic
261C8—21no—
262C8—972no—
263C8—1941no—
264C8D4×D9712yes—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—7768no—
271C22—972no—
272C22—3882no—
273C22—3882no—
274C22—3882no—
275C22—1941no—
276C22—1942no—
277C22—1942no—
278C22C2×D4×D9711yescharacteristic
279C4—21no—
280C4—42no—
281C4—972no—
282C4—3882no—
283C4—1941no—
284C4C2×D4×D9711yescharacteristic
285C4#41×D9711yescharacteristic
286C2—41no—
287C2—82no—
288C2—7762no—
289C2—3881no—
290C2—1942no—
291C2#121×D9711yescharacteristic
292C2#2026153912yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#469×C97C2
socleC22×C97C22×D4
hypercenter#469D97
hypocenterC97#3058
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D97C2semidirect
2. C2×#15690818C2semidirect
3. #20261301C2semidirect
4. #20260989C2semidirect
5. #20260988C2semidirect
6. #20260985C2semidirect
7. #20260970C2semidirect
8. #20260969C2semidirect
9. #20260770C2semidirect
10. #20260738C2semidirect
11. #469×C97C2semidirect
19. C2×#13095494C22semidirect
20. C22×#11715895C22semidirect
22. C2×C8×D97C22semidirect
25. #15690785C22semidirect
26. #15690784C22semidirect
27. #15690781C22semidirect
31. #15690743C22semidirect
32. #15690742C22semidirect
33. #15690732C22semidirect
34. #15690731C22semidirect
35. #15690730C22semidirect
37. C2×M16×C97C22semidirect
38. #121×C97C22semidirect
39. #106×C97C22semidirect
40. #102×C97C22semidirect
65. C8×D97D4semidirect
147. C97#3058semidirect
154. #469D97semidirect
168. C2×M16C2×D97semidirect
177. #121C2×D97semidirect
183. #106C2×D97semidirect
187. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D972
C2×#156908182
#202613012
#202609892
#202609882
#202609852
#202609702
#202609692
#202607702
#202607382
#469×C972
#305897