Finite Group Databasebeta

← #27151316

Subgroups — #27151316

Order 12416.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15690830C211yesmaximal, characteristic, complemented
2C22×C8×D97C211yesmaximal, characteristic, complemented
3#20261247C211yesmaximal, characteristic, complemented
4#20261242C211yesmaximal, characteristic, complemented
5#20260975C212yesmaximal, complemented
6#20260970C212yesmaximal, complemented
7#20260967C212yesmaximal, complemented
8#20260963C212yesmaximal, complemented
9#20260771C211yesmaximal, characteristic, complemented
10#20260732C211yesmaximal, characteristic, complemented
11#416×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14#15690830—24no—
15C2×#13095501C2211yescharacteristic, complemented
16C2×#13095499C2211yescharacteristic, complemented
17C2×#13095498C2211yescharacteristic, complemented
18C2×#13095496C2211yescharacteristic, complemented
19C22×#11715895C2211yescharacteristic
20C2×C8×D97—24no—
21C2×C8×D97C2212yes—
22C2×C8×D97C2212yes—
23#15690794—21no—
24#15690791—21no—
25#15690790C2212yes—
26#15690786C2212yes—
27#15690785C2212yes—
28#15690781C2212yes—
29#15690780—21no—
30#15690778—21no—
31C4×D388—22no—
32C4×C22⋊D97—22no—
33#15690769C2211yescharacteristic
34#15690743C2212yescomplemented
35#15690741C2212yescomplemented
36#15690735C2212yescomplemented
37#15690732C2212yescomplemented
38C2×#41×C97C2211yescharacteristic
39C22×C8×C97C2211yescharacteristic
40#117×C97C2211yescharacteristic
41#103×C97C2212yescomplemented
42#102×C97C2212yescomplemented
43C23×D97D411yescharacteristic
44#13095511—22no—
45#13095511—22no—
46C2×Dic194C2311yescharacteristic
47C2×D388—22no—
48C2×D388C2311yescharacteristic
49C2×C22⋊D97—21no—
50C2×C22⋊D97—22no—
51C22×Dic97D411yescharacteristic
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97—21no—
55C2×C4×D97D412yes—
56C2×C4×D97C2312yes—
57#13095501—22no—
58#13095499—22no—
59#13095498—22no—
60#13095496—22no—
61C2×#11715895D414yescomplemented
62C2×#11715895C2311yescharacteristic
63C2×#11715895C2311yescharacteristic
64C8×D97—24no—
65C8×D97—24no—
66C4⋊Dic97—22no—
67C4⋊Dic97C2312yes—
68#13095490—22no—
69C22⋊Dic97—21no—
70C22⋊Dic97—22no—
71#13095488—22no—
72#13095488—22no—
73#13095488—22no—
74C4×Dic97—21no—
75#41×C97—22no—
76#41×C97—22no—
77C2×Q8×C97C2311yescharacteristic
78C2×D4×C97—21no—
79C2×D4×C97C2311yescharacteristic
80C22×C4×C97—21no—
81C22×C4×C97C2311yescharacteristic
82C2×C8×C97—22no—
83C2×C8×C97C2312yes—
84C4⋊C4×C97C2312yes—
85C22⋊C4×C97—21no—
86C42×C97—21no—
87C23×C97—21no—
88C23×C97C2×D411yescharacteristic
89C22×D97—22no—
90C22×D97—24no—
91C22×D97—21no—
92C22×D97C2×D412yes—
93Q8×C97—21no—
94Q8×C97—21no—
95D4×C97—21no—
96D4×C97—21no—
97D4×C97—42no—
98C2×C4×C97—22no—
99C2×C4×C97—21no—
100C2×C4×C97—42no—
101C2×C4×C97C2×D412yes—
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97#4112yes—
104C2×C4×C97C2411yescharacteristic
105C22⋊D97—42no—
106C22⋊D97—44no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—21no—
110C2×Dic97C2×D412yes—
111D388—21no—
112D388—21no—
113D388—44no—
114C4×D97—24no—
115C4×D97—22no—
116C4×D97—22no—
117C4×D97—42no—
118Dic194—21no—
119Dic194—21no—
120C8×C97—22no—
121#11715895C2×D412yes—
122#11715895C2×D412yes—
123C22×C97—22no—
124C22×C97—41no—
125C22×C97—42no—
126C22×C97#12112yes—
127C22×C97C22×D411yescharacteristic
128C2×D97—28no—
129C2×D97—22no—
130C2×D97—22no—
131C2×D97—41no—
132C2×D97—42no—
133C2×D97#13514yescomplemented
134C4×C97—41no—
135C4×C97—42no—
136C4×C97#12112yes—
137C4×C97C22×D411yescharacteristic
138C4×C97C2×#4111yescharacteristic
139Dic97—22no—
140Dic97—41no—
141Dic97—42no—
142C2×C97—22no—
143C2×C97—81no—
144C2×C97C2×#12111yescharacteristic
145C2×C97C2×#13512yes—
146D97—24no—
147D97—81no—
148#3050—971nomaximal, Sylow
149C97#305011yescharacteristic, complemented, Sylow
150C2×#135—971no—
151C23×C8—971no—
152#525—971no—
153#525—971no—
154#467—972no—
155#466—972no—
156#465—972no—
157#465—972no—
158#427—971no—
159#416—971no—
160#416D9711yescharacteristic, complemented
161C2×#41—971no—
162C2×#41C2×D9711yescharacteristic
163C23×C4—971no—
164#135—1944no—
165C2×Q16—971no—
166C2×SD16—971no—
167C2×SD16—971no—
168C2×D8—971no—
169C22×C8—971no—
170C22×C8—972no—
171C22×C8—972no—
172C22×C8—1944no—
173C22×C8C2×D9711yescharacteristic
174#123—1941no—
175#123—1941no—
176C22⋊Q8—972no—
177C22⋊Q8—972no—
178#121—972no—
179#121—972no—
180#120—1941no—
181#120—1941no—
182C4×D4—1942no—
183C4×D4—1942no—
184#117—971no—
185#117C2×D9711yescharacteristic
186#107—972no—
187#106—972no—
188#103—972no—
189#103C2×D9712yescomplemented
190#102—972no—
191#102C2×D9712yescomplemented
192C24—971no—
193#41—22no—
194#41—22no—
195#41—1942no—
196#41—1942no—
197C2×Q8—971no—
198C2×Q8C22×D9711yescharacteristic
199C2×D4—21no—
200C2×D4—971no—
201C2×D4—1942no—
202C2×D4—1942no—
203C2×D4—1941no—
204C2×D4C22×D9711yescharacteristic
205C22×C4—21no—
206C22×C4—971no—
207C22×C4—972no—
208C22×C4—972no—
209C22×C4—1941no—
210C22×C4—1942no—
211C22×C4—1942no—
212C22×C4C22×D9711yescharacteristic
213Q16—1942no—
214SD16—1942no—
215SD16—1942no—
216D8—1942no—
217C2×C8—22no—
218C2×C8—974no—
219C2×C8—971no—
220C2×C8—971no—
221C2×C8—1944no—
222C2×C8—1944no—
223C2×C8C22×D9712yes—
224C4⋊C4—972no—
225C4⋊C4—1942no—
226C4⋊C4—1942no—
227C4⋊C4C22×D9712yes—
228C22⋊C4—21no—
229C22⋊C4—1942no—
230C22⋊C4—1942no—
231C22⋊C4—1942no—
232C22⋊C4—1942no—
233C22⋊C4—1941no—
234C42—21no—
235C42—1941no—
236C23—21no—
237C23—972no—
238C23—1944no—
239C23—1941no—
240C23—1942no—
241C23D4×D9711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—1941no—
245Q8—1941no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—3884no—
250D4—3884no—
251D4—3882no—
252D4—1941no—
253D4—1941no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—972no—
258C2×C4—3882no—
259C2×C4—1942no—
260C2×C4—1941no—
261C2×C4—1942no—
262C2×C4—1944no—
263C2×C4—1942no—
264C2×C4—1942no—
265C2×C4D4×D9712yes—
266C2×C4D4×D9711yescharacteristic
267C2×C4#1309551112yes—
268C2×C4C23×D9711yescharacteristic
269C8—22no—
270C8—972no—
271C8—972no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—974no—
276C22—3881no—
277C22—3882no—
278C22—1948no—
279C22—1942no—
280C22—1942no—
281C22#1569078312yes—
282C22C2×D4×D9711yescharacteristic
283C4—41no—
284C4—42no—
285C4—3881no—
286C4—3882no—
287C4—1942no—
288C4#1569078312yes—
289C4C2×D4×D9711yescharacteristic
290C4C2×#1309551111yescharacteristic
291C2—22no—
292C2—81no—
293C2—7761no—
294C2—1944no—
295C2C2×#1569078311yescharacteristic
296C2#135×D9712yes—
subgroupG/H
centerC2×C4D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#416×C97C2
socleC22×C97C22×D4
hypercenter#416D97
hypocenterC97#3050
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#15690830C2semidirect
2. C22×C8×D97C2semidirect
3. #20261247C2semidirect
4. #20261242C2semidirect
5. #20260975C2semidirect
6. #20260970C2semidirect
7. #20260967C2semidirect
8. #20260963C2semidirect
9. #20260771C2semidirect
10. #20260732C2semidirect
11. #416×C97C2semidirect
15. C2×#13095501C22semidirect
16. C2×#13095499C22semidirect
17. C2×#13095498C22semidirect
18. C2×#13095496C22semidirect
34. #15690743C22semidirect
35. #15690741C22semidirect
36. #15690735C22semidirect
37. #15690732C22semidirect
41. #103×C97C22semidirect
42. #102×C97C22semidirect
61. C2×#11715895D4semidirect
133. C2×D97#135semidirect
149. C97#3050semidirect
160. #416D97semidirect
189. #103C2×D97semidirect
191. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908302
C22×C8×D972
#202612472
#202612422
#202609752
#202609702
#202609672
#202609632
#202607712
#202607322
#416×C972
#305097