Finite Group Databasebeta

← #27153336

Subgroups — #27153336

Order 12416.

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

normal subgroups only

This group has 546 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×#15690835C212yesmaximal, complemented
2D4×D388C211yesmaximal, characteristic, complemented
3#20261109C211yesmaximal, characteristic, complemented
4#20260983C212yesmaximal, complemented
5#20260937C211yesmaximal, characteristic, complemented
6#20260930C211yesmaximal, characteristic, complemented
7#20260885C212yesmaximal, complemented
8#20260880C212yesmaximal, complemented
9C4×#13095496C211yesmaximal, characteristic, complemented
10#20260813C211yesmaximal, characteristic, complemented
11#547×C97C211yesmaximal, characteristic, complemented
12C22×D388C2212yes—
13C2×D4×D97—21no—
14#15690835—28no—
15C2×#13095498C2212yes—
16C2×#13095498C2212yescomplemented
17C2×#13095498C2212yescomplemented
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2212yescomplemented
20C2×#13095495C2212yescomplemented
21#15690807C2211yescharacteristic
22#15690785—21no—
23#15690784C2212yes—
24#15690780—22no—
25C4×D388C2211yescharacteristic
26#15690743C2211yescharacteristic, complemented
27C4⋊#11715895C2211yescharacteristic, complemented
28#15690734C2212yescomplemented
29#15690731C2211yescharacteristic, complemented
30#15690730C2211yescharacteristic
31#15690730C2212yescomplemented
32C22⋊#11715895C2212yescomplemented
33C4×#11715895C2211yescharacteristic, complemented
34C2×#41×C97C2212yescomplemented
35C4⋊Q8×C97C2211yescharacteristic, complemented
36#123×C97—22no—
37C22⋊Q8×C97C2212yescomplemented
38#121×C97—21no—
39C4×D4×C97C2211yescharacteristic
40C4×D4×C97C2211yescharacteristic, complemented
41C2×C4⋊C4×C97—21no—
42C23×D97—22no—
43C2×D388—24no—
44C2×D388—22no—
45C2×D388—21no—
46C2×D388—21no—
47C2×D388—42no—
48C2×D388C2312yes—
49C2×D388C2311yescharacteristic
50C2×C22⋊D97—22no—
51D4×D97—44no—
52C2×C4×D97—21no—
53#13095498—24no—
54#13095498—24no—
55#13095498—42no—
56#13095496—24no—
57#13095496D414yescomplemented
58#13095495—24no—
59C2×#11715895C2312yes—
60C2×#11715895C2311yescharacteristic
61C2×#11715895C2311yescharacteristic
62C4⋊Dic97C2311yescharacteristic
63#13095488—21no—
64#13095488—22no—
65#41×C97—24no—
66#41×C97—42no—
67C2×Q8×C97C2312yes—
68C2×D4×C97—22no—
69C2×D4×C97D411yescharacteristic
70C2×D4×C97C2311yescharacteristic
71C22×C4×C97—21no—
72C22×C4×C97—22no—
73C22×C4×C97C2312yes—
74C4⋊C4×C97—21no—
75C4⋊C4×C97—22no—
76C4⋊C4×C97—21no—
77C4⋊C4×C97D411yescharacteristic
78C4⋊C4×C97C2311yescharacteristic
79C22⋊C4×C97—22no—
80C22⋊C4×C97—21no—
81C22⋊C4×C97D412yes—
82C42×C97C2311yescharacteristic
83C23×C97—21no—
84C23×C97C2×D412yes—
85C22×D97—21no—
86C22×D97—22no—
87C22×D97—44no—
88C22×D97—44no—
89C22×D97—82no—
90Q8×C97—22no—
91Q8×C97—22no—
92D4×C97—22no—
93D4×C97—21no—
94D4×C97—44no—
95D4×C97C2×D412yescomplemented
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97—21no—
99C2×C4×C97—22no—
100C2×C4×C97—44no—
101C2×C4×C97—42no—
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97C2×D412yes—
104C2×C4×C97C2×D411yescharacteristic
105C2×C4×C97C2411yescharacteristic
106C22⋊D97—44no—
107C2×Dic97—21no—
108D388—22no—
109D388—21no—
110D388—22no—
111D388—44no—
112D388—42no—
113D388—81no—
114D388C2×D412yes—
115C4×D97—42no—
116#11715895—22no—
117#11715895—21no—
118#11715895C2×D412yescomplemented
119C22×C97—22no—
120C22×C97—22no—
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—41no—
124C22×C97—41no—
125C22×C97C22×D411yescharacteristic
126C2×D97—22no—
127C2×D97—41no—
128C2×D97—41no—
129C2×D97—42no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—88no—
133C4×C97—22no—
134C4×C97—21no—
135C4×C97—42no—
136C4×C97—41no—
137C4×C97AGL(1,ℤ/8)12yescomplemented
138C4×C97C22×D411yescharacteristic
139C4×C97H(5,𝔽2)11yescharacteristic
140Dic97—41no—
141C2×C97—42no—
142C2×C97—42no—
143C2×C97D4211yescharacteristic
144C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C97#57711yescharacteristic
146D97—42no—
147D97—82no—
148#3272—971nomaximal, Sylow
149C97#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—972no—
151#547D9711yescharacteristic, complemented
152D42—971no—
153#492—971no—
154#468—972no—
155#461—971no—
156#459—971no—
157#450—972no—
158#449—972no—
159C4×D8—971no—
160#436—971no—
161C2×#41C2×D9712yescomplemented
162C22×D4—972no—
163C22×D4—1941no—
164AGL(1,ℤ/8)—1948no—
165C2×SD16—972no—
166C2×SD16—972no—
167C2×SD16—972no—
168C2×D8—971no—
169C2×D8—972no—
170C2×M16—972no—
171C4⋊Q8C2×D9711yescharacteristic, complemented
172#127—971no—
173#123—22no—
174C22⋊Q8C2×D9712yescomplemented
175#121—21no—
176#121—972no—
177#121—1941no—
178#120—1942no—
179C4×D4—971no—
180C4×D4C2×D9711yescharacteristic
181C4×D4C2×D9711yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—971no—
184C4⋊C8—971no—
185#103—972no—
186#102—971no—
187#102—971no—
188#102—972no—
189C22⋊C8—972no—
190C4×C8—971no—
191C24—1942no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D9712yes—
195C2×D4—22no—
196C2×D4—971no—
197C2×D4—972no—
198C2×D4—3882no—
199C2×D4—3884no—
200C2×D4—1944no—
201C2×D4—1942no—
202C2×D4—1941no—
203C2×D4—1942no—
204C2×D4—1941no—
205C2×D4C22⋊D9711yescharacteristic
206C2×D4C22×D9711yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—1941no—
210C22×C4C22×D9712yes—
211SD16—3882no—
212SD16—1944no—
213SD16—1944no—
214D8—974no—
215D8—1944no—
216M16—1944no—
217C2×C8—971no—
218C2×C8—971no—
219C2×C8—972no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—971no—
224C4⋊C4C22⋊D9711yescharacteristic
225C4⋊C4C22×D9711yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—1941no—
229C22⋊C4—1942no—
230C22⋊C4C22⋊D9712yes—
231C42C22×D9711yescharacteristic
232C23—21no—
233C23—7762no—
234C23—3884no—
235C23—3884no—
236C23—1942no—
237C23—1941no—
238C23C2×C22⋊D9712yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—7761no—
245D4—972no—
246D4—3884no—
247D4—3884no—
248D4—3882no—
249D4—1942no—
250D4—1942no—
251D4—1941no—
252D4D4×D9712yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3882no—
260C2×C4—1941no—
261C2×C4C2×C22⋊D9711yescharacteristic
262C2×C4C2×C22⋊D9712yes—
263C2×C4C2×C22⋊D9711yescharacteristic
264C2×C4C23×D9711yescharacteristic
265C8—972no—
266C8—1942no—
267C8—1941no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—7768no—
275C22—3882no—
276C22—3882no—
277C22—3882no—
278C22—3881no—
279C22—3881no—
280C22—1942no—
281C22C22×C22⋊D9711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—3881no—
287C4#1569083512yescomplemented
288C4C2×D4×D9711yescharacteristic
289C4#1569085511yescharacteristic
290C2—42no—
291C2—42no—
292C2—7762no—
293C2—3882no—
294C2D4×C22⋊D9711yescharacteristic
295C2C2×#1569083511yescharacteristic
296C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#547×C97C2
socleC22×C97C22×D4
hypercenter#547D97
hypocenterC97#3272
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. D4×D388C2semidirect
3. #20261109C2semidirect
4. #20260983C2semidirect
5. #20260937C2semidirect
6. #20260930C2semidirect
7. #20260885C2semidirect
8. #20260880C2semidirect
9. C4×#13095496C2semidirect
10. #20260813C2semidirect
11. #547×C97C2semidirect
16. C2×#13095498C22semidirect
17. C2×#13095498C22semidirect
19. C2×#13095496C22semidirect
20. C2×#13095495C22semidirect
26. #15690743C22semidirect
27. C4⋊#11715895C22semidirect
28. #15690734C22semidirect
29. #15690731C22semidirect
31. #15690730C22semidirect
32. C22⋊#11715895C22semidirect
33. C4×#11715895C22semidirect
34. C2×#41×C97C22semidirect
35. C4⋊Q8×C97C22semidirect
37. C22⋊Q8×C97C22semidirect
40. C4×D4×C97C22semidirect
57. #13095496D4semidirect
95. D4×C97C2×D4semidirect
118. #11715895C2×D4semidirect
137. C4×C97AGL(1,ℤ/8)semidirect
149. C97#3272semidirect
151. #547D97semidirect
161. C2×#41C2×D97semidirect
171. C4⋊Q8C2×D97semidirect
174. C22⋊Q8C2×D97semidirect
181. C4×D4C2×D97semidirect
252. D4D4×D97semidirect
287. C4#15690835semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
D4×D3882
#202611092
#202609832
#202609372
#202609302
#202608852
#202608802
C4×#130954962
#202608132
#547×C972
#327297