Finite Group Databasebeta

← #27153554

Subgroups — #27153554

Order 12416.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D388C211yesmaximal, characteristic
2#20261083C211yesmaximal, characteristic, complemented
3#20261053C212yesmaximal, complemented
4#20261022C211yesmaximal, characteristic, complemented
5#20260966C212yesmaximal, complemented
6#20260935C211yesmaximal, characteristic, complemented
7#20260885C212yesmaximal, complemented
8C4×#13095498C211yesmaximal, characteristic, complemented
9D4×#11715895C211yesmaximal, characteristic, complemented
10C2×#15690730C212yesmaximal, complemented
11#554×C97C211yesmaximal, characteristic, complemented
12C22×D388C2212yes—
13C2×D4×D97—21no—
14C2×#13095498C2211yescharacteristic
15C2×#13095498C2212yescomplemented
16C22×#11715895C2212yescomplemented
17#15690807C2211yescharacteristic
18#15690785—21no—
19#15690784C2212yes—
20#15690780—22no—
21C4×D388C2211yescharacteristic
22#15690741C2212yescomplemented
23#15690741C2211yescharacteristic, complemented
24C4⋊#11715895C2211yescharacteristic, complemented
25#15690734C2211yescharacteristic, complemented
26#15690730C2212yes—
27#15690730C2211yescharacteristic
28#15690730C2212yescomplemented
29#15690730C2212yescomplemented
30#15690730C2212yescomplemented
31C22⋊#11715895C2212yescomplemented
32C4×#11715895C2211yescharacteristic, complemented
33C4⋊Q8×C97C2211yescharacteristic, complemented
34#125×C97—21no—
35C22⋊Q8×C97—22no—
36C22⋊Q8×C97C2212yescomplemented
37C4×Q8×C97C2211yescharacteristic, complemented
38C4×D4×C97—21no—
39C4×D4×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2212yescomplemented
41C23×D97—22no—
42C2×D388—24no—
43C2×D388—22no—
44C2×D388—21no—
45C2×D388—21no—
46C2×D388—42no—
47C2×D388C2312yes—
48C2×D388C2311yescharacteristic
49C2×C22⋊D97—22no—
50D4×D97—44no—
51C2×C4×D97—21no—
52#13095498—22no—
53#13095498—42no—
54C2×#11715895—42no—
55C2×#11715895C2311yescharacteristic
56C2×#11715895C2312yes—
57C2×#11715895C2311yescharacteristic
58C4⋊Dic97C2311yescharacteristic
59#13095488—21no—
60#13095488—22no—
61C2×Q8×C97—21no—
62C2×Q8×C97C2311yescharacteristic
63C2×D4×C97D411yescharacteristic
64C22×C4×C97—22no—
65C22×C4×C97C2312yes—
66C4⋊C4×C97—22no—
67C4⋊C4×C97—21no—
68C4⋊C4×C97—21no—
69C4⋊C4×C97—22no—
70C4⋊C4×C97D411yescharacteristic
71C4⋊C4×C97C2311yescharacteristic
72C4⋊C4×C97C2312yes—
73C22⋊C4×C97—22no—
74C22⋊C4×C97D412yes—
75C42×C97—21no—
76C42×C97C2311yescharacteristic
77C23×C97C2×D412yes—
78C22×D97—21no—
79C22×D97—22no—
80C22×D97—44no—
81C22×D97—44no—
82C22×D97—82no—
83Q8×C97—21no—
84Q8×C97—21no—
85Q8×C97—81no—
86D4×C97SD1614yescomplemented
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—82no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D412yes—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—44no—
98C2×Dic97—21no—
99D388—22no—
100D388—21no—
101D388—22no—
102D388—44no—
103D388—42no—
104D388—81no—
105D388#4112yes—
106C4×D97—42no—
107#11715895—21no—
108#11715895—21no—
109#11715895—22no—
110C22×C97—22no—
111C22×C97—22no—
112C22×C97C2×SD1614yes—
113C22×C97C22×D411yescharacteristic
114C2×D97—22no—
115C2×D97—41no—
116C2×D97—41no—
117C2×D97—42no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—88no—
121C4×C97—22no—
122C4×C97—21no—
123C4×C97—41no—
124C4×C97—41no—
125C4×C97—42no—
126C4×C97C2×SD1612yes—
127C4×C97C2×#4111yescharacteristic
128C4×C97H(5,𝔽2)11yescharacteristic
129Dic97—41no—
130C2×C97—24no—
131C2×C97#54611yescharacteristic
132C2×C97C22×SD1611yescharacteristic
133C2×C97#57611yescharacteristic
134D97—42no—
135D97—82no—
136#3295—971nomaximal, Sylow
137C97#329511yescharacteristic, complemented, Sylow
138#554D9711yescharacteristic, complemented
139D42—971no—
140#486—971no—
141#481—972no—
142#476—971no—
143#465—972no—
144#460—971no—
145#450—972no—
146C4×SD16—971no—
147C8×D4—971no—
148C2×#102—972no—
149C22×D4—972no—
150C22×D4—1941no—
151C2×SD16—972no—
152C2×SD16—971no—
153C22×C8—972no—
154C4⋊Q8C2×D9711yescharacteristic, complemented
155#127—971no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D9712yescomplemented
159#121—972no—
160#121—1941no—
161#120—1942no—
162C4×Q8C2×D9711yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—971no—
165C4×D4C2×D9711yescharacteristic
166C2×C4⋊C4C2×D9712yescomplemented
167#106—971no—
168#106—972no—
169C4⋊C8—971no—
170#103—971no—
171#102—971no—
172#102—972no—
173#102—972no—
174#102—972no—
175#102—972no—
176C22⋊C8—972no—
177C4×C8—971no—
178C24—1942no—
179C2×Q8—21no—
180C2×Q8C22×D9711yescharacteristic
181C2×D4—971no—
182C2×D4—972no—
183C2×D4—3882no—
184C2×D4—3884no—
185C2×D4—1944no—
186C2×D4—1942no—
187C2×D4—1941no—
188C2×D4—1942no—
189C2×D4—1941no—
190C2×D4C22⋊D9711yescharacteristic
191C22×C4—22no—
192C22×C4—1941no—
193C22×C4C22×D9712yes—
194SD16—3882no—
195SD16—1942no—
196C2×C8—971no—
197C2×C8—971no—
198C2×C8—972no—
199C2×C8—3882no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—971no—
205C4⋊C4C22⋊D9711yescharacteristic
206C4⋊C4C22×D9711yescharacteristic
207C4⋊C4C22×D9712yes—
208C22⋊C4—22no—
209C22⋊C4—1941no—
210C22⋊C4—1942no—
211C22⋊C4C22⋊D9712yes—
212C42—21no—
213C42C22×D9711yescharacteristic
214C23—7762no—
215C23—3884no—
216C23—3884no—
217C23—1942no—
218C23—1941no—
219C23C2×C22⋊D9712yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—7761no—
224D4—972no—
225D4—3884no—
226D4—3884no—
227D4—3882no—
228D4—1942no—
229D4—1942no—
230D4—1941no—
231D4#1309549814yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—3882no—
239C2×C4—1941no—
240C2×C4C2×C22⋊D9711yescharacteristic
241C2×C4C2×C22⋊D9712yes—
242C2×C4C2×C22⋊D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—1941no—
245C8—1941no—
246C8—1942no—
247C22—22no—
248C22—22no—
249C22—7768no—
250C22—3882no—
251C22—3882no—
252C22—3882no—
253C22—3881no—
254C22—3881no—
255C22—1942no—
256C22C2×#1309549814yes—
257C22C22×C22⋊D9711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—3881no—
264C4C2×#1309549812yes—
265C4#41×D9711yescharacteristic
266C4#1569085611yescharacteristic
267C2—24no—
268C2—7762no—
269C2—3882no—
270C2#2026137311yescharacteristic
271C2C22×#1309549811yescharacteristic
272C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#554×C97C2
socleC22×C97C22×D4
prime radicalD4×D388C2
hypercenter#554D97
hypocenterC97#3295
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #20261083C2semidirect
3. #20261053C2semidirect
4. #20261022C2semidirect
5. #20260966C2semidirect
6. #20260935C2semidirect
7. #20260885C2semidirect
8. C4×#13095498C2semidirect
9. D4×#11715895C2semidirect
10. C2×#15690730C2semidirect
11. #554×C97C2semidirect
15. C2×#13095498C22semidirect
16. C22×#11715895C22semidirect
22. #15690741C22semidirect
23. #15690741C22semidirect
24. C4⋊#11715895C22semidirect
25. #15690734C22semidirect
28. #15690730C22semidirect
29. #15690730C22semidirect
30. #15690730C22semidirect
31. C22⋊#11715895C22semidirect
32. C4×#11715895C22semidirect
33. C4⋊Q8×C97C22semidirect
36. C22⋊Q8×C97C22semidirect
37. C4×Q8×C97C22semidirect
40. C2×C4⋊C4×C97C22semidirect
86. D4×C97SD16semidirect
137. C97#3295semidirect
138. #554D97semidirect
154. C4⋊Q8C2×D97semidirect
158. C22⋊Q8C2×D97semidirect
162. C4×Q8C2×D97semidirect
166. C2×C4⋊C4C2×D97semidirect
231. D4#13095498semidirect

Maximal subgroups

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

subgroupindex
D4×D3882
#202610832
#202610532
#202610222
#202609662
#202609352
#202608852
C4×#130954982
D4×#117158952
C2×#156907302
#554×C972
#329597