Finite Group Databasebeta

← #27153663

Subgroups — #27153663

Order 12416.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#20261451C211yesmaximal, characteristic, complemented
2#20261388C211yesmaximal, characteristic, complemented
3#20261075C212yesmaximal, complemented
4#20261022C211yesmaximal, characteristic, complemented
5#20260967C212yesmaximal, complemented
6#20260937C211yesmaximal, characteristic, complemented
7#20260883C212yesmaximal, complemented
8C4×D776C211yesmaximal, characteristic, complemented
9C8×D388C211yesmaximal, characteristic, complemented
10#20260730C212yesmaximal, complemented
11#487×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×#13095498C2212yes—
14C2×D776C2211yescharacteristic
15C2×C8×D97C2212yes—
16#15690799—22no—
17#15690795—22no—
18#15690786C2212yes—
19#15690785—21no—
20#15690785—21no—
21#15690785C2212yes—
22C4×D388C2211yescharacteristic
23C4×D388C2211yescharacteristic
24C4×D388C2211yescharacteristic
25#15690768—21no—
26#15690768C2212yes—
27C4⋊C4×D97—21no—
28#15690744C2211yescharacteristic
29#15690741C2212yes—
30C4⋊#11715895C2211yescharacteristic
31#15690734C2212yes—
32#15690732C2211yescharacteristic
33#15690732C2211yescharacteristic
34#15690730C2212yes—
35#15690722C2212yes—
36C4⋊Q8×C97C2211yescharacteristic
37C4⋊Q8×C97C2211yescharacteristic
38#103×C97C2212yes—
39#103×C97C2212yes—
40C4×C8×C97C2211yescharacteristic
41#13095512—24no—
42#13095512—42no—
43C2×D388—22no—
44C2×D388—22no—
45C2×D388D411yescharacteristic
46C2×D388C2311yescharacteristic
47C2×D388C2311yescharacteristic
48C2×C4×D97—21no—
49C2×C4×D97—21no—
50C2×C4×D97—22no—
51C2×C4×D97C2312yes—
52#13095498—42no—
53D776—22no—
54C2×#11715895C2312yes—
55C8×D97—42no—
56C4⋊Dic97D411yescharacteristic
57C4⋊Dic97C2311yescharacteristic
58C4⋊Dic97C2311yescharacteristic
59#13095490—22no—
60#13095488—22no—
61#13095488—22no—
62#13095488—21no—
63#13095488—21no—
64#13095488—22no—
65#13095488D412yes—
66C4×Dic97—22no—
67C2×Q8×C97—21no—
68C2×Q8×C97C2312yes—
69C2×C8×C97C2311yescharacteristic
70C2×C8×C97C2311yescharacteristic
71C4⋊C4×C97—21no—
72C4⋊C4×C97—21no—
73C4⋊C4×C97—21no—
74C4⋊C4×C97C2312yes—
75C42×C97C2311yescharacteristic
76C22×D97—21no—
77C22×D97—21no—
78C22×D97C2×D412yes—
79Q8×C97—22no—
80Q8×C97—22no—
81Q8×C97—81no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97C2411yescharacteristic
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97C2×D412yes—
90D388—22no—
91D388—21no—
92D388—21no—
93D388—21no—
94D388—44no—
95D388—82no—
96D388#4112yes—
97C4×D97—22no—
98C4×D97—22no—
99C4×D97—44no—
100C4×D97—42no—
101C4×D97—42no—
102C8×C97—21no—
103C8×C97—21no—
104#11715895—22no—
105C22×C97C22×D411yescharacteristic
106C2×D97—22no—
107C2×D97—22no—
108C2×D97—22no—
109C2×D97—22no—
110C2×D97—41no—
111C2×D97—41no—
112C2×D97—41no—
113C2×D97—41no—
114C2×D97—41no—
115C4×C97—21no—
116C4×C97—42no—
117C4×C97—42no—
118C4×C97#13512yes—
119C4×C97C2×#4111yescharacteristic
120C4×C97H(5,𝔽2)11yescharacteristic
121Dic97—22no—
122Dic97—41no—
123Dic97—41no—
124C2×C97#54611yescharacteristic
125C2×C97C2×#13511yescharacteristic
126C2×C97#57711yescharacteristic
127D97—42no—
128D97—42no—
129D97—81no—
130#3303—971nomaximal, Sylow
131C97#330311yescharacteristic, complemented, Sylow
132#555—971no—
133#547—971no—
134#487D9711yescharacteristic, complemented
135#485—972no—
136#476—971no—
137#465—972no—
138#461—971no—
139#449—972no—
140C4×D8—971no—
141C8×D4—971no—
142#416—972no—
143C2×#41—972no—
144C2×SD16—972no—
145C2×D8—971no—
146C22×C8—972no—
147C4⋊Q8C2×D9711yescharacteristic
148C4⋊Q8C2×D9711yescharacteristic
149#124—1942no—
150#123—1942no—
151C22⋊Q8—972no—
152#121—972no—
153#121—1941no—
154#121—1941no—
155C4×D4—971no—
156C4×D4—971no—
157C4×D4—971no—
158#117—972no—
159#117—1941no—
160C2×C4⋊C4—1941no—
161#107—971no—
162#106—972no—
163C4⋊C8—971no—
164#103—972no—
165#103C2×D9712yes—
166#103C2×D9712yes—
167#102—971no—
168#102—971no—
169#102—972no—
170C22⋊C8—972no—
171C4×C8C2×D9711yescharacteristic
172#41—3882no—
173#41—1944no—
174C2×Q8—21no—
175C2×Q8C22×D9712yes—
176C2×D4—971no—
177C2×D4—971no—
178C2×D4—971no—
179C2×D4—1942no—
180C2×D4—1942no—
181C22×C4—972no—
182C22×C4—1942no—
183C22×C4—1941no—
184C22×C4—1941no—
185SD16—3882no—
186D8—1942no—
187C2×C8—972no—
188C2×C8—3882no—
189C2×C8C22×D9711yescharacteristic
190C2×C8C22×D9711yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—971no—
195C4⋊C4—971no—
196C4⋊C4—971no—
197C4⋊C4—1942no—
198C4⋊C4C22×D9712yes—
199C22⋊C4—972no—
200C22⋊C4—1942no—
201C22⋊C4—1942no—
202C22⋊C4—1942no—
203C22⋊C4—1941no—
204C22⋊C4—1941no—
205C42—1942no—
206C42C22×D9711yescharacteristic
207C23—972no—
208C23—1941no—
209C23—1941no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—7762no—
214D4—972no—
215D4—3884no—
216D4—1942no—
217D4—1941no—
218D4—1941no—
219D4—1941no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—972no—
223C2×C4—3884no—
224C2×C4—3882no—
225C2×C4—3882no—
226C2×C4—1941no—
227C2×C4—1941no—
228C2×C4—1942no—
229C2×C4—1942no—
230C2×C4D4×D9711yescharacteristic
231C2×C4D4×D9711yescharacteristic
232C2×C4C23×D9711yescharacteristic
233C8—21no—
234C8—21no—
235C8—1942no—
236C22—3881no—
237C22—3881no—
238C22—3881no—
239C22—3881no—
240C22—3881no—
241C22—1942no—
242C22—1942no—
243C22—1942no—
244C22—1942no—
245C22C2×D4×D9711yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—3881no—
250C4—3881no—
251C4—1942no—
252C4#1569083312yes—
253C4#41×D9711yescharacteristic
254C4#1569085611yescharacteristic
255C2—7761no—
256C2—3882no—
257C2—3882no—
258C2#2026137811yescharacteristic
259C2C2×#1569083311yescharacteristic
260C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#487×C97C2
socleC22×C97C22×D4
hypercenter#487D97
hypocenterC97#3303
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #20261451C2semidirect
2. #20261388C2semidirect
3. #20261075C2semidirect
4. #20261022C2semidirect
5. #20260967C2semidirect
6. #20260937C2semidirect
7. #20260883C2semidirect
8. C4×D776C2semidirect
9. C8×D388C2semidirect
10. #20260730C2semidirect
11. #487×C97C2semidirect
131. C97#3303semidirect
134. #487D97semidirect

Maximal subgroups

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

subgroupindex
#202614512
#202613882
#202610752
#202610222
#202609672
#202609372
#202608832
C4×D7762
C8×D3882
#202607302
#487×C972
#330397