Finite Group Databasebeta

← #27141961

Subgroups — #27141961

Order 12416.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#20261216C211yesmaximal, characteristic
2#20260775C211yesmaximal, characteristic, complemented
3C2×#15690738C211yesmaximal, characteristic
4#20260734C211yesmaximal, characteristic
5#20260504C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7#384×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C2×#13095495C2211yescharacteristic
11#15690775—21no—
12#15690772—21no—
13C4×C22⋊D97—22no—
14#15690769C2211yescharacteristic
15#15690768—21no—
16C4⋊C4×D97—21no—
17#15690743—21no—
18#15690741—21no—
19#15690738C414yescomplemented
20#15690733—21no—
21#15690730—21no—
22C2×C4⋊C4×C97—22no—
23C2×C4⋊C4×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25C2×C42×C97C2211yescharacteristic
26#13095511—21no—
27#13095511—21no—
28#13095511C2×C412yes—
29#13095511Q812yes—
30C2×Dic194D411yescharacteristic
31C2×D388D411yescharacteristic
32C2×C22⋊D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—22no—
35#13095495—21no—
36#13095495—21no—
37#13095495—21no—
38#13095495C2×C412yescomplemented
39C2×#11715895—21no—
40C2×#11715895—21no—
41C4⋊Dic97D411yescharacteristic
42C4⋊Dic97D411yescharacteristic
43#13095490—22no—
44C22⋊Dic97—21no—
45#13095488—22no—
46C4×Dic97—22no—
47C4×Dic97—21no—
48C22×C4×C97—21no—
49C22×C4×C97—21no—
50C22×C4×C97D412yes—
51C22×C4×C97C2311yescharacteristic
52C4⋊C4×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97—21no—
55C4⋊C4×C97—21no—
56C4⋊C4×C97—44no—
57C42×C97—21no—
58C42×C97C2×C412yescomplemented
59C23×C97C2×D411yescharacteristic
60C22×D97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—42no—
66C2×C4×C97—42no—
67C2×C4×C97—41no—
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×Q811yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C22⋊D97—22no—
75C22⋊D97—22no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79D388—21no—
80D388C4⋊C412yescomplemented
81C4×D97—22no—
82C4×D97—22no—
83C4×D97—82no—
84Dic194—21no—
85Dic194C4⋊C412yescomplemented
86#11715895—22no—
87#11715895—41no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97C4×D411yescharacteristic
91C22×C97#12011yescharacteristic
92C22×C97C22⋊Q811yescharacteristic
93C2×D97—22no—
94C2×D97—41no—
95C2×D97—41no—
96C4×C97—24no—
97C4×C97—42no—
98C4×C97—81no—
99C4×C97C2×C4⋊C411yescharacteristic
100C4×C97C4×D411yescharacteristic
101C4×C97C22⋊Q811yescharacteristic
102C4×C97#12311yescharacteristic
103Dic97—22no—
104Dic97—41no—
105Dic97—41no—
106C2×C97—21no—
107C2×C97—21no—
108C2×C97#38511yescharacteristic
109C2×C97#45311yescharacteristic
110C2×C97#45611yescharacteristic
111D97—42no—
112#1859—971nomaximal, Sylow
113C97#185911yescharacteristic, complemented, Sylow
114#520—971no—
115#428—971no—
116C2×#104—971no—
117#417—971no—
118#384D9711yescharacteristic, complemented
119#343—971no—
120#339—971no—
121C2×#41—971no—
122C2×M16—971no—
123C2×M16—971no—
124C4×Q8—1941no—
125C4×D4—1941no—
126C4×D4—1942no—
127#117—971no—
128#117—1941no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—1941no—
131C2×C4⋊C4C2×D9711yescharacteristic
132C2×C4⋊C4C2×D9711yescharacteristic
133C2×C42C2×D9711yescharacteristic
134#107—1941no—
135#106—1941no—
136#104—974no—
137#103—1941no—
138#102—1941no—
139#41—972no—
140#41—972no—
141#41—1941no—
142#41—1941no—
143C2×Q8—971no—
144C2×D4—971no—
145C2×D4—1941no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1942no—
149C22×C4—1941no—
150C22×C4C22⋊D9712yes—
151C22×C4C22×D9711yescharacteristic
152M16—972no—
153M16—1941no—
154M16—1941no—
155M16—1941no—
156C2×C8—1941no—
157C2×C8—1941no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—44no—
163C4⋊C4—971no—
164C4⋊C4—971no—
165C4⋊C4—1942no—
166C22⋊C4—1942no—
167C22⋊C4—1941no—
168C42—21no—
169C42—1942no—
170C42—1941no—
171C42C4×D9712yescomplemented
172C23—1941no—
173C23C2×C22⋊D9711yescharacteristic
174Q8—972no—
175Q8—1941no—
176D4—972no—
177D4—1942no—
178D4—1942no—
179D4—1941no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—7762no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4—1941no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4C2×C4×D9711yescharacteristic
194C2×C4D4×D9711yescharacteristic
195C2×C4D4×D9711yescharacteristic
196C2×C4Q8×D9711yescharacteristic
197C2×C4#1309551111yescharacteristic
198C2×C4#1309551211yescharacteristic
199C8—3881no—
200C8—1942no—
201C22—21no—
202C22—21no—
203C22—3881no—
204C22—3881no—
205C22—1942no—
206C22C4×C22⋊D9711yescharacteristic
207C22#1569077811yescharacteristic
208C22#1569078611yescharacteristic
209C4—24no—
210C4—42no—
211C4—81no—
212C4—3881no—
213C4—3881no—
214C4—1942no—
215C4C4⋊C4×D9711yescharacteristic
216C4#1569077211yescharacteristic
217C4#1569078811yescharacteristic
218C4#1569079511yescharacteristic
219C2—21no—
220C2—21no—
221C2—3882no—
222C2#2026060611yescharacteristic
223C2#2026090211yescharacteristic
224C2#2026091711yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#384×C97C2
socleC22×C97#120
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenter#384D97
hypocenterC97#1859
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #20260775C2semidirect
5. #20260504C2semidirect
6. #20260495C2semidirect
7. #384×C97C2semidirect
19. #15690738C4semidirect
38. #13095495C2×C4semidirect
58. C42×C97C2×C4semidirect
80. D388C4⋊C4semidirect
85. Dic194C4⋊C4semidirect
113. C97#1859semidirect
118. #384D97semidirect
171. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202612162
#202607752
C2×#156907382
#202607342
#202605042
#202604952
#384×C972
#185997