Finite Group Databasebeta

← #27141968

Subgroups — #27141968

Order 12416.

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

normal subgroups only

This group has 286 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#382×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×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24C2×C42×C97C2211yescharacteristic
25#95×C97—22no—
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—
56C42×C97—21no—
57C42×C97C2×C412yescomplemented
58C23×C97C2×D411yescharacteristic
59C22×D97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—42no—
65C2×C4×C97—42no—
66C2×C4×C97—41no—
67C2×C4×C97C22×C411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×Q811yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C22⋊D97—22no—
74C22⋊D97—22no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78D388—21no—
79D388C4⋊C412yes—
80C4×D97—22no—
81C4×D97—22no—
82C4×D97—82no—
83Dic194—21no—
84Dic194C4⋊C412yes—
85#11715895—22no—
86#11715895—41no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97C4×D411yescharacteristic
90C22×C97#12011yescharacteristic
91C22×C97C22⋊Q811yescharacteristic
92C2×D97—22no—
93C2×D97—41no—
94C2×D97—41no—
95C4×C97—42no—
96C4×C97—42no—
97C4×C97—81no—
98C4×C97C2×C4⋊C411yescharacteristic
99C4×C97C4×D411yescharacteristic
100C4×C97C22⋊Q811yescharacteristic
101C4×C97#12311yescharacteristic
102Dic97—22no—
103Dic97—41no—
104Dic97—41no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97#38511yescharacteristic
108C2×C97#45411yescharacteristic
109C2×C97#45511yescharacteristic
110D97—42no—
111#1860—971nomaximal, Sylow
112C97#186011yescharacteristic, complemented, Sylow
113#520—971no—
114#428—971no—
115C2×#104—971no—
116#417—971no—
117#382D9711yescharacteristic, complemented
118#343—971no—
119#339—971no—
120C2×#41—971no—
121C2×M16—971no—
122C2×M16—971no—
123C4×Q8—1941no—
124C4×D4—1941no—
125C4×D4—1942no—
126#117—971no—
127#117—1941no—
128C2×C4⋊C4—1941no—
129C2×C4⋊C4C2×D9711yescharacteristic
130C2×C4⋊C4C2×D9711yescharacteristic
131C2×C42C2×D9711yescharacteristic
132#107—1941no—
133#106—1941no—
134#104—974no—
135#103—1941no—
136#102—1941no—
137#95—22no—
138#41—972no—
139#41—972no—
140#41—1941no—
141#41—1941no—
142C2×Q8—971no—
143C2×D4—971no—
144C2×D4—1941no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—1942no—
148C22×C4—1941no—
149C22×C4C22⋊D9712yes—
150C22×C4C22×D9711yescharacteristic
151M16—972no—
152M16—1941no—
153M16—1941no—
154M16—1941no—
155C2×C8—1941no—
156C2×C8—1941no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—971no—
162C4⋊C4—971no—
163C4⋊C4—1942no—
164C22⋊C4—1942no—
165C22⋊C4—1941no—
166C42—21no—
167C42—1942no—
168C42—1941no—
169C42C4×D9712yescomplemented
170C23—1941no—
171C23C2×C22⋊D9711yescharacteristic
172Q8—972no—
173Q8—1941no—
174D4—972no—
175D4—1942no—
176D4—1942no—
177D4—1941no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—7762no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4D4×D9711yescharacteristic
194C2×C4Q8×D9711yescharacteristic
195C2×C4#1309551111yescharacteristic
196C2×C4#1309551211yescharacteristic
197C8—3881no—
198C8—1942no—
199C22—21no—
200C22—21no—
201C22—3881no—
202C22—3881no—
203C22—1942no—
204C22C4×C22⋊D9711yescharacteristic
205C22#1569077811yescharacteristic
206C22#1569078611yescharacteristic
207C4—42no—
208C4—42no—
209C4—81no—
210C4—3881no—
211C4—3881no—
212C4—1942no—
213C4C4⋊C4×D9711yescharacteristic
214C4#1569077211yescharacteristic
215C4#1569078811yescharacteristic
216C4#1569079511yescharacteristic
217C2—21no—
218C2—21no—
219C2—3882no—
220C2#2026060611yescharacteristic
221C2#2026090711yescharacteristic
222C2#2026091211yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#382×C97C2
socleC22×C97#120
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenter#382D97
hypocenterC97#1860
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #20260775C2semidirect
5. #20260504C2semidirect
6. #20260495C2semidirect
7. #382×C97C2semidirect
19. #15690738C4semidirect
38. #13095495C2×C4semidirect
57. C42×C97C2×C4semidirect
112. C97#1860semidirect
117. #382D97semidirect
169. 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
#382×C972
#186097