Finite Group Databasebeta

← #27157034

Subgroups — #27157034

Order 12416.

264 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 78 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#20261753C211yesmaximal, characteristic, complemented
2#20261736C211yesmaximal, characteristic, complemented
3Q8⋊#11715897C212yesmaximal, complemented
4#20261583C212yesmaximal, complemented
5C4×#13095512C211yesmaximal, characteristic
6C2×#13095519—21no—
7#15690877—21no—
8#15690877—21no—
9C4⋊#11715897C2212yes—
10#15690867—21no—
11C4×#11715897C2212yes—
12C2×#13095512C411yescharacteristic
13Q8×Dic97C2211yescharacteristic
14C4×D388—21no—
15C4×D388C411yescharacteristic
16#15690772—21no—
17#15690772C2211yescharacteristic
18#15690768—21no—
19#15690768C411yescharacteristic
20C42×D97—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic
23#13095519—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26#13095512—22no—
27#13095512C814yescomplemented
28C2×D388—21no—
29C2×D388C2×C411yescharacteristic
30C2×C4×D97—21no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C411yescharacteristic
38#13095488—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97—21no—
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97C2311yescharacteristic
46C2×Q8×C97C2×C411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97C2×C411yescharacteristic
49C42×C97—21no—
50C42×C97C2×C411yescharacteristic
51C22×D97—21no—
52C22×D97C22⋊C411yescharacteristic
53Q8×C97—21no—
54Q8×C97C2×C812yescomplemented
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64D388—24no—
65D388—21no—
66D388C2×C812yescomplemented
67C4×D97—24no—
68C4×D97—21no—
69C4×D97—21no—
70C4×D97—41no—
71C4×D97—42no—
72C4×D97—41no—
73C4×D97—41no—
74C4×D97C2×C812yes—
75C4×D97M1612yes—
76#11715897—42no—
77#11715897—42no—
78C22×C97C2×C22⋊C411yescharacteristic
79C2×D97—21no—
80C2×D97—21no—
81C2×D97—22no—
82C2×D97—41no—
83C2×D97—41no—
84C2×D97C22⋊C812yes—
85C4×C97—22no—
86C4×C97—21no—
87C4×C97—21no—
88C4×C97—41no—
89C4×C97C22×C811yescharacteristic
90C4×C97C2×M1611yescharacteristic
91Dic97—21no—
92Dic97—41no—
93Dic97C22⋊C812yes—
94Dic97AGL(1,ℤ/8)11yescharacteristic
95Dic97#13711yescharacteristic
96C2×C97C2×C22⋊C811yescharacteristic
97C2×C97#41711yescharacteristic
98C2×C97#42111yescharacteristic
99D97—22no—
100D97—42no—
101#1477—971nomaximal, Sylow
102C97#147711yescharacteristic, complemented, Sylow
103C4×#41—971no—
104#431—971no—
105C2×C4⋊C8—971no—
106Q8⋊C8—972no—
107D4⋊C8—972no—
108C2×#41—971no—
109C22×C8—1941no—
110C4×Q8—971no—
111C4×Q8#8760411yescharacteristic
112C4×D4—971no—
113C4×D4—971no—
114C4×D4—1941no—
115C4×D4—1941no—
116#117—971no—
117#117—1941no—
118C2×C42—971no—
119C2×C42—1941no—
120C4⋊C8—972no—
121C4⋊C8—1941no—
122C4⋊C8—1941no—
123C22⋊C8—1941no—
124C4×C8—972no—
125#41—974no—
126#41—1942no—
127C2×Q8C2×#8760411yescharacteristic
128C2×D4—971no—
129C2×D4—1941no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—1941no—
134C22×C4—1941no—
135C22×C4—1941no—
136C2×C8—3882no—
137C2×C8—1942no—
138C2×C8—1942no—
139C4⋊C4—21no—
140C4⋊C4—971no—
141C4⋊C4—1941no—
142C4⋊C4C2×#8760411yescharacteristic
143C22⋊C4—1941no—
144C22⋊C4—1941no—
145C22⋊C4—1941no—
146C42—21no—
147C42—971no—
148C42—971no—
149C42—971no—
150C42—1941no—
151C42—1941no—
152C42C2×#8760411yescharacteristic
153C23—971no—
154C23—1941no—
155Q8—21no—
156Q8#1309551912yescomplemented
157D4—972no—
158D4—1944no—
159D4—1941no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—971no—
163C2×C4—971no—
164C2×C4—972no—
165C2×C4—972no—
166C2×C4—971no—
167C2×C4—3882no—
168C2×C4—3881no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4—1941no—
172C2×C4—1941no—
173C2×C4—1944no—
174C2×C4—1941no—
175C2×C4C22⋊#8760411yescharacteristic
176C2×C4C22⋊#8760411yescharacteristic
177C2×C4C22×#8760411yescharacteristic
178C8—3882no—
179C8—3882no—
180C22—972no—
181C22—3881no—
182C22—3881no—
183C22—1942no—
184C22—1941no—
185C22—1941no—
186C22C2×C22⋊#8760411yescharacteristic
187C4—22no—
188C4—21no—
189C4—21no—
190C4—41no—
191C4—972no—
192C4—971no—
193C4—971no—
194C4—3881no—
195C4—1941no—
196C4C2×#1309551911yescharacteristic
197C4D97⋊M1611yescharacteristic
198C2—3882no—
199C2—1942no—
200C2#2026170311yescharacteristic
201C2#2026172611yescharacteristic
202C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C22×C8
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095512C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97#1477
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20261753C2semidirect
2. #20261736C2semidirect
3. Q8⋊#11715897C2semidirect
4. #20261583C2semidirect
27. #13095512C8semidirect
54. Q8×C97C2×C8semidirect
66. D388C2×C8semidirect
102. C97#1477semidirect
156. Q8#13095519semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202617362
Q8⋊#117158972
#202615832
C4×#130955122
#147797