Finite Group Databasebeta

← #27157025

Subgroups — #27157025

Order 12416.

280 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 94 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#20261736C211yesmaximal, characteristic
2C4×#13095519C211yesmaximal, characteristic
3#20261585C212yesmaximal, complemented
4Q8⋊#11715897C212yesmaximal, complemented
5C4×Q8×D97C211yesmaximal, characteristic
6C2×#13095519—21no—
7C2×#13095519—21no—
8#15690877—21no—
9C4⋊#11715897C2212yes—
10C4×#11715897—21no—
11C4×#11715897C2212yes—
12C2×Q8×D97C411yescharacteristic
13Q8×Dic97C2211yescharacteristic
14C4×Dic194—21no—
15C4×Dic194C411yescharacteristic
16#15690775—21no—
17#15690775C2211yescharacteristic
18C4⋊C4×D97—21no—
19C4⋊C4×D97C411yescharacteristic
20C42×D97—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic
23#13095519—24no—
24#13095519—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29Q8×D97—22no—
30Q8×D97C814yescomplemented
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#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41C4×Dic97—21no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97C2311yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C4⋊C4×C97—21no—
49C4⋊C4×C97C2×C411yescharacteristic
50C42×C97—21no—
51C42×C97C2×C411yescharacteristic
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×Dic97—21no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C4×D97—24no—
66C4×D97—22no—
67C4×D97—42no—
68C4×D97C2×C812yes—
69C4×D97M1612yes—
70C4×D97SD1612yes—
71C4×D97Q1612yes—
72Dic194—21no—
73Dic194—24no—
74Dic194C2×C812yescomplemented
75#11715897—24no—
76#11715897—42no—
77C22×C97C2×C22⋊C411yescharacteristic
78C2×D97C22⋊C812yes—
79C2×D97#10312yes—
80C2×D97#10412yes—
81C4×C97—22no—
82C4×C97—21no—
83C4×C97—41no—
84C4×C97#10312yes—
85C4×C97C22×C811yescharacteristic
86C4×C97C2×M1611yescharacteristic
87Dic97—22no—
88Dic97—21no—
89Dic97—41no—
90Dic97C22⋊C812yes—
91Dic97C2×SD1611yescharacteristic
92Dic97C2×Q1611yescharacteristic
93C2×C97C2×C22⋊C811yescharacteristic
94C2×C97C2×#10311yescharacteristic
95C2×C97C2×#10411yescharacteristic
96D97Q8⋊C814yescomplemented
97C2×Q8⋊C8—971nomaximal, Sylow
98C97C2×Q8⋊C811yescharacteristic, complemented, Sylow
99C2×C4×Q8—971no—
100C2×C4⋊C8—971no—
101C2×C4×C8—971no—
102Q8⋊C8—972no—
103Q8⋊C8—972no—
104C22×Q8—971no—
105C22×C8—1941no—
106C22×C8—1941no—
107C4×Q8—971no—
108C4×Q8—971no—
109C4×Q8—971no—
110C4×Q8—1941no—
111C4×Q8—1941no—
112C4×Q8#8760411yescharacteristic
113C2×C4⋊C4—971no—
114C2×C4⋊C4—1941no—
115C2×C42—971no—
116C2×C42—1941no—
117C4⋊C8—972no—
118C4⋊C8—1941no—
119C4×C8—972no—
120C4×C8—1941no—
121C2×Q8—974no—
122C2×Q8—971no—
123C2×Q8—1941no—
124C2×Q8—1942no—
125C2×Q8C2×#8760411yescharacteristic
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—971no—
129C22×C4—1941no—
130C22×C4—1941no—
131C2×C8—3882no—
132C2×C8—1942no—
133C2×C8—1942no—
134C2×C8—1944no—
135C4⋊C4—21no—
136C4⋊C4—971no—
137C4⋊C4—1941no—
138C4⋊C4—1941no—
139C4⋊C4—1941no—
140C4⋊C4—1941no—
141C4⋊C4C2×#8760411yescharacteristic
142C42—21no—
143C42—971no—
144C42—971no—
145C42—971no—
146C42—1941no—
147C42—1941no—
148C42—1941no—
149C42C2×#8760411yescharacteristic
150C23—971no—
151Q8—21no—
152Q8—972no—
153Q8—1944no—
154Q8—1941no—
155Q8#1309551912yescomplemented
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—972no—
161C2×C4—972no—
162C2×C4—972no—
163C2×C4—972no—
164C2×C4—971no—
165C2×C4—3882no—
166C2×C4—1942no—
167C2×C4—1944no—
168C2×C4—1941no—
169C2×C4—1941no—
170C2×C4C22⋊#8760411yescharacteristic
171C2×C4C22⋊#8760411yescharacteristic
172C2×C4C22×#8760411yescharacteristic
173C8—3882no—
174C8—1944no—
175C22—972no—
176C22—972no—
177C22—972no—
178C22C2×C22⋊#8760411yescharacteristic
179C4—22no—
180C4—21no—
181C4—41no—
182C4—972no—
183C4—971no—
184C4—971no—
185C4—3881no—
186C4—1942no—
187C4—1941no—
188C4#1569087312yes—
189C4C2×#1309551911yescharacteristic
190C4D97⋊M1611yescharacteristic
191C2—974no—
192C2#2026170311yescharacteristic
193C2C2×#1569087311yescharacteristic
194C2D97⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC4×C97C22×C8
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97C2×Q8⋊C8
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20261585C2semidirect
4. Q8⋊#11715897C2semidirect
30. Q8×D97C8semidirect
54. Q8×C97C2×C8semidirect
74. Dic194C2×C8semidirect
96. D97Q8⋊C8semidirect
98. C97C2×Q8⋊C8semidirect
155. Q8#13095519semidirect

Maximal subgroups

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

subgroupindex
#202617362
C4×#130955192
#202615852
Q8⋊#117158972
C4×Q8×D972
C2×Q8⋊C897