Finite Group Databasebeta

← #27157039

Subgroups — #27157039

Order 12416.

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

normal subgroups only

This group has 248 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#20261754C211yesmaximal, characteristic
2#20261753C211yesmaximal, characteristic
3#20261585C212yesmaximal, complemented
4Q8⋊#11715897C212yesmaximal, complemented
5C4×Q8×D97C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897C2212yes—
8#15690867—21no—
9#15690867—21no—
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
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×Dic194—21no—
26C2×Dic194C2×C411yescharacteristic
27Q8×D97—22no—
28Q8×D97C814yescomplemented
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97C2×C411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C4⋊Dic97—21no—
35C4⋊Dic97C2×C411yescharacteristic
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39C4×Dic97—21no—
40C4×Dic97—21no—
41C4×Dic97—21no—
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C2×Q8×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C42×C97—21no—
49C42×C97C2×C411yescharacteristic
50C22×D97C22⋊C411yescharacteristic
51Q8×C97—21no—
52Q8×C97C2×C812yes—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C2×D411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C4×D97—24no—
64C4×D97—22no—
65C4×D97—41no—
66C4×D97—42no—
67C4×D97C2×C812yes—
68C4×D97M1612yes—
69Dic194—21no—
70Dic194—24no—
71Dic194C2×C812yes—
72#11715897—42no—
73#11715897—42no—
74C22×C97C2×C22⋊C411yescharacteristic
75C2×D97—21no—
76C2×D97—21no—
77C2×D97C22⋊C812yes—
78C4×C97—22no—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—41no—
82C4×C97C22×C811yescharacteristic
83C4×C97C2×M1611yescharacteristic
84Dic97—22no—
85Dic97—21no—
86Dic97—41no—
87Dic97C22⋊C812yes—
88Dic97#13712yes—
89C2×C97C2×C22⋊C811yescharacteristic
90C2×C97#41911yescharacteristic
91C2×C97#42111yescharacteristic
92D97—22no—
93#1479—971nomaximal, Sylow
94C97#147911yescharacteristic, complemented, Sylow
95C2×C4×Q8—971no—
96#431—971no—
97#431—971no—
98Q8⋊C8—972no—
99Q8⋊C8—972no—
100C22×Q8—971no—
101C4×Q8—971no—
102C4×Q8—971no—
103C4×Q8—971no—
104C4×Q8—1941no—
105C4×Q8—1941no—
106C4×Q8#8760411yescharacteristic
107C2×C4⋊C4—971no—
108C2×C4⋊C4—1941no—
109C2×C42—971no—
110C2×C42—1941no—
111C4⋊C8—972no—
112C4⋊C8—1941no—
113C22⋊C8—1941no—
114C22⋊C8—1941no—
115C4×C8—972no—
116C4×C8—1941no—
117C2×Q8—974no—
118C2×Q8—971no—
119C2×Q8—1941no—
120C2×Q8—1942no—
121C2×Q8C2×#8760411yescharacteristic
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—1941no—
126C22×C4—1941no—
127C2×C8—1942no—
128C2×C8—1942no—
129C4⋊C4—21no—
130C4⋊C4—971no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4C2×#8760411yescharacteristic
136C42—21no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42—1941no—
141C42—1941no—
142C42—1941no—
143C42C2×#8760411yescharacteristic
144C23—971no—
145Q8—21no—
146Q8—972no—
147Q8—1944no—
148Q8—1941no—
149Q8#1309551912yes—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—971no—
153C2×C4—971no—
154C2×C4—972no—
155C2×C4—972no—
156C2×C4—971no—
157C2×C4—3881no—
158C2×C4—3882no—
159C2×C4—1942no—
160C2×C4—1944no—
161C2×C4—1941no—
162C2×C4—1941no—
163C2×C4C22⋊#8760411yescharacteristic
164C2×C4C22⋊#8760411yescharacteristic
165C2×C4C22×#8760411yescharacteristic
166C8—3882no—
167C8—3882no—
168C22—972no—
169C22—1941no—
170C22—1941no—
171C22C2×C22⋊#8760411yescharacteristic
172C4—22no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—972no—
177C4—972no—
178C4—3881no—
179C4—1942no—
180C4—1941no—
181C4C2×#1309551911yescharacteristic
182C4D97⋊M1611yescharacteristic
183C2—1942no—
184C2#2026170311yescharacteristic
185C2#2026172811yescharacteristic
186C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C22×C8
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97#1479
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
3. #20261585C2semidirect
4. Q8⋊#11715897C2semidirect
28. Q8×D97C8semidirect
94. C97#1479semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202617532
#202615852
Q8⋊#117158972
C4×Q8×D972
#147997