Finite Group Databasebeta

← #27151554

Subgroups — #27151554

Order 12416.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C4×#13095511C211yesmaximal, characteristic
2#20261039C214yesmaximal, complemented
3#20261032C214yesmaximal, complemented
4#20260775C212yesmaximal, complemented
5#20260759C211yesmaximal, characteristic, complemented
6#20260734C212yesmaximal
7C2×#125×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2212yes—
10C4×Dic194C2212yes—
11C4×D388C2212yes—
12C4×C22⋊D97—22no—
13#15690769C2211yescharacteristic
14#15690767—21no—
15C42×D97—21no—
16#15690743C2214yes—
17#15690741C2214yes—
18C4⋊#11715895C2214yes—
19#15690733C2214yes—
20#15690730C2214yes—
21#125×C97—21no—
22#125×C97—21no—
23#125×C97C2214yes—
24C2×C4⋊C4×C97—22no—
25C2×C4⋊C4×C97C2212yes—
26C2×C42×C97C2211yescharacteristic
27#13095511—22no—
28#13095511Q814yes—
29C2×Dic194C2311yescharacteristic
30C2×D388C2311yescharacteristic
31C2×C22⋊D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—22no—
34#13095495—24no—
35C2×#11715895C2314yes—
36C4⋊Dic97C2312yes—
37#13095490—22no—
38C22⋊Dic97—21no—
39#13095488—22no—
40C4×Dic97—21no—
41C4×Dic97—22no—
42C22×C4×C97—22no—
43C22×C4×C97D412yes—
44C22×C4×C97C2311yescharacteristic
45C4⋊C4×C97—24no—
46C4⋊C4×C97—24no—
47C4⋊C4×C97—22no—
48C4⋊C4×C97C2314yes—
49C42×C97D411yescharacteristic
50C42×C97D411yescharacteristic
51C42×C97C2312yes—
52C23×C97C2×D411yescharacteristic
53C22×D97—21no—
54C2×C4×C97—24no—
55C2×C4×C97—42no—
56C2×C4×C97—44no—
57C2×C4×C97C2×D412yes—
58C2×C4×C97C2×D412yes—
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×Q812yes—
61C2×C4×C97#4112yes—
62C2×C4×C97C2411yescharacteristic
63C22⋊D97—24no—
64C2×Dic97—21no—
65C2×Dic97—22no—
66D388—21no—
67D388C2×Q812yes—
68C4×D97—24no—
69C4×D97—44no—
70Dic194—21no—
71Dic194C2×Q812yes—
72#11715895—24no—
73C22×C97—22no—
74C22×C97C22⋊Q812yes—
75C22×C97C22×D411yescharacteristic
76C2×D97—22no—
77C2×D97—42no—
78C4×C97—22no—
79C4×C97—22no—
80C4×C97—44no—
81C4×C97C22⋊Q812yes—
82C4×C97C22×Q811yescharacteristic
83C4×C97C2×#4111yescharacteristic
84Dic97—22no—
85Dic97—42no—
86C2×C97—22no—
87C2×C97C2×C22⋊Q811yescharacteristic
88C2×C97#57512yes—
89D97—42no—
90#3080—971nomaximal, Sylow
91C97#308011yescharacteristic, complemented, Sylow
92C2×#125D9711yescharacteristic, complemented
93C4×#41—971no—
94#479—974no—
95#478—974no—
96#428—972no—
97#423—971no—
98#417—972no—
99C2×#41—971no—
100C2×M16—972no—
101#125—21no—
102#125—21no—
103#125C2×D9714yes—
104C4×Q8—972no—
105C4×D4—972no—
106C4×D4—1942no—
107#117—971no—
108#117—1941no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C2×D9712yes—
111C2×C42—1941no—
112C2×C42C2×D9711yescharacteristic
113#107—974no—
114#106—974no—
115C4⋊C8—974no—
116#103—974no—
117#102—974no—
118#41—974no—
119#41—1942no—
120C2×Q8—971no—
121C2×D4—971no—
122C2×D4—1941no—
123C22×C4—22no—
124C22×C4—1942no—
125C22×C4—1941no—
126C22×C4C22⋊D9712yes—
127C22×C4C22×D9711yescharacteristic
128M16—1944no—
129C2×C8—974no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—972no—
134C4⋊C4—1942no—
135C4⋊C4C22×D9714yes—
136C22⋊C4—1941no—
137C22⋊C4—1942no—
138C42—1941no—
139C42—1942no—
140C42C22⋊D9711yescharacteristic
141C42C22⋊D9711yescharacteristic
142C42C22×D9712yes—
143C23—1941no—
144C23C2×C22⋊D9711yescharacteristic
145Q8—972no—
146Q8—1941no—
147D4—972no—
148D4—1944no—
149D4—1941no—
150C2×C4—24no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—3884no—
154C2×C4—1942no—
155C2×C4—1941no—
156C2×C4—1944no—
157C2×C4C2×C22⋊D9712yes—
158C2×C4C2×C22⋊D9712yes—
159C2×C4C2×C22⋊D9711yescharacteristic
160C2×C4Q8×D9712yes—
161C2×C4#1309551212yes—
162C2×C4C23×D9711yescharacteristic
163C8—1944no—
164C22—22no—
165C22—3882no—
166C22—1942no—
167C22#1569078612yes—
168C22C22×C22⋊D9711yescharacteristic
169C4—22no—
170C4—22no—
171C4—44no—
172C4—3882no—
173C4—1942no—
174C4#1569078612yes—
175C4C2×Q8×D9711yescharacteristic
176C4C2×#1309551211yescharacteristic
177C2—22no—
178C2—3882no—
179C2C2×#1569078611yescharacteristic
180C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#125×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095511C22
hypercenterC2×#125D97
hypocenterC97#3080
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20261039C2semidirect
3. #20261032C2semidirect
4. #20260775C2semidirect
5. #20260759C2semidirect
7. C2×#125×C97C2semidirect
91. C97#3080semidirect
92. C2×#125D97semidirect

Maximal subgroups

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

subgroupindex
C4×#130955112
#202610392
#202610322
#202607752
#202607592
#202607342
C2×#125×C972
#308097