Finite Group Databasebeta

← #27157237

Subgroups — #27157237

Order 12416.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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
2#20261736C211yesmaximal, characteristic
3#20261595C212yesmaximal, complemented
4#20261594C212yesmaximal, complemented
5#20261289C211yesmaximal, characteristic
6C2×#13095519—21no—
7#15690877C2212yes—
8#15690877C2212yes—
9C4⋊#11715897—21no—
10#15690867—21no—
11C4×#11715897—21no—
12#15690809C2211yescharacteristic
13#15690808C2211yescharacteristic
14#15690800—21no—
15#15690790C411yescharacteristic
16#15690788—21no—
17#15690786C411yescharacteristic
18C4×Dic194—21no—
19C4×Dic194C411yescharacteristic
20#15690767—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic
23#13095519—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194—21no—
27C2×Dic194C2×C411yescharacteristic
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C42×C97—21no—
48C42×C97C2×C411yescharacteristic
49C22×D97C22⋊C411yescharacteristic
50Q8×C97—41no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22×C411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97C22⋊C411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60C2×Dic97C2×D411yescharacteristic
61C4×D97—22no—
62C4×D97—41no—
63C4×D97SD1612yes—
64C4×D97Q1612yes—
65Dic194—42no—
66Dic194—41no—
67#11715897—42no—
68#11715897—42no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—21no—
72C2×D97#10312yes—
73C4×C97—21no—
74C4×C97—21no—
75C4×C97—41no—
76C4×C97—41no—
77C4×C97#10312yes—
78Dic97—21no—
79Dic97—41no—
80Dic97—41no—
81Dic97C2×SD1611yescharacteristic
82Dic97C2×Q1611yescharacteristic
83Dic97#13512yes—
84C2×C97#41311yescharacteristic
85C2×C97C2×#10311yescharacteristic
86C2×C97#41611yescharacteristic
87D97—22no—
88#1542—971nomaximal, Sylow
89C97#154211yescharacteristic, complemented, Sylow
90#533—971no—
91#431—971no—
92C2×C4⋊C8—971no—
93#333—972no—
94#332—972no—
95C22×C8—1941no—
96C4⋊Q8—971no—
97C4⋊Q8—971no—
98#125—1941no—
99C22⋊Q8—971no—
100C22⋊Q8—971no—
101C22⋊Q8—1941no—
102C4×Q8—971no—
103C4×Q8—1941no—
104C4×Q8#8760411yescharacteristic
105#117—1941no—
106C2×C42—971no—
107C4⋊C8—972no—
108C4⋊C8—972no—
109C4⋊C8—1941no—
110C22⋊C8—1941no—
111C4×C8—1941no—
112C2×Q8—971no—
113C2×Q8—1941no—
114C2×Q8C2×#8760411yescharacteristic
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—971no—
118C2×C8—3882no—
119C2×C8—1942no—
120C2×C8—1942no—
121C4⋊C4—21no—
122C4⋊C4—971no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4C2×#8760411yescharacteristic
130C22⋊C4—1941no—
131C22⋊C4—1941no—
132C42—21no—
133C42—971no—
134C42—971no—
135C42—971no—
136C42—1941no—
137C42C2×#8760411yescharacteristic
138C23—971no—
139Q8—41no—
140Q8—3881no—
141Q8—3882no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—972no—
147C2×C4—972no—
148C2×C4—971no—
149C2×C4—3881no—
150C2×C4—1942no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4C22⋊#8760411yescharacteristic
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22×#8760411yescharacteristic
156C8—3882no—
157C8—3882no—
158C22—972no—
159C22—1941no—
160C22—1941no—
161C22C2×C22⋊#8760411yescharacteristic
162C4—21no—
163C4—21no—
164C4—41no—
165C4—41no—
166C4—971no—
167C4—971no—
168C4—972no—
169C4—3881no—
170C4—3881no—
171C4—1941no—
172C4#1569087312yes—
173C2—1942no—
174C2#2026172111yescharacteristic
175C2C2×#1569087311yescharacteristic
176C2#2026172511yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97#1542
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20261595C2semidirect
4. #20261594C2semidirect
89. C97#1542semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202617362
#202615952
#202615942
#202612892
#154297