Finite Group Databasebeta

← #27157928

Subgroups — #27157928

Order 12416.

260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 76 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#20261738C211yesmaximal, characteristic
2C4×#13095520C211yesmaximal, characteristic
3#20261615C214yesmaximal
4C2×#15690808C211yesmaximal, characteristic
5C2×#13095520C2212yes—
6C2×#13095520C2212yes—
7#15690877—21no—
8C4⋊#11715897—21no—
9#15690865—21no—
10C4×#11715897—21no—
11C22×Dic194C411yescharacteristic
12#15690808—21no—
13#15690808—21no—
14#15690808—21no—
15#15690808—21no—
16C2×#13095490C412yes—
17C2×#13095490C2212yes—
18C2×C4×Dic97C2211yescharacteristic
19C22×Q8×C97C411yescharacteristic
20#13095520—22no—
21#13095520—24no—
22#13095520Q814yescomplemented
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—42no—
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C412yes—
30C22×Dic97C2311yescharacteristic
31#13095490—22no—
32#13095490—22no—
33#13095490—22no—
34#13095490—22no—
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97D411yescharacteristic
38C4×Dic97D411yescharacteristic
39C2×Q8×C97—21no—
40C2×Q8×C97—21no—
41C2×Q8×C97—42no—
42C22×C4×C97C2×C412yes—
43C22×C4×C97C2×C411yescharacteristic
44C23×C97C22×C411yescharacteristic
45Q8×C97—42no—
46Q8×C97—42no—
47C2×C4×C97—22no—
48C2×C4×C97—22no—
49C2×C4×C97—42no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22⋊C411yescharacteristic
52C2×Dic97—22no—
53C2×Dic97—22no—
54C2×Dic97—42no—
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C2×D411yescharacteristic
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97C2×Q812yes—
60C2×Dic97#4112yes—
61Dic194—42no—
62Dic194—42no—
63#11715897—24no—
64#11715897—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97C4×Q812yes—
69C4×C97—21no—
70C4×C97—42no—
71C4×C97#10012yes—
72Dic97—21no—
73Dic97—42no—
74Dic97#10112yes—
75Dic97C22⋊Q812yes—
76Dic97#12411yescharacteristic
77Dic97C4⋊Q811yescharacteristic
78C2×C97—22no—
79C2×C97#39111yescharacteristic
80C2×C97C2×#10011yescharacteristic
81C2×C97C2×#10111yescharacteristic
82#1997—971nomaximal, Sylow
83C97#199711yescharacteristic, complemented, Sylow
84C2×C4⋊Q8—971no—
85#423—971no—
86C4×M16—971no—
87#343—974no—
88C22×Q8—971no—
89C22×Q8#8760411yescharacteristic
90C2×M16—972no—
91C2×M16—972no—
92C4⋊Q8—1941no—
93C4⋊Q8—1941no—
94C4⋊Q8—1941no—
95C4⋊Q8—1941no—
96C2×C4⋊C4—972no—
97C2×C4⋊C4—972no—
98C2×C42—971no—
99C4⋊C8—1941no—
100C4⋊C8—1941no—
101#97—1941no—
102C4×C8—1941no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—42no—
106C2×Q8—3882no—
107C2×Q8—1941no—
108C2×Q8—1941no—
109C22×C4—971no—
110C22×C4—971no—
111C22×C4—972no—
112C22×C4C2×#8760412yes—
113C22×C4C2×#8760411yescharacteristic
114M16—974no—
115M16—1944no—
116M16—1942no—
117C2×C8—1942no—
118C2×C8—1942no—
119C4⋊C4—1942no—
120C4⋊C4—1942no—
121C4⋊C4—1942no—
122C4⋊C4—1942no—
123C42—971no—
124C42—971no—
125C42—971no—
126C42—971no—
127C23C22×#8760411yescharacteristic
128Q8—42no—
129Q8—42no—
130Q8—3882no—
131Q8—3882no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—42no—
135C2×C4—971no—
136C2×C4—971no—
137C2×C4—971no—
138C2×C4—972no—
139C2×C4—972no—
140C2×C4—971no—
141C2×C4—3882no—
142C2×C4—1942no—
143C2×C4—1942no—
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C22⋊#8760411yescharacteristic
146C8—3882no—
147C8—1944no—
148C22—21no—
149C22—21no—
150C22C2×C22⋊#8760411yescharacteristic
151C22Q8×#8760412yes—
152C4—21no—
153C4—42no—
154C4—971no—
155C4—972no—
156C4—971no—
157C4—972no—
158C4—3882no—
159C4—1941no—
160C4#1569087012yes—
161C2—22no—
162C2(C2×#87604)⋊Q811yescharacteristic
163C2C2×#1569087011yescharacteristic
164C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutator, prime radicalC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
cosocleC22×Dic97C23
hypercenterC22×Q8#87604
hypocenterC97#1997
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
22. #13095520Q8semidirect
83. C97#1997semidirect

Maximal subgroups

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

subgroupindex
#202617382
C4×#130955202
#202616152
C2×#156908082
#199797