Finite Group Databasebeta

← #27157246

Subgroups — #27157246

Order 12416.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261740C211yesmaximal, characteristic
3#20261594C214yesmaximal, complemented
4C4⋊Q8×D97C211yesmaximal, characteristic
5C2×#13095521—21no—
6#15690877—21no—
7#15690877C2212yes—
8#15690877C2212yes—
9#15690867—21no—
10C4×#11715897—21no—
11C2×Q8×D97—21no—
12#15690810C411yescharacteristic
13#15690809—21no—
14#15690809C2212yes—
15#15690808—21no—
16C4⋊C4×D97—21no—
17C4⋊C4×D97C412yes—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic
20#13095521—42no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×Dic194—21no—
24C2×Dic194—22no—
25Q8×D97—42no—
26Q8×D97—42no—
27C2×C4×D97—22no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C412yes—
33#13095490—22no—
34#13095490—22no—
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97C2311yescharacteristic
38C2×Q8×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97C2×C412yes—
41C42×C97C2×C411yescharacteristic
42C22×D97C22⋊C411yescharacteristic
43Q8×C97—42no—
44C2×C4×C97—22no—
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—22no—
49C2×Dic97C22⋊C411yescharacteristic
50C2×Dic97C2×D411yescharacteristic
51C2×Dic97C2×D411yescharacteristic
52C4×D97—22no—
53C4×D97—22no—
54C4×D97—44no—
55C4×D97D812yes—
56C4×D97SD1612yes—
57Dic194—42no—
58Dic194—42no—
59Dic194—42no—
60#11715897—42no—
61#11715897—42no—
62C22×C97C2×C22⋊C411yescharacteristic
63C2×D97—21no—
64C2×D97—21no—
65C2×D97#10212yes—
66C4×C97—21no—
67C4×C97—21no—
68C4×C97—42no—
69C4×C97#10212yes—
70Dic97—42no—
71Dic97#10112yes—
72Dic97C2×D811yescharacteristic
73Dic97C2×SD1611yescharacteristic
74Dic97#13712yes—
75C2×C97C2×#10111yescharacteristic
76C2×C97C2×#10211yescharacteristic
77C2×C97#41911yescharacteristic
78D97—22no—
79#1546—971nomaximal, Sylow
80C97#154611yescharacteristic, complemented, Sylow
81C2×C4⋊Q8—971no—
82#431—971no—
83#423—971no—
84#332—974no—
85C22×Q8—1941no—
86C2×M16—1941no—
87C4⋊Q8—972no—
88C4⋊Q8—971no—
89C4⋊Q8—1941no—
90C4⋊Q8—1941no—
91C4⋊Q8#8760411yescharacteristic
92C2×C4⋊C4—972no—
93C2×C4⋊C4—1941no—
94C2×C42—971no—
95C4⋊C8—972no—
96C4⋊C8—972no—
97C4⋊C8—1941no—
98C22⋊C8—1941no—
99C4×C8—1941no—
100C2×Q8—21no—
101C2×Q8—3882no—
102C2×Q8—3882no—
103C2×Q8—1942no—
104C2×Q8—1941no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4—971no—
108C22×C4—1942no—
109M16—3882no—
110C2×C8—1942no—
111C2×C8—1942no—
112C4⋊C4—21no—
113C4⋊C4—972no—
114C4⋊C4—1942no—
115C4⋊C4—1942no—
116C4⋊C4—1941no—
117C4⋊C4C2×#8760412yes—
118C42—971no—
119C42—971no—
120C42—971no—
121C42C2×#8760411yescharacteristic
122C23—971no—
123Q8—42no—
124Q8—3882no—
125Q8—3882no—
126Q8—3882no—
127C2×C4—22no—
128C2×C4—971no—
129C2×C4—971no—
130C2×C4—971no—
131C2×C4—972no—
132C2×C4—972no—
133C2×C4—3884no—
134C2×C4—1942no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4C22⋊#8760411yescharacteristic
138C2×C4C22⋊#8760411yescharacteristic
139C2×C4C22×#8760411yescharacteristic
140C8—3882no—
141C8—3882no—
142C22—972no—
143C22—1941no—
144C22—1941no—
145C22C2×C22⋊#8760411yescharacteristic
146C4—21no—
147C4—21no—
148C4—42no—
149C4—972no—
150C4—971no—
151C4—971no—
152C4—972no—
153C4—3882no—
154C4#1569087212yes—
155C2—1942no—
156C2D97⋊#10111yescharacteristic
157C2C2×#1569087211yescharacteristic
158C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#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#1546
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20261594C2semidirect
80. C97#1546semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202617402
#202615942
C4⋊Q8×D972
#154697