Finite Group Databasebeta

← #27157252

Subgroups — #27157252

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—44no—
26C2×C4×D97—22no—
27C2×C4×D97C2×C411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C412yes—
32#13095490—22no—
33#13095490—22no—
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97C2311yescharacteristic
37C2×Q8×C97—21no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97C2×C412yes—
40C42×C97C2×C411yescharacteristic
41C22×D97C22⋊C411yescharacteristic
42Q8×C97—42no—
43C2×C4×C97—22no—
44C2×C4×C97C22⋊C411yescharacteristic
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22×C411yescharacteristic
47C2×Dic97—22no—
48C2×Dic97C22⋊C411yescharacteristic
49C2×Dic97C2×D411yescharacteristic
50C2×Dic97C2×D411yescharacteristic
51C4×D97—22no—
52C4×D97—22no—
53C4×D97—44no—
54C4×D97SD1612yes—
55C4×D97Q1612yes—
56Dic194—42no—
57Dic194—44no—
58#11715897—42no—
59#11715897—42no—
60C22×C97C2×C22⋊C411yescharacteristic
61C2×D97—21no—
62C2×D97—21no—
63C2×D97#10312yes—
64C4×C97—21no—
65C4×C97—21no—
66C4×C97—42no—
67C4×C97#10312yes—
68Dic97—42no—
69Dic97#10112yes—
70Dic97C2×SD1611yescharacteristic
71Dic97C2×Q1611yescharacteristic
72Dic97AGL(1,ℤ/8)12yes—
73C2×C97C2×#10111yescharacteristic
74C2×C97C2×#10311yescharacteristic
75C2×C97#41811yescharacteristic
76D97—22no—
77#1549—971nomaximal, Sylow
78C97#154911yescharacteristic, complemented, Sylow
79C2×C4⋊Q8—971no—
80#431—971no—
81#423—971no—
82#332—974no—
83C22×Q8—1941no—
84C2×M16—1941no—
85C4⋊Q8—972no—
86C4⋊Q8—971no—
87C4⋊Q8—1941no—
88C4⋊Q8—1941no—
89C4⋊Q8#8760411yescharacteristic
90C2×C4⋊C4—972no—
91C2×C4⋊C4—1941no—
92C2×C42—971no—
93C4⋊C8—972no—
94C4⋊C8—972no—
95C4⋊C8—1941no—
96C22⋊C8—1941no—
97C4×C8—1941no—
98C2×Q8—21no—
99C2×Q8—3884no—
100C2×Q8—1942no—
101C2×Q8—1941no—
102C22×C4—971no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—1942no—
106M16—3882no—
107C2×C8—1942no—
108C2×C8—1942no—
109C4⋊C4—21no—
110C4⋊C4—972no—
111C4⋊C4—1942no—
112C4⋊C4—1942no—
113C4⋊C4—1941no—
114C4⋊C4C2×#8760412yes—
115C42—971no—
116C42—971no—
117C42—971no—
118C42C2×#8760411yescharacteristic
119C23—971no—
120Q8—42no—
121Q8—3882no—
122Q8—3884no—
123C2×C4—22no—
124C2×C4—971no—
125C2×C4—971no—
126C2×C4—971no—
127C2×C4—972no—
128C2×C4—972no—
129C2×C4—3884no—
130C2×C4—1942no—
131C2×C4—1942no—
132C2×C4—1942no—
133C2×C4C22⋊#8760411yescharacteristic
134C2×C4C22⋊#8760411yescharacteristic
135C2×C4C22×#8760411yescharacteristic
136C8—3882no—
137C8—3882no—
138C22—972no—
139C22—1941no—
140C22—1941no—
141C22C2×C22⋊#8760411yescharacteristic
142C4—21no—
143C4—21no—
144C4—42no—
145C4—972no—
146C4—971no—
147C4—971no—
148C4—972no—
149C4—3882no—
150C4#1569087312yes—
151C2—1942no—
152C2D97⋊#10111yescharacteristic
153C2C2×#1569087311yescharacteristic
154C2#2026172711yescharacteristic
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#1549
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
78. C97#1549semidirect

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
#154997