Finite Group Databasebeta

← #27140833

Subgroups — #27140833

Order 12416.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261460C211yesmaximal, characteristic, complemented
2#20261083C211yesmaximal, characteristic
3#20260936C211yesmaximal, characteristic
4#20260936C211yesmaximal, characteristic
5#20260465C211yesmaximal, characteristic, complemented
6#20260465C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8C2×#13095500—21no—
9C2×#13095500—21no—
10#15690808—21no—
11#15690807C2211yescharacteristic
12#15690801—21no—
13#15690800—22no—
14Q8×Dic97—21no—
15C4×Dic194C2211yescharacteristic
16C4×Dic194C2211yescharacteristic
17#15690775—21no—
18#15690732—21no—
19#15690732—21no—
20#15690730—22no—
21C4×#11715895C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194D411yescharacteristic
26C2×Dic194D411yescharacteristic
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30#13095500—42no—
31#13095500—42no—
32C2×#11715895—21no—
33C4⋊Dic97—22no—
34C4⋊Dic97D411yescharacteristic
35C4⋊Dic97D411yescharacteristic
36#13095490—22no—
37#13095490—22no—
38#13095490—21no—
39#13095490—21no—
40C4×Dic97—22no—
41C4×Dic97—21no—
42C4×Dic97—21no—
43C2×Q8×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D412yes—
48C42×C97C2311yescharacteristic
49C22×D97—41no—
50Q8×C97—41no—
51Q8×C97—41no—
52C2×C4×C97—22no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60D388—42no—
61D388—41no—
62D388—41no—
63D388—41no—
64D388—41no—
65Dic194—21no—
66Dic194—21no—
67Dic194—21no—
68Dic194SD1612yescomplemented
69C8×C97—41no—
70C8×C97—41no—
71#11715895—41no—
72C22×C97#12011yescharacteristic
73C2×D97—81no—
74C2×D97—81no—
75C2×D97—81no—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97C2×SD1611yescharacteristic
79C4×C97#13511yescharacteristic
80C4×C97AGL(1,ℤ/8)11yescharacteristic
81C4×C97#13711yescharacteristic
82Dic97—22no—
83Dic97—41no—
84Dic97—41no—
85Dic97—41no—
86C2×C97#44811yescharacteristic
87C2×C97#44911yescharacteristic
88C2×C97#45411yescharacteristic
89D97—161no—
90#1626—971nomaximal, Sylow
91C97#162611yescharacteristic, complemented, Sylow
92#557—971no—
93#486—971no—
94#460—971no—
95#460—971no—
96#332D9711yescharacteristic, complemented
97Q8⋊C8—971no—
98Q8⋊C8—971no—
99C2×SD16—1941no—
100C2×SD16—1941no—
101C4⋊Q8—1941no—
102C4⋊Q8C2×D9711yescharacteristic
103#127—971no—
104#125—1941no—
105#125—1942no—
106C4×Q8—971no—
107C4×Q8—971no—
108C4×Q8—1941no—
109C4×Q8—1941no—
110C4⋊C8C2×D9711yescharacteristic
111C4⋊C8C2×D9711yescharacteristic
112#102—1941no—
113#102—1941no—
114#102—1942no—
115C4×C8—971no—
116C2×Q8—21no—
117C2×Q8—971no—
118C2×Q8—971no—
119C2×D4—1941no—
120C2×D4—1941no—
121C2×D4—1941no—
122SD16—3882no—
123SD16—3882no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—1941no—
127C4⋊C4—21no—
128C4⋊C4—971no—
129C4⋊C4—971no—
130C4⋊C4—1942no—
131C4⋊C4—1942no—
132C4⋊C4—1942no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4D38812yes—
136C42—1942no—
137C42—1941no—
138C42—1941no—
139C42C22×D9711yescharacteristic
140C23—3881no—
141Q8—41no—
142Q8—41no—
143Q8—972no—
144Q8—1941no—
145Q8—1941no—
146Q8—1941no—
147D4—3882no—
148D4—3881no—
149D4—3881no—
150D4—3881no—
151D4—3881no—
152C2×C4—22no—
153C2×C4—1941no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×D38811yescharacteristic
160C8—41no—
161C8—41no—
162C8—3881no—
163C22—7761no—
164C22—7761no—
165C22—7761no—
166C22#1569078011yescharacteristic
167C4—21no—
168C4—42no—
169C4—3881no—
170C4—3881no—
171C4—3881no—
172C4—1942no—
173C4SD16×D9711yescharacteristic
174C4#1569083111yescharacteristic
175C4#1569083411yescharacteristic
176C4#1569084211yescharacteristic
177C2—15521no—
178C2#2026087511yescharacteristic
179C2#2026088111yescharacteristic
180C2#2026091011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97#120
prime radical#15690807C22
hypercenter#332D97
hypocenterC97#1626
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #20261460C2semidirect
5. #20260465C2semidirect
6. #20260465C2semidirect
7. #332×C97C2semidirect
68. Dic194SD16semidirect
91. C97#1626semidirect
96. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202614602
#202610832
#202609362
#202604652
#332×C972
#162697