Finite Group Databasebeta

← #27140904

Subgroups — #27140904

Order 12416.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic194C211yesmaximal, characteristic
2#20261086C211yesmaximal, characteristic, complemented
3#20261031C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic
5#20260482C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic
8C22×Dic194—21no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690809—21no—
12#15690789—21no—
13#15690787—22no—
14C4×Dic194C2211yescharacteristic
15D4×Dic97—21no—
16#15690744—21no—
17#15690741—21no—
18C4⋊#11715895C2211yescharacteristic
19#15690735—21no—
20#15690735—22no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—42no—
29C2×Dic194—42no—
30C2×Dic194D411yescharacteristic
31C22×Dic97—22no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97D411yescharacteristic
37C4⋊Dic97D412yes—
38#13095490—22no—
39#13095490—21no—
40C22⋊Dic97—22no—
41C4×Dic97—21no—
42C2×D4×C97D411yescharacteristic
43C22×C4×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C22⋊C4×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—21no—
50D4×C97—21no—
51D4×C97Q1612yes—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—22no—
61C2×Dic97—44no—
62C2×Dic97—41no—
63Dic194—21no—
64Dic194—44no—
65Dic194—41no—
66Dic194—41no—
67Dic194—42no—
68Dic194SD1612yes—
69C8×C97—41no—
70C8×C97—41no—
71#11715895—41no—
72C22×C97—22no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97#12011yescharacteristic
76C4×C97—21no—
77C4×C97—41no—
78C4×C97C2×SD1611yescharacteristic
79C4×C97C2×Q1611yescharacteristic
80C4×C97AGL(1,ℤ/8)11yescharacteristic
81C4×C97#13711yescharacteristic
82Dic97—22no—
83Dic97—42no—
84Dic97—41no—
85Dic97—81no—
86C2×C97—42no—
87C2×C97#45011yescharacteristic
88C2×C97#45111yescharacteristic
89C2×C97#45411yescharacteristic
90#1637—971nomaximal, Sylow
91C97#163711yescharacteristic, complemented, Sylow
92D4×Q8—971no—
93#487—971no—
94#477—971no—
95#476—971no—
96#332—971no—
97Q8⋊C8—971no—
98D4⋊C8D9711yescharacteristic
99C22×Q8—1941no—
100C4⋊Q8—971no—
101C4⋊Q8—971no—
102C4⋊Q8—1941no—
103C22⋊Q8—1942no—
104C22⋊Q8—1941no—
105C4×Q8—971no—
106C4×D4—1941no—
107C4×D4C2×D9711yescharacteristic
108#107—1941no—
109#106—1941no—
110C4⋊C8—971no—
111C4⋊C8C2×D9711yescharacteristic
112#103—1942no—
113#103—1941no—
114#102—1941no—
115C4×C8C2×D9711yescharacteristic
116C2×Q8—971no—
117C2×Q8—3882no—
118C2×Q8—3882no—
119C2×Q8—1942no—
120C2×Q8—1941no—
121C2×Q8—1941no—
122C2×D4D38811yescharacteristic
123C22×C4—21no—
124C22×C4—1942no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—1941no—
128C4⋊C4—972no—
129C4⋊C4—971no—
130C4⋊C4—1942no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4D38811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—1942no—
138C42—1941no—
139C42C22×D9711yescharacteristic
140C23—21no—
141Q8—972no—
142Q8—3882no—
143Q8—3884no—
144Q8—3881no—
145Q8—3881no—
146Q8—1941no—
147D4—21no—
148D4Dic38812yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3881no—
153C2×C4—3884no—
154C2×C4—1941no—
155C2×C4—1942no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×D38811yescharacteristic
160C8—41no—
161C8—41no—
162C8—3881no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1569078011yescharacteristic
167C4—21no—
168C4—41no—
169C4—7761no—
170C4—3882no—
171C4—3881no—
172C4—1942no—
173C4SD16×D9711yescharacteristic
174C4C2×Dic38811yescharacteristic
175C4#1569083411yescharacteristic
176C4#1569084311yescharacteristic
177C2—42no—
178C2#2026088711yescharacteristic
179C2#2026089311yescharacteristic
180C2#2026090911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4⋊C8D97
hypocenterC97#1637
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20261086C2semidirect
3. #20261031C2semidirect
5. #20260482C2semidirect
6. #20260467C2semidirect
91. C97#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1942
#202610862
#202610312
#202610252
#202604822
#202604672
D4⋊C8×C972
#163797