Finite Group Databasebeta

← #27141009

Subgroups — #27141009

Order 12416.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261455C211yesmaximal, characteristic
2#20261133C211yesmaximal, characteristic, complemented
3#20261015C211yesmaximal, characteristic
4#20260946C211yesmaximal, characteristic, complemented
5#20260480C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic
8C2×Dic388—21no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690803—22no—
12#15690797—21no—
13#15690789—21no—
14C4×Dic194C2211yescharacteristic
15#15690775—21no—
16#15690769—21no—
17#15690743—21no—
18#15690742—22no—
19C4⋊#11715895C2211yescharacteristic
20#15690735—21no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194D411yescharacteristic
28Dic388—42no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97D411yescharacteristic
34C4⋊Dic97D412yes—
35#13095490—22no—
36#13095490—21no—
37C22⋊Dic97—22no—
38C22⋊Dic97—22no—
39C4×Dic97—22no—
40C4×Dic97—21no—
41C2×D4×C97D411yescharacteristic
42C22×C4×C97—21no—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49D4×C97—21no—
50D4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—22no—
60C2×Dic97—41no—
61Dic194—21no—
62Dic194—21no—
63Dic194—41no—
64Dic194—41no—
65Dic194—41no—
66Dic194—41no—
67C8×C97—22no—
68C8×C97—41no—
69#11715895—41no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97#12011yescharacteristic
74C4×C97—21no—
75C4×C97—41no—
76C4×C97#13511yescharacteristic
77C4×C97#13511yescharacteristic
78C4×C97AGL(1,ℤ/8)11yescharacteristic
79C4×C97#13711yescharacteristic
80Dic97—41no—
81Dic97—42no—
82Dic97—41no—
83Dic97—81no—
84C2×C97—81no—
85C2×C97#44911yescharacteristic
86C2×C97#45211yescharacteristic
87C2×C97#45611yescharacteristic
88#1652—971nomaximal, Sylow
89C97#165211yescharacteristic, complemented, Sylow
90#556—971no—
91#498—971no—
92#474—971no—
93#462—971no—
94#332—971no—
95Q8⋊C8—971no—
96D4⋊C8D9711yescharacteristic
97C2×Q16—1941no—
98C4⋊Q8—971no—
99C4⋊Q8—971no—
100#126—1942no—
101#124—1941no—
102C22⋊Q8—1941no—
103C4×Q8—971no—
104C4×Q8—1941no—
105C4×D4C2×D9711yescharacteristic
106#117—1941no—
107#107—1941no—
108#106—1942no—
109C4⋊C8—971no—
110C4⋊C8C2×D9711yescharacteristic
111#103—1941no—
112#102—1941no—
113C4×C8C2×D9711yescharacteristic
114C2×Q8—971no—
115C2×Q8—1941no—
116C2×Q8—1941no—
117C2×D4D38811yescharacteristic
118C22×C4—21no—
119Q16—3882no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1941no—
123C4⋊C4—971no—
124C4⋊C4—972no—
125C4⋊C4—1942no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4D38811yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—1942no—
133C22⋊C4—1942no—
134C42—1942no—
135C42—1941no—
136C42C22×D9711yescharacteristic
137C23—21no—
138Q8—3881no—
139Q8—3881no—
140Q8—3881no—
141Q8—3881no—
142Q8—1941no—
143Q8—1941no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3881no—
150C2×C4—1941no—
151C2×C4—1942no—
152C2×C4—1941no—
153C2×C4D4×D9711yescharacteristic
154C2×C4D4×D9711yescharacteristic
155C2×C4C2×D38811yescharacteristic
156C8—22no—
157C8—41no—
158C8—3881no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#1569078011yescharacteristic
163C4—21no—
164C4—41no—
165C4—7761no—
166C4—3881no—
167C4—3882no—
168C4—3881no—
169C4#1569083111yescharacteristic
170C4#1569083211yescharacteristic
171C4#1569083411yescharacteristic
172C4#1569084311yescharacteristic
173C2—81no—
174C2#2026088111yescharacteristic
175C2#2026089911yescharacteristic
176C2#2026091911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4⋊C8D97
hypocenterC97#1652
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. #20261133C2semidirect
4. #20260946C2semidirect
5. #20260480C2semidirect
6. #20260467C2semidirect
89. C97#1652semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202611332
#202610152
#202609462
#202604802
#202604672
D4⋊C8×C972
#165297