Finite Group Databasebeta

← #27141090

Subgroups — #27141090

Order 12416.

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

normal subgroups only

This group has 204 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#20261111C211yesmaximal, characteristic, complemented
2#20260931C211yesmaximal, characteristic, complemented
3#20260927C211yesmaximal, characteristic, complemented
4#20260800C211yesmaximal, characteristic, complemented
5#20260487C211yesmaximal, characteristic, complemented
6#20260478C211yesmaximal, characteristic, complemented
7#331×C97C211yesmaximal, characteristic, complemented
8C2×D776—21no—
9C2×#13095496—21no—
10C2×#13095496—21no—
11C2×#13095496—21no—
12C2×C8×D97—21no—
13#15690807C2211yescharacteristic
14C4×D388C2211yescharacteristic
15C4⋊#11715895C2211yescharacteristic
16#15690732—21no—
17#15690731—21no—
18#15690722—21no—
19C4×#11715895C2211yescharacteristic
20#127×C97C2211yescharacteristic
21C4⋊C8×C97C2211yescharacteristic
22C4⋊C8×C97C2211yescharacteristic
23C2×D388—21no—
24C2×D388—21no—
25C2×D388—21no—
26C2×D388D411yescharacteristic
27C2×C4×D97—21no—
28D776—42no—
29#13095496—42no—
30#13095496—42no—
31#13095496—42no—
32C2×#11715895—21no—
33C2×#11715895D411yescharacteristic
34C2×#11715895D411yescharacteristic
35C8×D97—42no—
36C4⋊Dic97D411yescharacteristic
37#13095488—21no—
38C2×D4×C97—21no—
39C2×D4×C97—21no—
40C2×D4×C97—21no—
41C2×C8×C97—21no—
42C2×C8×C97—21no—
43C42×C97C2311yescharacteristic
44C23×C97—41no—
45C22×D97—21no—
46C22×D97—41no—
47D4×C97—41no—
48D4×C97—41no—
49D4×C97—41no—
50D4×C97—41no—
51D4×C97—81no—
52C2×C4×C97C2×D411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97#4111yescharacteristic
55C2×Dic97—21no—
56D388—21no—
57D388—41no—
58D388—41no—
59D388—41no—
60D388—41no—
61D388—81no—
62D388D812yescomplemented
63C4×D97—41no—
64C4×D97—41no—
65C8×C97—41no—
66C8×C97—41no—
67#11715895—21no—
68#11715895—41no—
69#11715895D812yescomplemented
70C22×C97—81no—
71C22×C97—81no—
72C22×C97—81no—
73C22×C97#12111yescharacteristic
74C2×D97—22no—
75C2×D97—41no—
76C2×D97—41no—
77C2×D97—81no—
78C2×D97—81no—
79C2×D97—81no—
80C4×C97—21no—
81C4×C97C2×D811yescharacteristic
82C4×C97C2×D811yescharacteristic
83C4×C97#13511yescharacteristic
84C4×C97AGL(1,ℤ/8)11yescharacteristic
85Dic97—41no—
86C2×C97—161no—
87C2×C97#45911yescharacteristic
88C2×C97#46611yescharacteristic
89C2×C97#47211yescharacteristic
90D97—42no—
91D97—161no—
92#1667—971nomaximal, Sylow
93C97#166711yescharacteristic, complemented, Sylow
94#493—971no—
95#459—971no—
96#459—971no—
97C8×D4—971no—
98#335—971no—
99#331—971no—
100#331D9711yescharacteristic, complemented
101C2×D8—1941no—
102C2×D8—1941no—
103C2×D8—1941no—
104C2×D8—1941no—
105C22×C8—1941no—
106#127—971no—
107#127C2×D9711yescharacteristic
108C4×D4—971no—
109C4⋊C8—971no—
110C4⋊C8C2×D9711yescharacteristic
111C4⋊C8C2×D9711yescharacteristic
112#102—1941no—
113#102—1941no—
114C22⋊C8—1941no—
115C4×C8—971no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—971no—
120C2×D4—1941no—
121C2×D4—1941no—
122C2×D4—1941no—
123C22×C4—1941no—
124D8—3882no—
125D8—3882no—
126D8—3882no—
127D8—3882no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—971no—
131C2×C8—971no—
132C2×C8—3882no—
133C2×C8—1941no—
134C4⋊C4—971no—
135C22⋊C4—1941no—
136C42C22×D9711yescharacteristic
137C23—41no—
138C23—3881no—
139C23—1941no—
140D4—41no—
141D4—41no—
142D4—41no—
143D4—41no—
144D4—81no—
145D4—7761no—
146D4—972no—
147D4—3881no—
148D4—3881no—
149D4—3881no—
150D4—3881no—
151D4—1941no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—1941no—
155C2×C4D4×D9711yescharacteristic
156C2×C4D4×D9711yescharacteristic
157C2×C4#1309551111yescharacteristic
158C8—41no—
159C8—41no—
160C8—972no—
161C8—3881no—
162C8—1941no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—7761no—
167C22—7761no—
168C22—7761no—
169C22—3881no—
170C22—3881no—
171C22—1942no—
172C22#1569078311yescharacteristic
173C4—21no—
174C4—3881no—
175C4D8×D9711yescharacteristic
176C4D8×D9711yescharacteristic
177C4#1569083111yescharacteristic
178C4#1569083711yescharacteristic
179C2—161no—
180C2—15521no—
181C2—3882no—
182C2#2026092911yescharacteristic
183C2#2026097011yescharacteristic
184C2#2026100411yescharacteristic
subgroupG/H
centerC22#15690783
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#331×C97C2
socleC22×C97#121
hypercenter#331D97
hypocenterC97#1667
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #20261111C2semidirect
2. #20260931C2semidirect
3. #20260927C2semidirect
4. #20260800C2semidirect
5. #20260487C2semidirect
6. #20260478C2semidirect
7. #331×C97C2semidirect
62. D388D8semidirect
69. #11715895D8semidirect
93. C97#1667semidirect
100. #331D97semidirect

Maximal subgroups

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

subgroupindex
#202611112
#202609312
#202609272
#202608002
#202604872
#202604782
#331×C972
#166797