Finite Group Databasebeta

← #27158036

Subgroups — #27158036

Order 12416.

262 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 52 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#20261722C212yesmaximal, complemented
2#20261644C212yesmaximal, complemented
3#20261643C212yesmaximal, complemented
4#20261298C211yesmaximal, characteristic
5C2×#13095520—22no—
6#15690871C2214yescomplemented
7#15690870—22no—
8C2×#13095510C2211yescharacteristic
9C2×D4×D97C411yescharacteristic
10#15690806C411yescharacteristic
11#15690798C2211yescharacteristic
12#15690797C2211yescharacteristic
13#15690782—22no—
14#15690778—22no—
15#15690767C411yescharacteristic
16#127×C97C411yescharacteristic
17#13095520—24no—
18#13095520—42no—
19C2×#11715897—42no—
20C23×D97—21no—
21#13095510—42no—
22C2×Dic194C2311yescharacteristic
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—22no—
25C2×C22⋊D97—21no—
26C2×C22⋊D97D412yes—
27D4×D97—42no—
28C22×Dic97D412yes—
29C2×C4×D97C2×C411yescharacteristic
30#13095490—21no—
31C22⋊Dic97—22no—
32#13095488—21no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic
35C2×D4×C97—22no—
36C2×D4×C97C2×C411yescharacteristic
37C2×D4×C97C2×C411yescharacteristic
38C42×C97C2×C411yescharacteristic
39C23×C97—21no—
40C23×C97C22⋊C412yes—
41C22×D97—21no—
42C22×D97—42no—
43C22×D97—82no—
44C22×D97C22⋊C411yescharacteristic
45D4×C97—41no—
46D4×C97—41no—
47D4×C97—82no—
48C2×C4×C97—21no—
49C2×C4×C97C22×C411yescharacteristic
50C22⋊D97—42no—
51C22⋊D97—42no—
52C22⋊D97—82no—
53C2×Dic97—22no—
54C2×Dic97—22no—
55C2×Dic97—21no—
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C2×D412yes—
58D388—41no—
59C4×D97—22no—
60Dic194—41no—
61#11715897—44no—
62C22×C97—22no—
63C22×C97—42no—
64C22×C97—42no—
65C22×C97—41no—
66C22×C97C2×C22⋊C411yescharacteristic
67C2×D97—21no—
68C2×D97—41no—
69C2×D97—42no—
70C2×D97—41no—
71C2×D97—84no—
72C4×C97—21no—
73C4×C97—41no—
74Dic97—22no—
75Dic97—41no—
76Dic97#9912yes—
77C2×C97—21no—
78C2×C97—42no—
79C2×C97—81no—
80C2×C97C2×#9911yescharacteristic
81D97—41no—
82D97—81no—
83#2134—971nomaximal, Sylow
84C97#213411yescharacteristic, complemented, Sylow
85#535—971no—
86#413—972no—
87#355—972no—
88#355—972no—
89C2×#41—971no—
90C22×D4—971no—
91C2×M16—1942no—
92#127—971no—
93#127#8760411yescharacteristic
94#124—971no—
95#124—971no—
96#121—1942no—
97#120—1942no—
98#117—971no—
99#101—974no—
100#100—1942no—
101C24—1941no—
102#41—3882no—
103C2×Q8—971no—
104C2×D4—22no—
105C2×D4—972no—
106C2×D4—971no—
107C2×D4—3882no—
108C2×D4—1941no—
109C2×D4—1942no—
110C2×D4C2×#8760411yescharacteristic
111C2×D4C2×#8760411yescharacteristic
112C22×C4—972no—
113C22×C4—971no—
114M16—3882no—
115M16—1944no—
116C2×C8—3882no—
117C4⋊C4—1941no—
118C22⋊C4—1942no—
119C22⋊C4—1941no—
120C22⋊C4—1942no—
121C42—971no—
122C42C2×#8760411yescharacteristic
123C23—21no—
124C23—7762no—
125C23—971no—
126C23—3882no—
127C23—1941no—
128C23C22⋊#8760412yes—
129Q8—3881no—
130D4—41no—
131D4—41no—
132D4—82no—
133D4—7762no—
134D4—3881no—
135D4—3882no—
136D4—3882no—
137C2×C4—21no—
138C2×C4—971no—
139C2×C4—972no—
140C2×C4—1942no—
141C2×C4—1942no—
142C2×C4—1942no—
143C2×C4—1941no—
144C2×C4C22×#8760411yescharacteristic
145C8—3884no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22—41no—
150C22—7764no—
151C22—3882no—
152C22—3881no—
153C22—3881no—
154C22—1941no—
155C22C2×C22⋊#8760411yescharacteristic
156C4—21no—
157C4—41no—
158C4—972no—
159C4—3881no—
160C4—1942no—
161C2—21no—
162C2—42no—
163C2—81no—
164C2—7761no—
165C2—3881no—
166C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC2×C97C2×#99
prime radical#20261298C2
cosocleC2×Dic194C23
hypercenter#127#87604
hypocenterC97#2134
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261722C2semidirect
2. #20261644C2semidirect
3. #20261643C2semidirect
6. #15690871C22semidirect
84. C97#2134semidirect

Maximal subgroups

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

subgroupindex
#202617222
#202616442
#202616432
#202612982
#213497