Finite Group Databasebeta

← #27157087

Subgroups — #27157087

Order 12416.

230 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 64 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#20261756C211yesmaximal, characteristic, complemented
2C2×C4×#11715897C211yesmaximal, characteristic, complemented
3#20261585C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5#20261266C211yesmaximal, characteristic
6C22×#11715897—21no—
7C4⋊#11715897C2212yes—
8C22⋊#11715897—21no—
9#15690865—21no—
10C4×#11715897—21no—
11C4×#11715897C2212yes—
12#15690803—21no—
13#15690801C411yescharacteristic
14#15690797C411yescharacteristic
15#15690791—21no—
16#15690789C411yescharacteristic
17#15690775C2211yescharacteristic
18D4×Dic97—21no—
19D4×Dic97C2211yescharacteristic
20#15690766—21no—
21C2×C4×Dic97C2211yescharacteristic
22#121×C97C411yescharacteristic
23C2×#11715897—24no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194C2×C411yescharacteristic
27C22×Dic97—21no—
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C4×Dic97—21no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×D4×C97—21no—
45C2×D4×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C23×C97—21no—
50C23×C97C22⋊C411yescharacteristic
51D4×C97—22no—
52D4×C97—42no—
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×Dic97—22no—
58C2×Dic97—22no—
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97—42no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66Dic194—22no—
67#11715897—24no—
68#11715897—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97#10412yes—
75C22×C97C2×C22⋊C411yescharacteristic
76C4×C97—21no—
77C4×C97—41no—
78C4×C97#13112yes—
79Dic97—21no—
80Dic97—21no—
81Dic97—21no—
82Dic97—41no—
83Dic97—41no—
84Dic97#13512yes—
85C2×C97—22no—
86C2×C97—81no—
87C2×C97#40811yescharacteristic
88C2×C97#41611yescharacteristic
89C2×C97C2×#10411yescharacteristic
90#1494—971nomaximal, Sylow
91C97#149411yescharacteristic, complemented, Sylow
92#529—971no—
93#432—971no—
94C2×C4×C8—971no—
95Q8⋊C8—972no—
96D4⋊C8—972no—
97C22×C8—1941no—
98#126—1941no—
99#125—971no—
100#124—971no—
101#123—1941no—
102C22⋊Q8—971no—
103#121#8760411yescharacteristic
104C4×Q8—971no—
105C4×D4—971no—
106C4×D4—1941no—
107#117—1941no—
108C2×C42—971no—
109C4⋊C8—972no—
110C22⋊C8—1941no—
111#97—1941no—
112C4×C8—972no—
113C4×C8—1941no—
114C2×Q8—971no—
115C2×D4—21no—
116C2×D4C2×#8760411yescharacteristic
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—1941no—
120C22×C4C2×#8760411yescharacteristic
121C2×C8—1942no—
122C2×C8—1942no—
123C2×C8—1944no—
124C4⋊C4—971no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4C2×#8760411yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—1941no—
132C22⋊C4—1941no—
133C22⋊C4—1941no—
134C22⋊C4—1941no—
135C42—971no—
136C42—971no—
137C42—971no—
138C42—971no—
139C42—1941no—
140C23—21no—
141C23C22⋊#8760411yescharacteristic
142Q8—1942no—
143D4—22no—
144D4—42no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—971no—
151C2×C4—3882no—
152C2×C4—1942no—
153C2×C4—1941no—
154C2×C4—1941no—
155C2×C4—1942no—
156C2×C4C22⋊#8760411yescharacteristic
157C2×C4C22×#8760411yescharacteristic
158C8—3882no—
159C8—1944no—
160C22—21no—
161C22—21no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1569087412yes—
166C22C2×C22⋊#8760411yescharacteristic
167C4—21no—
168C4—41no—
169C4—972no—
170C4—3881no—
171C4—3881no—
172C4—1941no—
173C4—1941no—
174C4—1941no—
175C4#1569089712yes—
176C2—22no—
177C2—81no—
178C2#2026170811yescharacteristic
179C2#2026172511yescharacteristic
180C2C2×#1569087411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting, prime radical#121×C97C4
socleC22×C97C2×C22⋊C4
cosocleC4×Dic97C23
hypercenter#121#87604
hypocenterC97#1494
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. C2×C4×#11715897C2semidirect
3. #20261585C2semidirect
4. D4⋊#11715897C2semidirect
91. C97#1494semidirect

Maximal subgroups

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

subgroupindex
#202617562
C2×C4×#117158972
#202615852
D4⋊#117158972
#202612662
#149497