Finite Group Databasebeta

← #27140233

Subgroups — #27140233

Order 12416.

240 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 71 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#20261287C211yesmaximal, characteristic
2#20260785C211yesmaximal, characteristic
3#20260586C211yesmaximal, characteristic
4#20260473C211yesmaximal, characteristic
5#20260472C211yesmaximal, characteristic
6#20260471C211yesmaximal, characteristic
7#328×C97C211yesmaximal, characteristic
8#15690809C411yescharacteristic, complemented
9#15690808C411yescharacteristic, complemented
10#15690800—21no—
11#15690789C2211yescharacteristic
12#15690787—21no—
13Q8×Dic97C411yescharacteristic, complemented
14#15690775—21no—
15#15690775C411yescharacteristic, complemented
16#15690766—21no—
17C2×#13095490—21no—
18C2×C4×Dic97—21no—
19C2×C4×Dic97C2211yescharacteristic
20#15690739—21no—
21C22⋊#11715895C2211yescharacteristic
22C8×Dic97—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C411yescharacteristic
35#13095490—24no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41C22⋊Dic97—21no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C4×Dic97—42no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×Q8×C97C2×C411yescharacteristic
50C22×C4×C97—21no—
51C22×C4×C97C2311yescharacteristic
52C2×C8×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C22⋊C4×C97—21no—
56C23×C97C2×D411yescharacteristic
57Q8×C97—41no—
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×Dic97—22no—
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—41no—
71C2×Dic97—41no—
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C22⋊C411yescharacteristic
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C22⋊C411yescharacteristic
76C2×Dic97SD1612yes—
77C2×Dic97Q1612yes—
78Dic194—42no—
79Dic194—41no—
80C8×C97—41no—
81#11715895—41no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97C2×C22⋊C411yescharacteristic
85C22×C97C2×SD1611yescharacteristic
86C22×C97C2×Q1611yescharacteristic
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—42no—
90C4×C97—41no—
91C4×C97—41no—
92Dic97—21no—
93Dic97—42no—
94Dic97—41no—
95Dic97—41no—
96Dic97#10312yes—
97Dic97#10312yes—
98Dic97#10412yes—
99C2×C97—21no—
100C2×C97—21no—
101C2×C97#41011yescharacteristic
102C2×C97C2×#10311yescharacteristic
103C2×C97C2×#10411yescharacteristic
104#1503—971nomaximal, Sylow
105C97#150311yescharacteristic, complemented, Sylow
106#533—971no—
107#431—971no—
108C4×C4⋊C4—971no—
109#328—971no—
110#328—971no—
111#328—971no—
112#328D9711yescharacteristic
113C4⋊Q8—971no—
114C4⋊Q8—971no—
115#125—1941no—
116C22⋊Q8—971no—
117C22⋊Q8—1941no—
118C22⋊Q8C2×D9711yescharacteristic
119C4×Q8—971no—
120C4×Q8—971no—
121C4×Q8—1941no—
122#117—1941no—
123C2×C4⋊C4—1941no—
124C2×C42—971no—
125C2×C42—1941no—
126C4⋊C8—1941no—
127C22⋊C8—971no—
128C22⋊C8C2×D9711yescharacteristic
129C4×C8—1941no—
130#95—971no—
131#95C2×D9711yescharacteristic
132C2×Q8—971no—
133C2×Q8—1941no—
134C2×Q8C4×D9711yescharacteristic
135C22×C4—21no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139C22×C4C22×D9711yescharacteristic
140C2×C8—21no—
141C2×C8—1941no—
142C4⋊C4—21no—
143C4⋊C4—971no—
144C4⋊C4—1941no—
145C4⋊C4—1941no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C4⋊C4—1941no—
149C4⋊C4—1944no—
150C4⋊C4—1941no—
151C4⋊C4C4×D9711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—1941no—
154C42—971no—
155C42—971no—
156C42—971no—
157C42—971no—
158C42—3882no—
159C42—1941no—
160C42—1941no—
161C23D4×D9711yescharacteristic
162Q8—41no—
163Q8—3881no—
164Q8—3882no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—972no—
175C2×C4—972no—
176C2×C4—971no—
177C2×C4—3881no—
178C2×C4—3881no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4C2×C4×D9711yescharacteristic
184C2×C4D4×D9711yescharacteristic
185C8—41no—
186C8—3881no—
187C22—21no—
188C22—21no—
189C22C22⋊C4×D9711yescharacteristic
190C22SD16×D9711yescharacteristic
191C22Q16×D9711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—972no—
198C4—972no—
199C4—972no—
200C4—3882no—
201C4—3881no—
202C4—3881no—
203C4—1941no—
204C2—21no—
205C2—21no—
206C2#2026070711yescharacteristic
207C2#103×D9711yescharacteristic
208C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97#1503
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
8. #15690809C4semidirect
9. #15690808C4semidirect
13. Q8×Dic97C4semidirect
15. #15690775C4semidirect
105. C97#1503semidirect

Maximal subgroups

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

subgroupindex
#202612872
#202607852
#202605862
#202604732
#202604722
#202604712
#328×C972
#150397