Finite Group Databasebeta

← #27157069

Subgroups — #27157069

Order 12416.

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

normal subgroups only

This group has 246 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
2#20261752C211yesmaximal, characteristic, complemented
3#20261585C212yesmaximal, complemented
4#20261583C212yesmaximal, complemented
5#20261203C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897C2212yes—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10#15690865—21no—
11C4×#11715897C2212yes—
12C2×#13095511C411yescharacteristic
13#15690775C411yescharacteristic
14#15690775C2211yescharacteristic
15#15690772C411yescharacteristic
16#15690772C2211yescharacteristic
17#15690770—21no—
18#15690770—21no—
19#15690767—21no—
20C2×C4×Dic97C2211yescharacteristic
21C42×D97—21no—
22#117×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25#13095511—22no—
26#13095511—22no—
27C2×Dic194C2×C411yescharacteristic
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35#13095490—21no—
36#13095490—21no—
37#13095488—21no—
38#13095488—21no—
39C4×Dic97—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C4⋊C4×C97C2×C411yescharacteristic
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97—21no—
50C23×C97C22⋊C411yescharacteristic
51C22×D97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C22⋊D97—24no—
59C2×Dic97—21no—
60C2×Dic97—22no—
61C2×Dic97—41no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66D388—21no—
67D388M1612yes—
68C4×D97—24no—
69C4×D97—42no—
70C4×D97—42no—
71Dic194—21no—
72Dic194M1612yes—
73#11715897—42no—
74#11715897—42no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97C2×C22⋊C411yescharacteristic
79C2×D97—22no—
80C2×D97—41no—
81C2×D97—41no—
82C4×C97—21no—
83C4×C97—41no—
84C4×C97—41no—
85C4×C97C2×M1611yescharacteristic
86C4×C97C2×M1611yescharacteristic
87Dic97—22no—
88Dic97—21no—
89Dic97—21no—
90Dic97—21no—
91Dic97#13512yes—
92C2×C97—41no—
93C2×C97#40711yescharacteristic
94C2×C97#41611yescharacteristic
95C2×C97#42111yescharacteristic
96D97—42no—
97#1488—971nomaximal, Sylow
98C97#148811yescharacteristic, complemented, Sylow
99C4×#41—971no—
100#432—971no—
101#431—971no—
102Q8⋊C8—972no—
103D4⋊C8—972no—
104C2×#41—971no—
105C4×Q8—971no—
106C4×Q8—971no—
107C4×D4—971no—
108C4×D4—971no—
109C4×D4—1941no—
110C4×D4—1941no—
111#117—1941no—
112#117#8760411yescharacteristic
113C2×C42—971no—
114C2×C42—1941no—
115C4⋊C8—972no—
116C4⋊C8—1941no—
117C22⋊C8—1941no—
118C22⋊C8—1941no—
119#97—1941no—
120C4×C8—972no—
121#41—1942no—
122#41—1942no—
123C2×Q8—971no—
124C2×D4—971no—
125C2×D4—1941no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—1941no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4C2×#8760411yescharacteristic
132C2×C8—1942no—
133C2×C8—1942no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4C2×#8760411yescharacteristic
137C4⋊C4C2×#8760411yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—1941no—
140C22⋊C4—1941no—
141C42—21no—
142C42—971no—
143C42—971no—
144C42—971no—
145C42—971no—
146C42—1941no—
147C42—1941no—
148C23—1941no—
149C23C22⋊#8760411yescharacteristic
150Q8—972no—
151Q8—1941no—
152D4—972no—
153D4—1944no—
154D4—1941no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4—3881no—
166C2×C4—1944no—
167C2×C4—1942no—
168C2×C4—1941no—
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22×#8760411yescharacteristic
171C8—3882no—
172C8—3882no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—3881no—
177C22—3881no—
178C22—1942no—
179C22C2×C22⋊#8760411yescharacteristic
180C4—21no—
181C4—41no—
182C4—41no—
183C4—972no—
184C4—1942no—
185C4—1941no—
186C4—1941no—
187C4—1941no—
188C4D97⋊M1611yescharacteristic
189C4D97⋊M1611yescharacteristic
190C2—41no—
191C2—3882no—
192C2#2026170711yescharacteristic
193C2#2026172511yescharacteristic
194C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095511C4
cosocleC4×Dic97C23
hypercenter#117#87604
hypocenterC97#1488
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. #20261752C2semidirect
3. #20261585C2semidirect
4. #20261583C2semidirect
98. C97#1488semidirect

Maximal subgroups

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

subgroupindex
#202617562
#202617522
#202615852
#202615832
#202612032
#148897