Finite Group Databasebeta

← #27157534

Subgroups — #27157534

Order 12416.

272 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 104 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#20261736C211yesmaximal, characteristic
2C2×#15690864C211yesmaximal, characteristic
3C2×C8×#87604C211yesmaximal, characteristic
4C4×C4⋊#87604C211yesmaximal, characteristic
5#20261600C211yesmaximal, characteristic
6#20261600C211yesmaximal, characteristic
7#97×D97C211yesmaximal, characteristic
8C2×#13095519C2211yescharacteristic
9C2×#13095519C2211yescharacteristic
10C2×C4⋊#87604C412yes—
11C2×C4×#87604C2211yescharacteristic
12C2×C4×#87604C2211yescharacteristic
13#15690877C412yescomplemented
14C4⋊#11715897C412yescomplemented
15#15690864—22no—
16C8×#87604—22no—
17(C2×#87604)⋊C4C412yes—
18C2×C8×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20C42×D97C2211yescharacteristic
21#15690721C411yescharacteristic, complemented
22#15690720C411yescharacteristic, complemented
23#15690720C411yescharacteristic, complemented
24#97×C97C411yescharacteristic, complemented
25C22×#87604C2×C412yes—
26C22×#87604C2×C412yes—
27#13095519—21no—
28#13095519—21no—
29#13095519—21no—
30#13095519—21no—
31C2×#11715897C2×C412yes—
32C2×#11715897C2×C412yes—
33C4⋊#87604—24no—
34C4×#87604—21no—
35C4×#87604—21no—
36C4×#87604—21no—
37C4×#87604—21no—
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97C2×C411yescharacteristic
40C2×C4×D97C2311yescharacteristic
41C2×#11715895C2×C411yescharacteristic
42C2×#11715895C2×C411yescharacteristic
43C8×D97—21no—
44C8×D97—21no—
45C8×D97D412yes—
46C8×D97Q812yes—
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97C2×C411yescharacteristic
49C4×Dic97C2×C411yescharacteristic
50C2×C8×C97C2×C411yescharacteristic
51C2×C8×C97C2×C411yescharacteristic
52C42×C97C2×C411yescharacteristic
53C22×D97C22×C411yescharacteristic
54C2×#87604—22no—
55C2×#87604—22no—
56C2×#87604—22no—
57C2×#87604—22no—
58C2×#87604—22no—
59C2×#87604M1614yes—
60C2×C4×C97C4211yescharacteristic
61C2×C4×C97C4211yescharacteristic
62C2×C4×C97C22×C411yescharacteristic
63C2×Dic97C4211yescharacteristic
64C2×Dic97C4211yescharacteristic
65C2×Dic97C22×C411yescharacteristic
66C4×D97—22no—
67C4×D97—22no—
68C4×D97C2×D411yescharacteristic
69C4×D97C2×Q811yescharacteristic
70C4×D97#4111yescharacteristic
71C4×D97#4111yescharacteristic
72#11715897—22no—
73#11715897—22no—
74C8×C97—21no—
75C8×C97C4⋊C412yescomplemented
76#11715895—21no—
77#11715895C4⋊C412yescomplemented
78C22×C97C2×C4211yescharacteristic
79C2×D97C4×D411yescharacteristic
80C2×D97C4×Q811yescharacteristic
81C2×D97C2×M1612yes—
82C2×D97#13112yes—
83#87604—24no—
84#87604—42no—
85C4×C97—21no—
86C4×C97—21no—
87C4×C97C2×C4⋊C411yescharacteristic
88C4×C97#11711yescharacteristic
89Dic97—21no—
90Dic97—21no—
91Dic97C4×D411yescharacteristic
92Dic97C4×Q811yescharacteristic
93C2×C97C4×C4⋊C411yescharacteristic
94C2×C97C4×M1611yescharacteristic
95C2×C97#40511yescharacteristic
96D97#43512yes—
97D97#44612yes—
98#1771—971nomaximal, Sylow
99C97#177111yescharacteristic, complemented, Sylow
100C2×C4⋊C8—971no—
101C2×#97—971no—
102C2×#97—971no—
103C2×C4×C8—971no—
104C4×C4⋊C4—971no—
105#336—971no—
106#336—971no—
107C22×C8—971no—
108C22×C8—971no—
109C22×C8—971no—
110C22×C8—971no—
111C2×C4⋊C4—972no—
112C2×C42—971no—
113C2×C42—971no—
114C2×C42—971no—
115C4⋊C8—972no—
116C4⋊C8—972no—
117#97—971no—
118#97—971no—
119#97—971no—
120#97—1942no—
121#97#8760411yescharacteristic, complemented
122C4×C8—1942no—
123#95—972no—
124C22×C4—971no—
125C22×C4—972no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—972no—
129C2×C8—971no—
130C2×C8—972no—
131C2×C8—972no—
132C2×C8—972no—
133C2×C8—972no—
134C2×C8—971no—
135C2×C8—1941no—
136C2×C8—1941no—
137C2×C8—1941no—
138C2×C8—1941no—
139C2×C8—1941no—
140C2×C8—1941no—
141C2×C8C2×#8760411yescharacteristic
142C2×C8C2×#8760411yescharacteristic
143C4⋊C4—1944no—
144C42—971no—
145C42—971no—
146C42—971no—
147C42—1941no—
148C42—1941no—
149C42—1941no—
150C42—1941no—
151C42C2×#8760411yescharacteristic
152C23—971no—
153C2×C4—971no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—971no—
158C2×C4—974no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—1942no—
162C2×C4—1942no—
163C2×C4—1942no—
164C2×C4—1942no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4C4×#8760411yescharacteristic
169C2×C4C4×#8760411yescharacteristic
170C2×C4C22×#8760411yescharacteristic
171C8—21no—
172C8—972no—
173C8—1942no—
174C8—1942no—
175C8—1941no—
176C8C4⋊#8760412yescomplemented
177C22—971no—
178C22—971no—
179C22—972no—
180C22—972no—
181C22C2×C4×#8760411yescharacteristic
182C4—21no—
183C4—21no—
184C4—971no—
185C4—971no—
186C4—3882no—
187C4—1944no—
188C4—1941no—
189C4—1941no—
190C4C2×C4⋊#8760411yescharacteristic
191C4#1569088911yescharacteristic
192C2—972no—
193C2—972no—
194C2C4×C4⋊#8760411yescharacteristic
195C2M16×#8760411yescharacteristic
196C2#2026170111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×#87604
commutator, socleC22×C97C2×C42
Fitting#97×C97C4
prime radicalC22×D97C22×C4
cosocleC2×C4×D97C23
hypercenter#97#87604
hypocenterC97#1771
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
13. #15690877C4semidirect
14. C4⋊#11715897C4semidirect
21. #15690721C4semidirect
22. #15690720C4semidirect
23. #15690720C4semidirect
24. #97×C97C4semidirect
75. C8×C97C4⋊C4semidirect
77. #11715895C4⋊C4semidirect
99. C97#1771semidirect
121. #97#87604semidirect
176. C8C4⋊#87604semidirect

Maximal subgroups

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

subgroupindex
#202617362
C2×#156908642
C2×C8×#876042
C4×C4⋊#876042
#202616002
#97×D972
#177197