Finite Group Databasebeta

← #27157875

Subgroups — #27157875

Order 12416.

408 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×#15690868C214yesmaximal, complemented
2#20261677C211yesmaximal, characteristic, complemented
3C4×C22⋊#87604C211yesmaximal, characteristic, complemented
4C2×#15690781C211yesmaximal, characteristic
5C2×C4⋊#87604—21no—
6C2×C22⋊#87604C2212yes—
7C2×C22⋊#87604C2212yes—
8C2×C4×#87604—21no—
9#15690868—28no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4—21no—
12C22×C22⋊D97C2212yes—
13C22×C4×D97C2211yescharacteristic
14#15690781—21no—
15#15690781—21no—
16#15690781—21no—
17#15690781—21no—
18C2×C4⋊Dic97C411yescharacteristic, complemented
19C2×C22⋊Dic97C412yescomplemented
20C22×D4×C97C411yescharacteristic, complemented
21C22×#87604—22no—
22C22×#87604—22no—
23C4⋊#87604—42no—
24C22⋊#87604—22no—
25C22⋊#87604—24no—
26C22⋊#87604D414yescomplemented
27C4×#87604—42no—
28C23×D97C2311yescharacteristic
29C2×C22⋊D97—22no—
30C2×C22⋊D97—22no—
31C2×C22⋊D97—24no—
32C2×C22⋊D97—24no—
33C22×Dic97C2×C412yes—
34C22×Dic97C2×C411yescharacteristic
35C2×C4×D97—42no—
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C4⋊Dic97—21no—
41C4⋊Dic97—21no—
42C22⋊Dic97—22no—
43C22⋊Dic97—22no—
44C24×C97C2×C412yes—
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97—42no—
48C22×C4×C97C2×C411yescharacteristic
49C23×C97—22no—
50C23×C97—24no—
51C23×C97—42no—
52C23×C97—42no—
53C23×C97C22×C411yescharacteristic
54C22×D97—41no—
55C22×D97—41no—
56C22×D97C2×D411yescharacteristic
57C22×D97C2×D411yescharacteristic
58C22×D97C2×D412yes—
59C22×D97#4112yes—
60C2×#87604—24no—
61C2×#87604—42no—
62C2×#87604—42no—
63C2×#87604—42no—
64C2×#87604—42no—
65C2×#87604—42no—
66D4×C97—42no—
67D4×C97—42no—
68C2×C4×C97—22no—
69C2×C4×C97C22⋊C411yescharacteristic
70C2×C4×C97C22⋊C411yescharacteristic
71C22⋊D97—44no—
72C22⋊D97—42no—
73C22⋊D97—42no—
74C2×Dic97—22no—
75C2×Dic97—24no—
76C2×Dic97—41no—
77C2×Dic97C22⋊C411yescharacteristic
78C2×Dic97C22⋊C411yescharacteristic
79C4×D97—22no—
80C4×D97—22no—
81C4×D97—41no—
82C4×D97—41no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—42no—
86C22×C97—42no—
87C22×C97—42no—
88C22×C97—84no—
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97C4×D412yes—
91C2×D97—21no—
92C2×D97—21no—
93C2×D97—21no—
94C2×D97—21no—
95C2×D97—42no—
96C2×D97—42no—
97C2×D97#12112yes—
98C2×D97#12411yescharacteristic
99C2×D97#12711yescharacteristic
100#87604—44no—
101#87604—82no—
102C4×C97—21no—
103C4×C97#9912yescomplemented
104Dic97—21no—
105Dic97—21no—
106Dic97—42no—
107C2×C97—22no—
108C2×C97—82no—
109C2×C97#39011yescharacteristic
110C2×C97C2×#9911yescharacteristic
111C2×C97#41011yescharacteristic
112D97—22no—
113D97—21no—
114D97—21no—
115#1964—971nomaximal, Sylow
116C97#196411yescharacteristic, complemented, Sylow
117C2×#121—971no—
118C2×#99—974no—
119#380—971no—
120C4×C22⋊C4—971no—
121C22×D4—972no—
122C22×D4#8760411yescharacteristic, complemented
123C23×C4—971no—
124#121—1941no—
125#121—1941no—
126#121—1941no—
127#121—1941no—
128C2×C4⋊C4—971no—
129C2×C4⋊C4—1941no—
130C2×C22⋊C4—972no—
131C2×C22⋊C4—972no—
132C2×C22⋊C4—972no—
133C2×C42—1941no—
134#99—1948no—
135#95—1941no—
136#95—1941no—
137C24—971no—
138C24C2×#8760412yes—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—1944no—
143C2×D4—1942no—
144C2×D4—1942no—
145C2×D4—1944no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—971no—
149C22×C4—971no—
150C22×C4—971no—
151C22×C4—972no—
152C22×C4—3882no—
153C22×C4—1942no—
154C22×C4—1942no—
155C22×C4C2×#8760411yescharacteristic
156C4⋊C4—3882no—
157C4⋊C4—1941no—
158C4⋊C4—1941no—
159C22⋊C4—974no—
160C22⋊C4—1942no—
161C22⋊C4—1942no—
162C22⋊C4—1944no—
163C22⋊C4—1942no—
164C42—3882no—
165C23—22no—
166C23—24no—
167C23—42no—
168C23—42no—
169C23—972no—
170C23—972no—
171C23—971no—
172C23—971no—
173C23—3881no—
174C23—3881no—
175C23C22×#8760411yescharacteristic
176D4—42no—
177D4—42no—
178D4—3882no—
179D4—3884no—
180D4—3882no—
181C2×C4—22no—
182C2×C4—971no—
183C2×C4—971no—
184C2×C4—3881no—
185C2×C4—3881no—
186C2×C4—3882no—
187C2×C4—3882no—
188C2×C4—3882no—
189C2×C4—3882no—
190C2×C4—3882no—
191C2×C4—3881no—
192C2×C4—1944no—
193C2×C4—1942no—
194C2×C4—1944no—
195C2×C4—1942no—
196C2×C4—1942no—
197C2×C4C22⋊#8760411yescharacteristic
198C2×C4C22⋊#8760411yescharacteristic
199C22—21no—
200C22—21no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—84no—
205C22—971no—
206C22—972no—
207C22—971no—
208C22—3882no—
209C22—3882no—
210C22—1941no—
211C22—1941no—
212C22—1941no—
213C22—1941no—
214C22C2×C22⋊#8760411yescharacteristic
215C22D4×#8760412yes—
216C4—21no—
217C4—7762no—
218C4—3884no—
219C4—3882no—
220C4—1941no—
221C4—1941no—
222C4#1569086812yescomplemented
223C2—22no—
224C2—82no—
225C2—1941no—
226C2—1942no—
227C2—1941no—
228C2#2026168711yescharacteristic
229C2C2×#1569086811yescharacteristic
230C2#2026171511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690781C2
cosocleC23×D97C23
hypercenterC22×D4#87604
hypocenterC97#1964
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690868C2semidirect
2. #20261677C2semidirect
3. C4×C22⋊#87604C2semidirect
18. C2×C4⋊Dic97C4semidirect
19. C2×C22⋊Dic97C4semidirect
20. C22×D4×C97C4semidirect
26. C22⋊#87604D4semidirect
103. C4×C97#99semidirect
116. C97#1964semidirect
122. C22×D4#87604semidirect
222. C4#15690868semidirect

Maximal subgroups

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

subgroupindex
C2×#156908682
#202616772
C4×C22⋊#876042
C2×#156907812
#196497