Finite Group Databasebeta

← #27157712

Subgroups — #27157712

Order 12416.

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

normal subgroups only

This group has 432 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
1C22×#13095519C211yesmaximal, characteristic, complemented
2C2×#15690867C212yesmaximal, complemented
3C2×#15690867C211yesmaximal, characteristic, complemented
4#20261601C212yesmaximal, complemented
5C2×#15690772C211yesmaximal, characteristic
6C2×#13095519—24no—
7C2×#13095519—22no—
8C22×#11715897C2212yes—
9C22×#11715897C2212yes—
10#15690867—24no—
11#15690867—22no—
12C22×D388C411yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15#15690772—22no—
16#15690772—22no—
17C2×#13095488C411yescharacteristic
18C2×#13095488C411yescharacteristic
19C2×C4×Dic97C2211yescharacteristic
20C2×C4⋊C4×C97C411yescharacteristic
21#13095519—28no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C2×#11715897D414yescomplemented
28C23×D97C2×C411yescharacteristic
29C23×D97C2×C411yescharacteristic
30C2×D388—22no—
31C2×D388—22no—
32C2×D388C814yescomplemented
33C22×Dic97C2×C411yescharacteristic
34C22×Dic97C2311yescharacteristic
35C2×C4×D97—22no—
36C2×C4×D97—22no—
37C2×C4×D97—21no—
38C2×C4×D97—21no—
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41C2×C4×D97D412yes—
42C2×C4×D97D412yes—
43#13095488—22no—
44#13095488—22no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C22×C4×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C4⋊C4×C97—22no—
51C23×C97C22×C411yescharacteristic
52C22×D97—22no—
53C22×D97—22no—
54C22×D97—21no—
55C22×D97—22no—
56C22×D97—21no—
57C22×D97—22no—
58C22×D97C22⋊C412yes—
59C22×D97C2×C812yes—
60C22×D97C2×C812yescomplemented
61C22×D97M1612yes—
62C2×C4×C97—21no—
63C2×C4×C97—22no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—22no—
67C2×C4×C97C22⋊C412yes—
68C2×C4×C97C2×C812yescomplemented
69C2×Dic97—21no—
70C2×Dic97—22no—
71C2×Dic97C2×D412yes—
72C2×Dic97C2×D411yescharacteristic
73C2×Dic97C2×D411yescharacteristic
74C2×Dic97#4112yes—
75D388—28no—
76C4×D97—22no—
77C4×D97—24no—
78C4×D97—22no—
79C4×D97—41no—
80C4×D97—42no—
81C4×D97—41no—
82#11715897—24no—
83#11715897—42no—
84C22×C97C2×C22⋊C411yescharacteristic
85C22×C97C4×D412yes—
86C22×C97C22×C811yescharacteristic
87C22×C97C2×M1611yescharacteristic
88C22×C97#13112yes—
89C2×D97—22no—
90C2×D97—22no—
91C2×D97—24no—
92C2×D97—24no—
93C2×D97—22no—
94C2×D97—24no—
95C2×D97—41no—
96C2×D97—41no—
97C2×D97—42no—
98C2×D97C22⋊C814yescomplemented
99C4×C97—22no—
100C4×C97—22no—
101C4×C97—41no—
102Dic97—41no—
103Dic97#12011yescharacteristic
104Dic97#12112yes—
105Dic97#12311yescharacteristic
106C2×C97#38611yescharacteristic
107C2×C97C2×C22⋊C811yescharacteristic
108C2×C97#40811yescharacteristic
109C2×C97C8×D412yes—
110C2×C97#43512yes—
111D97—24no—
112D97—42no—
113#1869—971nomaximal, Sylow
114C97#186911yescharacteristic, complemented, Sylow
115C23×C8—971no—
116C2×C4×D4—971no—
117C2×C22⋊C8—972no—
118C2×C22⋊C8—971no—
119#336—972no—
120C22×D4—971no—
121C23×C4—971no—
122C23×C4—971no—
123C22×C8—972no—
124C22×C8—972no—
125C22×C8—1944no—
126C22×C8—1942no—
127C4×D4—1942no—
128C4×D4—1942no—
129C2×C4⋊C4#8760411yescharacteristic
130C2×C22⋊C4—971no—
131C2×C22⋊C4—971no—
132C2×C42—971no—
133C22⋊C8—1944no—
134C22⋊C8—1942no—
135C24—971no—
136C24—971no—
137C2×D4—974no—
138C2×D4—1942no—
139C2×D4—1942no—
140C22×C4—971no—
141C22×C4—972no—
142C22×C4—972no—
143C22×C4—971no—
144C22×C4—1942no—
145C22×C4—1941no—
146C22×C4—1941no—
147C22×C4—1942no—
148C22×C4—1942no—
149C22×C4—1942no—
150C22×C4C2×#8760411yescharacteristic
151C22×C4C2×#8760411yescharacteristic
152C22×C4C2×#8760411yescharacteristic
153C2×C8—974no—
154C2×C8—1942no—
155C2×C8—1942no—
156C2×C8—1942no—
157C2×C8—1942no—
158C2×C8—1942no—
159C2×C8—1948no—
160C4⋊C4—22no—
161C22⋊C4—1942no—
162C22⋊C4—1942no—
163C42—1941no—
164C42—1941no—
165C23—972no—
166C23—972no—
167C23—972no—
168C23—972no—
169C23—1941no—
170C23—1942no—
171C23—1941no—
172C23—1942no—
173C23—1942no—
174C23—1942no—
175C23C22×#8760411yescharacteristic
176D4—1948no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—971no—
183C2×C4—971no—
184C2×C4—972no—
185C2×C4—972no—
186C2×C4—3882no—
187C2×C4—3881no—
188C2×C4—3881no—
189C2×C4—1942no—
190C2×C4—1944no—
191C2×C4—1942no—
192C2×C4—1941no—
193C2×C4—1942no—
194C2×C4C22⋊#8760412yes—
195C2×C4#1309551912yescomplemented
196C8—3882no—
197C8—1944no—
198C22—974no—
199C22—3881no—
200C22—3881no—
201C22—3882no—
202C22—1942no—
203C22—1944no—
204C22—1942no—
205C22—1942no—
206C22—1944no—
207C22—1944no—
208C22C2×C22⋊#8760411yescharacteristic
209C22D4×#8760412yes—
210C22C2×#1309551911yescharacteristic
211C22D97⋊M1611yescharacteristic
212C22#1569089812yes—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—972no—
217C4—971no—
218C4—971no—
219C4—3881no—
220C2—3882no—
221C2—1944no—
222C2#2026168311yescharacteristic
223C2#2026170311yescharacteristic
224C2#2026171011yescharacteristic
225C2#2026176312yes—
226C2#2026176512yes—
subgroupG/H
center, FrattiniC23C22×#87604
commutatorC22×C97C22×C8
FittingC2×C4⋊C4×C97C4
socleC23×C97C22×C4
prime radicalC22×D388C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1869
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13095519C2semidirect
2. C2×#15690867C2semidirect
3. C2×#15690867C2semidirect
4. #20261601C2semidirect
27. C2×#11715897D4semidirect
32. C2×D388C8semidirect
60. C22×D97C2×C8semidirect
68. C2×C4×C97C2×C8semidirect
98. C2×D97C22⋊C8semidirect
114. C97#1869semidirect
195. C2×C4#13095519semidirect

Maximal subgroups

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

subgroupindex
C22×#130955192
C2×#156908672
#202616012
C2×#156907722
#186997