Finite Group Databasebeta

← #27157859

Subgroups — #27157859

Order 12416.

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

normal subgroups only

This group has 396 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
1C2×#15690877C211yesmaximal, characteristic, complemented
2C2×#15690867C212yesmaximal, complemented
3C2×C22⋊#11715897C212yesmaximal, complemented
4C2×C4×#11715897C211yesmaximal, characteristic, complemented
5C2×#15690770C211yesmaximal, characteristic
6C22×#11715897C2212yes—
7C22×#11715897C2212yes—
8#15690877—22no—
9#15690867—24no—
10C22⋊#11715897—24no—
11C4×#11715897—22no—
12C22×C22⋊D97C411yescharacteristic
13C23×Dic97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15#15690770—22no—
16#15690770—22no—
17C2×#13095490C411yescharacteristic
18C2×#13095488C411yescharacteristic
19C2×C4×Dic97C2211yescharacteristic
20C2×C22⋊C4×C97C411yescharacteristic
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897D414yescomplemented
27C23×D97C2×C411yescharacteristic
28C2×C22⋊D97—22no—
29C2×C22⋊D97—22no—
30C2×C22⋊D97C814yescomplemented
31C22×Dic97—21no—
32C22×Dic97—21no—
33C22×Dic97—22no—
34C22×Dic97—22no—
35C22×Dic97C2×C411yescharacteristic
36C22×Dic97C2×C411yescharacteristic
37C22×Dic97C2311yescharacteristic
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97—22no—
41C2×C4×D97—22no—
42#13095490—22no—
43#13095488—22no—
44C4×Dic97D412yes—
45C4×Dic97D412yes—
46C24×C97C2×C411yescharacteristic
47C22×C4×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22⋊C4×C97—22no—
50C23×C97—21no—
51C23×C97—22no—
52C23×C97—21no—
53C23×C97—22no—
54C23×C97C2×C812yes—
55C23×C97C22×C411yescharacteristic
56C22×D97—21no—
57C22×D97—22no—
58C22×D97—21no—
59C22×D97—22no—
60C22×D97C2×C812yes—
61C2×C4×C97—22no—
62C2×C4×C97—21no—
63C2×C4×C97—22no—
64C2×C4×C97C22⋊C412yes—
65C22⋊D97—28no—
66C2×Dic97—21no—
67C2×Dic97—22no—
68C2×Dic97—42no—
69C2×Dic97—41no—
70C2×Dic97—41no—
71C2×Dic97C22⋊C412yes—
72C2×Dic97C2×C812yes—
73C2×Dic97M1612yes—
74C2×Dic97C2×D411yescharacteristic
75C2×Dic97C2×D412yes—
76C2×Dic97C2×D411yescharacteristic
77C2×Dic97#4112yes—
78C4×D97—42no—
79C4×D97—41no—
80C4×D97—41no—
81#11715897—24no—
82#11715897—42no—
83C22×C97—22no—
84C22×C97—24no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97—42no—
88C22×C97C2×C22⋊C411yescharacteristic
89C22×C97C4×D412yes—
90C22×C97C22×C811yescharacteristic
91C22×C97C2×M1611yescharacteristic
92C22×C97#13112yes—
93C2×D97—22no—
94C2×D97—24no—
95C2×D97—41no—
96C2×D97—41no—
97C2×D97—42no—
98C4×C97—22no—
99C4×C97—41no—
100Dic97—41no—
101Dic97C22⋊C814yescomplemented
102Dic97#12112yes—
103Dic97#12411yescharacteristic
104Dic97#12711yescharacteristic
105C2×C97—42no—
106C2×C97#39011yescharacteristic
107C2×C97C2×C22⋊C811yescharacteristic
108C2×C97#40811yescharacteristic
109C2×C97C8×D412yes—
110C2×C97#43612yes—
111D97—42no—
112#1954—971nomaximal, Sylow
113C97#195411yescharacteristic, complemented, Sylow
114C2×C4×D4—971no—
115C2×C4⋊C8—971no—
116C2×C22⋊C8—972no—
117C2×C22⋊C8—972no—
118C2×C4×C8—971no—
119C22×D4—971no—
120C23×C4—971no—
121C23×C4—971no—
122C22×C8—972no—
123C22×C8—972no—
124C4×D4—1942no—
125C4×D4—1942no—
126C2×C4⋊C4—971no—
127C2×C22⋊C4—971no—
128C2×C22⋊C4#8760411yescharacteristic
129C2×C42—971no—
130C4⋊C8—1942no—
131C22⋊C8—1944no—
132C22⋊C8—1944no—
133C4×C8—1942no—
134C24—971no—
135C24C2×#8760411yescharacteristic
136C2×D4—974no—
137C2×D4—1942no—
138C2×D4—1942no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—1942no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4—1942no—
146C22×C4—1942no—
147C22×C4—1941no—
148C22×C4—1941no—
149C22×C4—1942no—
150C22×C4C2×#8760411yescharacteristic
151C22×C4C2×#8760411yescharacteristic
152C2×C8—974no—
153C2×C8—1942no—
154C2×C8—1942no—
155C2×C8—1942no—
156C2×C8—1942no—
157C2×C8—1942no—
158C4⋊C4—1942no—
159C22⋊C4—22no—
160C22⋊C4—1942no—
161C42—972no—
162C42—972no—
163C23—21no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—972no—
168C23—1941no—
169C23—1942no—
170C23—1941no—
171C23—1942no—
172C23#1309551912yes—
173C23C22×#8760411yescharacteristic
174D4—1948no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—972no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—972no—
182C2×C4—972no—
183C2×C4—972no—
184C2×C4—972no—
185C2×C4—3881no—
186C2×C4—3882no—
187C2×C4—3881no—
188C2×C4—3881no—
189C2×C4—3882no—
190C2×C4—3881no—
191C2×C4—1941no—
192C2×C4—1942no—
193C2×C4C22⋊#8760412yes—
194C8—3882no—
195C8—1944no—
196C22—22no—
197C22—24no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—3881no—
202C22—3881no—
203C22—3882no—
204C22—1942no—
205C22—1944no—
206C22C2×C22⋊#8760411yescharacteristic
207C22D4×#8760412yes—
208C22C2×#1309551911yescharacteristic
209C22D97⋊M1611yescharacteristic
210C22#1569089712yes—
211C4—22no—
212C4—41no—
213C4—971no—
214C4—972no—
215C4—971no—
216C4—974no—
217C4—3881no—
218C2—42no—
219C2—3882no—
220C2#2026168711yescharacteristic
221C2#2026170311yescharacteristic
222C2#2026170811yescharacteristic
223C2#2026176212yes—
224C2#2026176812yes—
subgroupG/H
center, FrattiniC23C22×#87604
commutatorC22×C97C22×C8
FittingC2×C22⋊C4×C97C4
socleC23×C97C22×C4
prime radicalC22×C22⋊D97C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1954
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15690877C2semidirect
2. C2×#15690867C2semidirect
3. C2×C22⋊#11715897C2semidirect
4. C2×C4×#11715897C2semidirect
26. C2×#11715897D4semidirect
30. C2×C22⋊D97C8semidirect
101. Dic97C22⋊C8semidirect
113. C97#1954semidirect

Maximal subgroups

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

subgroupindex
C2×#156908772
C2×#156908672
C2×C22⋊#117158972
C2×C4×#117158972
C2×#156907702
#195497