Finite Group Databasebeta

← #27142195

Subgroups — #27142195

Order 12416.

378 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15690769C211yesmaximal, characteristic, complemented
2C2×#15690725C212yesmaximal
3#20260598C211yesmaximal, characteristic, complemented
4#20260502C212yesmaximal, complemented
5C2×#123×C97C211yesmaximal, characteristic
6#15690769—24no—
7C2×C4⋊Dic97—21no—
8C2×C22⋊Dic97C2212yes—
9C2×C22⋊Dic97C2212yes—
10C2×C4×Dic97—21no—
11#15690725—24no—
12#15690717—22no—
13C22×D4×C97C2211yescharacteristic
14C23×C4×C97C2211yescharacteristic
15#123×C97—21no—
16#123×C97—21no—
17#123×C97—22no—
18C2×C4⋊C4×C97C412yescomplemented
19C2×C22⋊C4×C97C412yescomplemented
20C2×C22⋊C4×C97C2211yescharacteristic
21C22×Dic97—22no—
22C22×Dic97—22no—
23C4⋊Dic97—24no—
24C22⋊Dic97—22no—
25C22⋊Dic97—22no—
26C22⋊Dic97—22no—
27C22⋊Dic97D414yes—
28C4×Dic97—24no—
29C24×C97C2×C411yescharacteristic
30C24×C97C2311yescharacteristic
31C2×D4×C97—21no—
32C2×D4×C97—21no—
33C2×D4×C97—22no—
34C2×D4×C97—22no—
35C22×C4×C97—22no—
36C22×C4×C97—22no—
37C22×C4×C97C2×C411yescharacteristic
38C22×C4×C97C2×C412yes—
39C22×C4×C97C2×C412yes—
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D412yes—
43C4⋊C4×C97—22no—
44C4⋊C4×C97—22no—
45C22⋊C4×C97—21no—
46C22⋊C4×C97—21no—
47C22⋊C4×C97—22no—
48C22⋊C4×C97—22no—
49C23×C97—21no—
50C23×C97—22no—
51C23×C97—42no—
52C23×C97—42no—
53C23×C97C22×C411yescharacteristic
54C23×C97C2×D412yes—
55C23×C97C2×D411yescharacteristic
56C23×C97C2×D411yescharacteristic
57C23×C97#4112yes—
58D4×C97—41no—
59D4×C97—42no—
60D4×C97—41no—
61C2×C4×C97—24no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—24no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—22no—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97—42no—
71C2×C4×C97C22⋊C412yes—
72C2×C4×C97C22⋊C412yes—
73C2×Dic97—24no—
74C2×Dic97—44no—
75C2×Dic97—44no—
76C2×Dic97—42no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—22no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97—41no—
83C22×C97—42no—
84C22×C97—82no—
85C22×C97C2×C22⋊C411yescharacteristic
86C22×C97C4×D412yes—
87C22×C97#12011yescharacteristic
88C22×C97#12112yes—
89C22×C97#12311yescharacteristic
90C4×C97—22no—
91C4×C97—22no—
92C4×C97—41no—
93C4×C97—42no—
94Dic97—44no—
95Dic97—82no—
96C2×C97—21no—
97C2×C97—22no—
98C2×C97—22no—
99C2×C97—21no—
100C2×C97—81no—
101C2×C97#38611yescharacteristic
102C2×C97#41012yes—
103#1899—971nomaximal, Sylow
104C97#189911yescharacteristic, complemented, Sylow
105C2×#123D9711yescharacteristic
106C2×#117—971no—
107C2×#99—972no—
108#381—971no—
109#342—972no—
110C22×D4C2×D9711yescharacteristic
111C23×C4C2×D9711yescharacteristic
112#123—22no—
113#123—21no—
114#123—21no—
115#117—1944no—
116C2×C4⋊C4—1941no—
117C2×C4⋊C4Dic9712yescomplemented
118C2×C22⋊C4—972no—
119C2×C22⋊C4—972no—
120C2×C22⋊C4Dic9712yescomplemented
121C2×C22⋊C4C2×D9711yescharacteristic
122C2×C42—1941no—
123#99—1944no—
124#95—1942no—
125C24C2×Dic9711yescharacteristic
126C24C22×D9711yescharacteristic
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—1942no—
134C22×C4—1942no—
135C22×C4C2×Dic9711yescharacteristic
136C22×C4C2×Dic9712yes—
137C22×C4C2×Dic9712yes—
138C22×C4C22⋊D9712yes—
139C22×C4C22⋊D9711yescharacteristic
140C22×C4C22⋊D9711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—1944no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—974no—
149C22⋊C4—1942no—
150C22⋊C4—1942no—
151C22⋊C4—1942no—
152C42—1944no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23C22×Dic9711yescharacteristic
158C23D4×D9712yes—
159C23C2×C22⋊D9711yescharacteristic
160C23C2×C22⋊D9711yescharacteristic
161C23#1309551012yes—
162D4—41no—
163D4—42no—
164D4—41no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—3884no—
176C2×C4—3882no—
177C2×C4—3884no—
178C2×C4—1944no—
179C2×C4C22⋊Dic9712yes—
180C2×C4C22⋊Dic9712yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—42no—
188C22—82no—
189C22C2×C22⋊Dic9711yescharacteristic
190C22D4×Dic9712yes—
191C22#1569077811yescharacteristic
192C22#1569078212yes—
193C22#1569079111yescharacteristic
194C4—22no—
195C4—22no—
196C4—41no—
197C4—42no—
198C4—7762no—
199C4—3884no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2#2026061511yescharacteristic
206C2#2026070912yes—
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC23×C97C22×C4
FrattiniC24C22×D97
FittingC2×#123×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D4×C97C22
cosocleC24×C97C23
hypercenterC2×#123D97
hypocenterC97#1899
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#15690769C2semidirect
3. #20260598C2semidirect
4. #20260502C2semidirect
18. C2×C4⋊C4×C97C4semidirect
19. C2×C22⋊C4×C97C4semidirect
104. C97#1899semidirect
117. C2×C4⋊C4Dic97semidirect
120. C2×C22⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
C2×#156907252
#202605982
#202605022
C2×#123×C972
#189997