Finite Group Databasebeta

← #27142722

Subgroups — #27142722

Order 12416.

324 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 101 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#20260595C211yesmaximal, characteristic
2C4×C22⋊Dic97C211yesmaximal, characteristic
3#20260502C214yesmaximal
4C2×C22⋊Q8×C97C211yesmaximal, characteristic
5C2×C4⋊Dic97—21no—
6C2×C22⋊Dic97C2212yes—
7C2×C22⋊Dic97C2212yes—
8C2×C4×Dic97—21no—
9#15690717—21no—
10#15690717—21no—
11C22×Q8×C97C411yescharacteristic, complemented
12C23×C4×C97C2211yescharacteristic
13C22⋊Q8×C97—21no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97—21no—
16C22⋊Q8×C97—21no—
17C2×C4⋊C4×C97C411yescharacteristic, complemented
18C2×C4⋊C4×C97C412yescomplemented
19C2×C22⋊C4×C97C2212yes—
20C22×Dic97—22no—
21C22×Dic97—22no—
22C4⋊Dic97—42no—
23C22⋊Dic97—22no—
24C22⋊Dic97—24no—
25C22⋊Dic97Q814yescomplemented
26C4×Dic97—42no—
27C24×C97C2311yescharacteristic
28C2×Q8×C97—21no—
29C2×Q8×C97—21no—
30C2×Q8×C97—42no—
31C22×C4×C97—42no—
32C22×C4×C97C2×C411yescharacteristic
33C22×C4×C97C2×C412yes—
34C22×C4×C97C2×C412yes—
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97D411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C4⋊C4×C97—21no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97—22no—
43C4⋊C4×C97—22no—
44C22⋊C4×C97—22no—
45C22⋊C4×C97—22no—
46C23×C97—41no—
47C23×C97—41no—
48C23×C97C22×C411yescharacteristic
49C23×C97C2×D411yescharacteristic
50C23×C97C2×D411yescharacteristic
51C23×C97C2×Q812yes—
52C23×C97#4112yes—
53Q8×C97—42no—
54Q8×C97—42no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—42no—
64C2×C4×C97—42no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C22⋊C411yescharacteristic
69C2×Dic97—24no—
70C2×Dic97—42no—
71C2×Dic97—42no—
72C2×Dic97—42no—
73C2×Dic97—42no—
74C2×Dic97—42no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97C4×Q812yes—
85C22×C97C22⋊Q812yes—
86C22×C97#12411yescharacteristic
87C22×C97C4⋊Q811yescharacteristic
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—42no—
92C4×C97—42no—
93C4×C97#9912yes—
94Dic97—44no—
95Dic97—82no—
96C2×C97—22no—
97C2×C97—21no—
98C2×C97—22no—
99C2×C97—21no—
100C2×C97#39111yescharacteristic
101C2×C97C2×#9911yescharacteristic
102C2×C97#41011yescharacteristic
103#1996—971nomaximal, Sylow
104C97#199611yescharacteristic, complemented, Sylow
105C2×C22⋊Q8D9711yescharacteristic
106#380—971no—
107C4×C22⋊C4—971no—
108#342—974no—
109C22×Q8Dic9711yescharacteristic, complemented
110C23×C4C2×D9711yescharacteristic
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C2×C4⋊C4—1941no—
116C2×C4⋊C4Dic9711yescharacteristic, complemented
117C2×C4⋊C4Dic9712yescomplemented
118C2×C22⋊C4—972no—
119C2×C22⋊C4—972no—
120C2×C22⋊C4C2×D9712yes—
121C2×C42—1941no—
122#95—1941no—
123#95—1941no—
124C24C22×D9711yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—42no—
129C22×C4—1942no—
130C22×C4—1942no—
131C22×C4C2×Dic9711yescharacteristic
132C22×C4C2×Dic9712yes—
133C22×C4C2×Dic9712yes—
134C22×C4C2×Dic9711yescharacteristic
135C22×C4C22⋊D9711yescharacteristic
136C22×C4C22⋊D9711yescharacteristic
137C22×C4C22⋊D9711yescharacteristic
138C22×C4C22⋊D9711yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—3882no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—974no—
147C22⋊C4—1944no—
148C22⋊C4—1942no—
149C42—3882no—
150C23—41no—
151C23—41no—
152C23C22×Dic9711yescharacteristic
153C23C2×C22⋊D9711yescharacteristic
154C23C2×C22⋊D9711yescharacteristic
155C23Q8×D9712yes—
156C23#1309551212yes—
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—3882no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—3882no—
173C2×C4—3882no—
174C2×C4—1944no—
175C2×C4C22⋊Dic9711yescharacteristic
176C2×C4C22⋊Dic9711yescharacteristic
177C2×C4C22⋊Dic9711yescharacteristic
178C2×C4C22⋊Dic9711yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22C2×C22⋊Dic9711yescharacteristic
188C22Q8×Dic9712yes—
189C22#1569078612yes—
190C22#1569079911yescharacteristic
191C22#1569080811yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—42no—
196C4—42no—
197C4—7762no—
198C4—3884no—
199C4#1569072512yes—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2#2026064211yescharacteristic
205C2C2×#1569072511yescharacteristic
206C2#2026070911yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC23×C97C22×C4
FrattiniC24C22×D97
FittingC2×C22⋊Q8×C97C2
socleC22×C97C2×C22⋊C4
prime radical, cosocleC24×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97#1996
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
11. C22×Q8×C97C4semidirect
17. C2×C4⋊C4×C97C4semidirect
18. C2×C4⋊C4×C97C4semidirect
25. C22⋊Dic97Q8semidirect
104. C97#1996semidirect
109. C22×Q8Dic97semidirect
116. C2×C4⋊C4Dic97semidirect
117. C2×C4⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202605952
C4×C22⋊Dic972
#202605022
C2×C22⋊Q8×C972
#199697