Finite Group Databasebeta

← #27140256

Subgroups — #27140256

Order 12416.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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#20260699C211yesmaximal, characteristic
2C4×C22⋊Dic97C211yesmaximal, characteristic
3#20260472C214yesmaximal, complemented
4C2×C22⋊Q8×C97C211yesmaximal, characteristic
5C2×#13095495—21no—
6C2×C22⋊Dic97—21no—
7C2×C4×Dic97—21no—
8C22⋊#11715895—21no—
9C22⋊#11715895C2212yes—
10#15690717C2212yes—
11C22×Q8×C97C411yescharacteristic, complemented
12C23×C4×C97C2211yescharacteristic
13C22⋊Q8×C97—21no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97C412yescomplemented
16C22⋊Q8×C97C2212yes—
17C2×C4⋊C4×C97—21no—
18C2×C4⋊C4×C97C411yescharacteristic, complemented
19C2×C22⋊C4×C97—21no—
20C22×Dic97—22no—
21#13095495—42no—
22C2×#11715895—22no—
23C22⋊Dic97—24no—
24C4×Dic97—42no—
25C24×C97D411yescharacteristic
26C2×Q8×C97—21no—
27C2×Q8×C97—42no—
28C2×Q8×C97C2×C412yes—
29C22×C4×C97—21no—
30C22×C4×C97—21no—
31C22×C4×C97—42no—
32C22×C4×C97C2×C411yescharacteristic
33C22×C4×C97C2×C411yescharacteristic
34C22×C4×C97D411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C4⋊C4×C97—22no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97—22no—
41C4⋊C4×C97C2×C412yes—
42C22⋊C4×C97—22no—
43C22⋊C4×C97—22no—
44C23×C97—22no—
45C23×C97—22no—
46C23×C97—41no—
47C23×C97C22⋊C411yescharacteristic
48C23×C97C22⋊C411yescharacteristic
49C23×C97C2×D411yescharacteristic
50Q8×C97—42no—
51Q8×C97—42no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—42no—
61C2×C4×C97—42no—
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×Dic97—24no—
67C2×Dic97—42no—
68C2×Dic97—42no—
69#11715895—42no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—44no—
78C22×C97#10412yes—
79C22×C97C2×C22⋊C411yescharacteristic
80C22×C97#13712yes—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—42no—
85C4×C97—42no—
86C4×C97#9912yes—
87Dic97—44no—
88C2×C97—22no—
89C2×C97—22no—
90C2×C97—41no—
91C2×C97C2×#9911yescharacteristic
92C2×C97#41911yescharacteristic
93C2×C97C2×#10411yescharacteristic
94#1508—971nomaximal, Sylow
95C97#150811yescharacteristic, complemented, Sylow
96C2×C22⋊Q8D9711yescharacteristic
97#407—971no—
98C4×C22⋊C4—971no—
99#328—974no—
100C22×Q8Dic9711yescharacteristic, complemented
101C23×C4C2×D9711yescharacteristic
102C2×M16—1941no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8Dic9712yescomplemented
106C22⋊Q8C2×D9712yes—
107C2×C4⋊C4—21no—
108C2×C4⋊C4Dic9711yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4—1941no—
111C2×C42—1941no—
112C22⋊C8—972no—
113C22⋊C8—1941no—
114#95—972no—
115C24C22⋊D9711yescharacteristic
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×Dic9712yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—1942no—
123C22×C4C2×Dic9711yescharacteristic
124C22×C4C2×Dic9711yescharacteristic
125C22×C4C22⋊D9711yescharacteristic
126C22×C4C22⋊D9711yescharacteristic
127C22×C4C22⋊D9711yescharacteristic
128C22×C4C22×D9711yescharacteristic
129M16—3882no—
130C2×C8—1942no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4C2×Dic9712yes—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—1944no—
138C42—3882no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23C22⋊Dic9711yescharacteristic
143C23C22⋊Dic9711yescharacteristic
144C23C2×C22⋊D9711yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—3882no—
158C2×C4—3882no—
159C2×C4—1944no—
160C2×C4C22⋊Dic9711yescharacteristic
161C2×C4C22⋊Dic9711yescharacteristic
162C2×C4C22×Dic9711yescharacteristic
163C2×C4C2×C22⋊D9711yescharacteristic
164C8—3882no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22#1569073712yes—
174C22C2×C22⋊Dic9711yescharacteristic
175C22#1569084112yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—42no—
180C4—42no—
181C4—3884no—
182C4#1569072512yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#1569072511yescharacteristic
187C2#2026074411yescharacteristic
188C2C2×#1569073711yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊Q8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97#1508
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20260472C2semidirect
11. C22×Q8×C97C4semidirect
15. C22⋊Q8×C97C4semidirect
18. C2×C4⋊C4×C97C4semidirect
95. C97#1508semidirect
100. C22×Q8Dic97semidirect
105. C22⋊Q8Dic97semidirect
108. C2×C4⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606992
C4×C22⋊Dic972
#202604722
C2×C22⋊Q8×C972
#150897