Finite Group Databasebeta

← #27140308

Subgroups — #27140308

Order 12416.

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

normal subgroups only

This group has 288 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×C22⋊#11715895C211yesmaximal, characteristic
2#20260595C211yesmaximal, characteristic
3#20260472C214yesmaximal, complemented
4C2×C22⋊Q8×C97C211yesmaximal, characteristic
5C22×#11715895—21no—
6C2×C4⋊Dic97—21no—
7C2×C22⋊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—
21C2×#11715895—22no—
22C2×#11715895—42no—
23C4⋊Dic97—42no—
24C22⋊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—41no—
45C23×C97—41no—
46C23×C97C22⋊C411yescharacteristic
47C23×C97C22⋊C411yescharacteristic
48C23×C97SD1612yes—
49C23×C97Q1612yes—
50C23×C97C2×D411yescharacteristic
51Q8×C97—42no—
52Q8×C97—42no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—42no—
62C2×C4×C97—42no—
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×Dic97—42no—
68C2×Dic97—42no—
69C2×Dic97—42no—
70#11715895—42no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97#10312yes—
80C22×C97#10312yes—
81C22×C97C2×C22⋊C411yescharacteristic
82C22×C97C2×SD1611yescharacteristic
83C22×C97C2×Q1611yescharacteristic
84C4×C97—21no—
85C4×C97—21no—
86C4×C97—21no—
87C4×C97—42no—
88C4×C97—42no—
89C4×C97#9912yes—
90Dic97—82no—
91C2×C97—21no—
92C2×C97—22no—
93C2×C97—22no—
94C2×C97—21no—
95C2×C97C2×#9911yescharacteristic
96C2×C97C2×#10311yescharacteristic
97C2×C97#42111yescharacteristic
98#1520—971nomaximal, Sylow
99C97#152011yescharacteristic, complemented, Sylow
100C2×C22⋊Q8D9711yescharacteristic
101C2×C22⋊C8—971no—
102#380—971no—
103#328—974no—
104C22×Q8Dic9711yescharacteristic, complemented
105C23×C4C2×D9711yescharacteristic
106C22×C8—1941no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8Dic9712yescomplemented
110C22⋊Q8C2×D9712yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—1941no—
113C2×C4⋊C4Dic9711yescharacteristic, complemented
114C2×C22⋊C4—21no—
115C2×C22⋊C4—1941no—
116C22⋊C8—972no—
117C22⋊C8—1941no—
118#95—972no—
119C24C22⋊D9711yescharacteristic
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×Dic9712yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—1942no—
127C22×C4C2×Dic9711yescharacteristic
128C22×C4C2×Dic9711yescharacteristic
129C22×C4C22⋊D9711yescharacteristic
130C22×C4C22⋊D9711yescharacteristic
131C22×C4C22⋊D9711yescharacteristic
132C22×C4C22×D9711yescharacteristic
133C2×C8—3882no—
134C2×C8—1942no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—3882no—
139C4⋊C4C2×Dic9712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—3882no—
143C23—41no—
144C23—41no—
145C23C22⋊Dic9711yescharacteristic
146C23C22⋊Dic9711yescharacteristic
147C23#1309549812yes—
148C23#1309550112yes—
149C23C2×C22⋊D9711yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—3882no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4C22⋊Dic9711yescharacteristic
166C2×C4C22⋊Dic9711yescharacteristic
167C2×C4C22×Dic9711yescharacteristic
168C2×C4C2×C22⋊D9711yescharacteristic
169C8—3882no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#1569073412yes—
179C22#1569073412yes—
180C22C2×C22⋊Dic9711yescharacteristic
181C22C2×#1309549811yescharacteristic
182C22C2×#1309550111yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—7762no—
189C4#1569072512yes—
190C2—21no—
191C2—21no—
192C2—22no—
193C2—22no—
194C2C2×#1569072511yescharacteristic
195C2C2×#1569073411yescharacteristic
196C2#2026075211yescharacteristic
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#1520
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
99. C97#1520semidirect
104. C22×Q8Dic97semidirect
109. C22⋊Q8Dic97semidirect
113. C2×C4⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#117158952
#202605952
#202604722
C2×C22⋊Q8×C972
#152097