Finite Group Databasebeta

← C2×#20260472 #27140217

Subgroups — C2×#20260472 #27140217

Order 12416.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C2×#15690717C211yesmaximal, characteristic
3#20260472C214yesmaximal, direct product factor
4C2×C22⋊Q8×C97C211yesmaximal, characteristic
5C23×Dic97—21no—
6C22×#11715895—21no—
7C22⋊#11715895—21no—
8C22⋊#11715895C2212yes—
9#15690717—21no—
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—24no—
21C22×Dic97—22no—
22C22×Dic97—42no—
23C22×Dic97—42no—
24C2×#11715895—22no—
25C2×#11715895—42no—
26C24×C97D411yescharacteristic
27C2×Q8×C97—21no—
28C2×Q8×C97—42no—
29C2×Q8×C97C2×C412yescomplemented
30C22×C4×C97—21no—
31C22×C4×C97—21no—
32C22×C4×C97—42no—
33C22×C4×C97C2×C411yescharacteristic
34C22×C4×C97C2×C411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97D411yescharacteristic
38C22×C4×C97C2311yescharacteristic
39C4⋊C4×C97—22no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97—22no—
42C4⋊C4×C97C2×C412yescomplemented
43C22⋊C4×C97—22no—
44C22⋊C4×C97—22no—
45C23×C97—22no—
46C23×C97—22no—
47C23×C97C22⋊C411yescharacteristic
48C23×C97C22⋊C411yescharacteristic
49C23×C97SD1612yes—
50C23×C97Q1612yes—
51C23×C97C2×D411yescharacteristic
52Q8×C97—42no—
53Q8×C97—42no—
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—41no—
62C2×C4×C97—42no—
63C2×C4×C97—42no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×Dic97—24no—
69C2×Dic97—48no—
70C2×Dic97—42no—
71C2×Dic97—42no—
72#11715895—42no—
73C22×C97—24no—
74C22×C97—24no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97#9912yes—
80C22×C97#10312yes—
81C22×C97#10312yes—
82C22×C97#10412yes—
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97C2×SD1611yescharacteristic
85C22×C97C2×Q1611yescharacteristic
86C4×C97—21no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—42no—
91C4×C97—42no—
92Dic97—44no—
93C2×C97—21no—
94C2×C97—22no—
95C2×C97—21no—
96C2×C97#32814yescomplemented
97C2×C97C2×#9911yescharacteristic
98C2×C97C2×#10311yescharacteristic
99C2×C97C2×#10411yescharacteristic
100C2×#328—971nomaximal, Sylow
101C97C2×#32811yescharacteristic, complemented, Sylow
102C2×C22⋊Q8D9711yescharacteristic
103C2×C22⋊C8—971no—
104C2×#95—971no—
105#328—974no—
106C22×Q8Dic9711yescharacteristic, complemented
107C23×C4—1941no—
108C23×C4C2×D9711yescharacteristic
109C22×C8—1941no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C22⋊Q8Dic9712yescomplemented
113C22⋊Q8C2×D9712yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4Dic9711yescharacteristic, complemented
116C2×C22⋊C4—21no—
117C22⋊C8—972no—
118C22⋊C8—1941no—
119#95—972no—
120#95—1941no—
121C24C22⋊D9711yescharacteristic
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8C2×Dic9712yescomplemented
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—3882no—
129C22×C4—3882no—
130C22×C4—1944no—
131C22×C4—1942no—
132C22×C4C2×Dic9711yescharacteristic
133C22×C4C2×Dic9711yescharacteristic
134C22×C4C22⋊D9711yescharacteristic
135C22×C4C22⋊D9711yescharacteristic
136C22×C4C22⋊D9711yescharacteristic
137C22×C4C22×D9711yescharacteristic
138C2×C8—3882no—
139C2×C8—1942no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic9712yescomplemented
144C22⋊C4—22no—
145C22⋊C4—22no—
146C23—22no—
147C23—22no—
148C23C22⋊Dic9711yescharacteristic
149C23C22⋊Dic9711yescharacteristic
150C23#1309549812yes—
151C23#1309550112yes—
152C23C2×C22⋊D9711yescharacteristic
153Q8—42no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—3882no—
166C2×C4—3888no—
167C2×C4—3882no—
168C2×C4—1944no—
169C2×C4C22⋊Dic9711yescharacteristic
170C2×C4C22⋊Dic9711yescharacteristic
171C2×C4C22×Dic9711yescharacteristic
172C2×C4C2×C22⋊D9711yescharacteristic
173C8—3882no—
174C22—21no—
175C22—21no—
176C22—24no—
177C22—24no—
178C22—21no—
179C22—21no—
180C22#1569072512yes—
181C22#1569073412yes—
182C22#1569073412yes—
183C22#1569073712yes—
184C22C2×C22⋊Dic9711yescharacteristic
185C22C2×#1309549811yescharacteristic
186C22C2×#1309550111yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—42no—
193C4—3884no—
194C2—21no—
195C2—21no—
196C2—22no—
197C2#2026047214yesdirect product factor
198C2C2×#1569072511yescharacteristic
199C2C2×#1569073411yescharacteristic
200C2C2×#1569073711yescharacteristic
subgroupG/H
centerC23C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C22⋊C4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#328
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20260472C2direct (N × H)
11. C22×Q8×C97C4semidirect
15. C22⋊Q8×C97C4semidirect
18. C2×C4⋊C4×C97C4semidirect
29. C2×Q8×C97C2×C4semidirect
42. C4⋊C4×C97C2×C4semidirect
96. C2×C97#328semidirect
101. C97C2×#328semidirect
106. C22×Q8Dic97semidirect
112. C22⋊Q8Dic97semidirect
115. C2×C4⋊C4Dic97semidirect
124. C2×Q8C2×Dic97semidirect
143. C4⋊C4C2×Dic97semidirect
197. C2#20260472direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊#117158952
C2×#156907172
#202604722
C2×C22⋊Q8×C972
C2×#32897