Finite Group Databasebeta

← C2×#324×C211 #41009077

Subgroups — C2×#324×C211 #41009077

Order 27008.

328 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22×C4⋊C4×C211C211yesmaximal, characteristic
2C2×C22⋊C8×C211C212yesmaximal
3#324×C211C214yesmaximal, direct product factor
4C23×C4×C211C411yescharacteristic
5C23×C4×C211C411yescharacteristic
6C23×C4×C211C2211yescharacteristic
7C22×C8×C211—22no—
8C2×C4⋊C4×C211—22no—
9C2×C4⋊C4×C211—22no—
10C2×C4⋊C4×C211C412yes—
11C2×C4⋊C4×C211C2212yes—
12C22⋊C8×C211—22no—
13C22⋊C8×C211C2214yes—
14C24×C211C2×C411yescharacteristic
15C22×C4×C211—21no—
16C22×C4×C211—21no—
17C22×C4×C211—22no—
18C22×C4×C211—21no—
19C22×C4×C211—21no—
20C22×C4×C211—42no—
21C22×C4×C211—42no—
22C22×C4×C211C814yescomplemented
23C22×C4×C211C2×C412yes—
24C22×C4×C211C2×C412yes—
25C22×C4×C211C2×C411yescharacteristic
26C22×C4×C211D412yes—
27C22×C4×C211D412yes—
28C22×C4×C211C2311yescharacteristic
29C2×C8×C211—24no—
30C2×C8×C211—44no—
31C4⋊C4×C211—24no—
32C4⋊C4×C211—24no—
33C23×C211—22no—
34C23×C211—22no—
35C23×C211C22⋊C411yescharacteristic
36C23×C211C22⋊C411yescharacteristic
37C23×C211C2×C812yes—
38C23×C211M1612yes—
39C23×C211C22×C411yescharacteristic
40C2×C4×C211—24no—
41C2×C4×C211—24no—
42C2×C4×C211—24no—
43C2×C4×C211—22no—
44C2×C4×C211—22no—
45C2×C4×C211—22no—
46C2×C4×C211—41no—
47C2×C4×C211—42no—
48C2×C4×C211—41no—
49C2×C4×C211—42no—
50C2×C4×C211—42no—
51C2×C4×C211—42no—
52C2×C4×C211C22⋊C412yes—
53C2×C4×C211C2×C814yescomplemented
54C2×C4×C211C2×D412yes—
55C8×C211—44no—
56C22×C211—24no—
57C22×C211—24no—
58C22×C211—21no—
59C22×C211—21no—
60C22×C211—21no—
61C22×C211—21no—
62C22×C211C22⋊C812yes—
63C22×C211C22⋊C812yes—
64C22×C211#9912yes—
65C22×C211#10112yes—
66C22×C211C2×C22⋊C411yescharacteristic
67C22×C211C22×C811yescharacteristic
68C22×C211C2×M1611yescharacteristic
69C4×C211—24no—
70C4×C211—22no—
71C4×C211—22no—
72C4×C211—42no—
73C2×C211—22no—
74C2×C211—21no—
75C2×C211—21no—
76C2×C211#32414yesdirect product factor
77C2×C211C2×C22⋊C811yescharacteristic
78C2×C211C2×#9911yescharacteristic
79C2×C211C2×#10111yescharacteristic
80C211C2×#32411yescharacteristic, direct product factor, Sylow
81C2×#324C21111yesmaximal, characteristic, direct product factor, Sylow
82C22×C4⋊C4C2×C21111yescharacteristic
83C2×C22⋊C8C2×C21112yes—
84#324C2×C21114yesdirect product factor
85C23×C4C4×C21111yescharacteristic
86C23×C4C4×C21111yescharacteristic
87C23×C4C22×C21111yescharacteristic
88C22×C8—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4C4×C21112yes—
92C2×C4⋊C4C22×C21112yes—
93C22⋊C8—22no—
94C22⋊C8C22×C21114yes—
95C24C2×C4×C21111yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4C8×C21114yescomplemented
104C22×C4C2×C4×C21112yes—
105C22×C4C2×C4×C21112yes—
106C22×C4C2×C4×C21111yescharacteristic
107C22×C4D4×C21112yes—
108C22×C4D4×C21112yes—
109C22×C4C23×C21111yescharacteristic
110C2×C8—24no—
111C2×C8—44no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C23—22no—
115C23—22no—
116C23C22⋊C4×C21111yescharacteristic
117C23C22⋊C4×C21111yescharacteristic
118C23C2×C8×C21112yes—
119C23M16×C21112yes—
120C23C22×C4×C21111yescharacteristic
121C2×C4—24no—
122C2×C4—24no—
123C2×C4—24no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4C22⋊C4×C21112yes—
134C2×C4C2×C8×C21114yescomplemented
135C2×C4C2×D4×C21112yes—
136C8—44no—
137C22—24no—
138C22—24no—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22C22⋊C8×C21112yes—
144C22C22⋊C8×C21112yes—
145C22#99×C21112yes—
146C22#101×C21112yes—
147C22C2×C22⋊C4×C21111yescharacteristic
148C22C22×C8×C21111yescharacteristic
149C22C2×M16×C21111yescharacteristic
150C4—24no—
151C4—22no—
152C4—22no—
153C4—42no—
154C2—22no—
155C2—21no—
156C2—21no—
157C2#324×C21114yesdirect product factor
158C2C2×C22⋊C8×C21111yescharacteristic
159C2C2×#99×C21111yescharacteristic
160C2C2×#101×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C22⋊C4
commutatorC22C22×C8×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #324×C211C2direct (N × H)
22. C22×C4×C211C8semidirect
53. C2×C4×C211C2×C8semidirect
76. C2×C211#324direct (N × H)
80. C211C2×#324direct (N × H)
81. C2×#324C211direct (N × H)
84. #324C2×C211direct (N × H)
103. C22×C4C8×C211semidirect
134. C2×C4C2×C8×C211semidirect
157. C2#324×C211direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×C2112
C2×C22⋊C8×C2112
#324×C2112
C2×#324211