Finite Group Databasebeta

← C2×#7366 #149609

Subgroups — C2×#7366 #149609

Order 512.

346 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1835C211yesmaximal, characteristic
2C2×#1324C212yesmaximal
3#7366C214yesmaximal, direct product factor
4C22×C4×C8C411yescharacteristic
5C2×#382C411yescharacteristic
6C2×#382C2211yescharacteristic
7#1835—22no—
8#1835C412yes—
9#1835C2212yes—
10C2×#340—22no—
11C2×#324—22no—
12#1324—22no—
13#1324C2214yes—
14C23×C8—21no—
15C22×C4⋊C4—21no—
16C22×C4⋊C4D412yes—
17C22×C42C2×C411yescharacteristic
18C2×C22⋊C8—42no—
19C2×C4×C8—21no—
20C2×C4×C8—22no—
21C2×C4×C8C814yescomplemented
22C2×C4×C8C2×C412yes—
23#382—21no—
24#382—41no—
25#382—41no—
26#382C2×C412yes—
27#382C2×C411yescharacteristic
28#382D412yes—
29#382C2311yescharacteristic
30C2×#95—41no—
31C2×#95—41no—
32#340—24no—
33#340—24no—
34#324—24no—
35#324—42no—
36C23×C4—21no—
37C23×C4—22no—
38C23×C4—41no—
39C23×C4C22⋊C411yescharacteristic
40C22×C8—24no—
41C22×C8—22no—
42C22×C8—41no—
43C22×C8—41no—
44C22×C8—42no—
45C22×C8—82no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4C22⋊C412yes—
53C2×C4⋊C4C2×D412yes—
54C2×C42—21no—
55C2×C42—22no—
56C2×C42C22⋊C411yescharacteristic
57C2×C42C2×C812yes—
58C2×C42M1612yes—
59C2×C42C22×C411yescharacteristic
60C22⋊C8—44no—
61C22⋊C8—82no—
62C4×C8—24no—
63C4×C8—22no—
64C4×C8C2×C814yescomplemented
65#95—41no—
66#95—42no—
67#95—41no—
68#95—42no—
69#95—42no—
70C24#9911yescharacteristic
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—44no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—84no—
85C22×C4—84no—
86C22×C4—82no—
87C22×C4C22⋊C812yes—
88C22×C4#9911yescharacteristic
89C22×C4#10112yes—
90C22×C4C2×C22⋊C411yescharacteristic
91C2×C8—24no—
92C2×C8—44no—
93C2×C8—44no—
94C2×C8—42no—
95C2×C8—42no—
96C2×C8—84no—
97C2×C8—164no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—44no—
101C4⋊C4—82no—
102C4⋊C4—82no—
103C42—22no—
104C42—22no—
105C42—21no—
106C42—21no—
107C42C22⋊C812yes—
108C42C22×C811yescharacteristic
109C42C2×M1611yescharacteristic
110C23—24no—
111C23#32412yes—
112C23#35411yescharacteristic
113C23#35411yescharacteristic
114C23#35512yes—
115C23C2×#9911yescharacteristic
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—22no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—44no—
130C2×C4—44no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—84no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—84no—
137C2×C4—82no—
138C2×C4—162no—
139C2×C4—164no—
140C2×C4#32412yes—
141C2×C4C2×C22⋊C811yescharacteristic
142C2×C4C2×#10111yescharacteristic
143C8—44no—
144C8—164no—
145C22—22no—
146C22—22no—
147C22—24no—
148C22—24no—
149C22#132412yes—
150C22#132412yes—
151C22#141414yes—
152C22C2×#32411yescharacteristic
153C22C2×#35411yescharacteristic
154C22C2×#35511yescharacteristic
155C4—21no—
156C4—22no—
157C4—21no—
158C4—42no—
159C4—41no—
160C4—41no—
161C4—82no—
162C4—82no—
163C4—162no—
164C2—21no—
165C2—21no—
166C2—22no—
167C2#736614yesdirect product factor
168C2C2×#132411yescharacteristic
169C2C2×#141412yes—
subgroupG/H
center, socleC23#354
commutatorC42C22×C8
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7366C2direct (N × H)
21. C2×C4×C8C8semidirect
64. C4×C8C2×C8semidirect
167. C2#7366direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#18352
C2×#13242
#73662