Finite Group Databasebeta

← C2×#7358 #149581

Subgroups — C2×#7358 #149581

Order 512.

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

normal subgroups only

This group has 418 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×#1992C211yesmaximal, characteristic
2C2×#1325C212yesmaximal
3#7358C214yesmaximal, direct product factor
4C22×#125C411yescharacteristic
5C22×#97C411yescharacteristic
6C2×#384C2211yescharacteristic
7#1992—22no—
8#1992C412yes—
9#1992C2212yes—
10C2×#340—22no—
11C2×#324—22no—
12#1325—22no—
13#1325C2214yes—
14C23×C8—21no—
15C2×#125—21no—
16C2×#125—22no—
17C2×#125—41no—
18C2×#125—42no—
19C2×#125C814yescomplemented
20C2×#125C2×C412yes—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4—41no—
24C22×C4⋊C4D412yes—
25C22×C42C2×C411yescharacteristic
26C2×C22⋊C8—42no—
27C2×#97—21no—
28C2×#97—42no—
29C2×#97C2×C412yes—
30#384—41no—
31#384C2×C411yescharacteristic
32#384D412yes—
33#384C2311yescharacteristic
34#340—24no—
35#340—24no—
36#324—24no—
37#324—42no—
38C23×C4—21no—
39C23×C4—22no—
40C23×C4—41no—
41C23×C4C22⋊C411yescharacteristic
42C22×C8—24no—
43C22×C8—22no—
44C22×C8—41no—
45C22×C8—41no—
46C22×C8—42no—
47C22×C8—82no—
48#125—24no—
49#125—22no—
50#125—44no—
51#125—42no—
52#125—44no—
53#125—42no—
54#125C2×C814yescomplemented
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—44no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—44no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—84no—
70C2×C4⋊C4C22⋊C412yes—
71C2×C4⋊C4C2×D412yes—
72C2×C42—21no—
73C2×C42—22no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C2×C812yes—
76C2×C42M1612yes—
77C2×C42C22×C411yescharacteristic
78C22⋊C8—44no—
79C22⋊C8—82no—
80#97—42no—
81#97—42no—
82C24#9911yescharacteristic
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—44no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—82no—
99C22×C4—82no—
100C22×C4—84no—
101C22×C4C22⋊C812yes—
102C22×C4#9911yescharacteristic
103C22×C4#10012yes—
104C22×C4C2×C22⋊C411yescharacteristic
105C2×C8—24no—
106C2×C8—44no—
107C2×C8—44no—
108C2×C8—42no—
109C2×C8—42no—
110C2×C8—84no—
111C2×C8—164no—
112C4⋊C4—24no—
113C4⋊C4—44no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—48no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—42no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C4⋊C4—84no—
123C4⋊C4—82no—
124C42—22no—
125C42—22no—
126C42—21no—
127C42—21no—
128C42C22⋊C812yes—
129C42C22×C811yescharacteristic
130C42C2×M1611yescharacteristic
131C23—22no—
132C23—22no—
133C23#32312yes—
134C23#35411yescharacteristic
135C23#35411yescharacteristic
136C23#35612yes—
137C23C2×#9911yescharacteristic
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—88no—
161C2×C4—84no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—162no—
165C2×C4#32312yes—
166C2×C4C2×C22⋊C811yescharacteristic
167C2×C4C2×#10011yescharacteristic
168C8—44no—
169C8—164no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—24no—
175C22—24no—
176C22#131912yes—
177C22#131912yes—
178C22#141212yes—
179C22#141312yes—
180C22C2×#32311yescharacteristic
181C22C2×#35411yescharacteristic
182C22C2×#35611yescharacteristic
183C4—21no—
184C4—22no—
185C4—21no—
186C4—42no—
187C4—41no—
188C4—41no—
189C4—84no—
190C4—82no—
191C4—82no—
192C2—21no—
193C2—22no—
194C2—21no—
195C2#735814yesdirect product factor
196C2C2×#131911yescharacteristic
197C2C2×#141211yescharacteristic
198C2C2×#141311yescharacteristic
subgroupG/H
center, socleC23#354
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7358C2direct (N × H)
19. C2×#125C8semidirect
54. #125C2×C8semidirect
195. C2#7358direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#19922
C2×#13252
#73582