Finite Group Databasebeta

← #93017

Subgroups — #93017

Order 512.

246 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#12976C211yesmaximal, characteristic
2#8620C211yesmaximal, characteristic
3#8608C211yesmaximal, characteristic
4C23×C16C411yescharacteristic, complemented
5C2×C8×D4C2211yescharacteristic
6C2×C22⋊C16—21no—
7C2×C22⋊C16C411yescharacteristic, complemented
8#1807—21no—
9#1799—21no—
10#1568—21no—
11#1466—21no—
12#1374—21no—
13#1290C412yescomplemented
14#1289C412yes—
15C23×C8C2×C411yescharacteristic
16C23×C8D411yescharacteristic
17C2×C4×D4D411yescharacteristic
18C22×C16—21no—
19C22×C16—21no—
20C22×C16—42no—
21C22×C16—41no—
22C8×D4—41no—
23C8×D4—41no—
24#432—22no—
25C2×#106—41no—
26C2×C4⋊C8C2×C411yescharacteristic
27C2×#99—41no—
28C2×C22⋊C8C2×C411yescharacteristic
29C2×C22⋊C8Q811yescharacteristic
30C2×C4×C8D411yescharacteristic
31#380—22no—
32C22⋊C16—42no—
33C22⋊C16—41no—
34#342—41no—
35#340—41no—
36#335—41no—
37#334—41no—
38#323—41no—
39#323—41no—
40C22×D4SD1611yescharacteristic
41C23×C4C22⋊C411yescharacteristic
42C23×C4C4⋊C411yescharacteristic
43C22×C8—21no—
44C22×C8—21no—
45C22×C8—21no—
46C22×C8—21no—
47C22×C8—41no—
48C22×C8—41no—
49C22×C8—41no—
50C22×C8—41no—
51C22×C8C4211yescharacteristic
52C22×C8C22⋊C411yescharacteristic
53C22×C8C4⋊C411yescharacteristic
54C22×C8C4⋊C411yescharacteristic
55C4×D4—41no—
56C4×D4—41no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4Q1611yescharacteristic
59C2×C22⋊C4—42no—
60C2×C22⋊C4D811yescharacteristic
61C2×C22⋊C4SD1611yescharacteristic
62C2×C42C22⋊C411yescharacteristic
63C2×C16—22no—
64C2×C16—41no—
65C2×C16—42no—
66C2×C16—41no—
67C2×C16—42no—
68C2×C16—41no—
69C2×C16—41no—
70C2×C16—41no—
71#106—81no—
72#106—81no—
73C4⋊C8—41no—
74C4⋊C8—42no—
75#99—82no—
76C22⋊C8—41no—
77C22⋊C8—41no—
78C22⋊C8—42no—
79#97—42no—
80C4×C8—21no—
81C4×C8—21no—
82#95—42no—
83C24#10211yescharacteristic
84C24#10611yescharacteristic
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—41no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—82no—
97C22×C4#9511yescharacteristic
98C22×C4#10211yescharacteristic
99C22×C4#10311yescharacteristic
100C22×C4#10311yescharacteristic
101C22×C4#10711yescharacteristic
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—42no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—81no—
116C2×C8—82no—
117C2×C8#9911yescharacteristic
118C2×C8#9911yescharacteristic
119C4⋊C4—41no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—82no—
125C22⋊C4—82no—
126C42—21no—
127C42—21no—
128C16—42no—
129C16—81no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23#34011yescharacteristic
139C23#35912yes—
140C23#36512yes—
141D4—44no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—162no—
160C2×C4—162no—
161C2×C4—162no—
162C2×C4H(3,ℤ/4)11yescharacteristic
163C2×C4#34211yescharacteristic
164C8—21no—
165C8—41no—
166C8—41no—
167C8—41no—
168C8—162no—
169C8AΣL(1,𝔽16)11yescharacteristic
170C8#35211yescharacteristic
171C22—21no—
172C22—21no—
173C22—22no—
174C22—22no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22#128911yescharacteristic
184C22#138211yescharacteristic
185C22#138311yescharacteristic
186C4—21no—
187C4—41no—
188C4—41no—
189C4—41no—
190C4—41no—
191C4—322no—
192C4#139011yescharacteristic
193C4#139511yescharacteristic
194C2—21no—
195C2—21no—
196C2—42no—
197C2—42no—
198C2#722811yescharacteristic
199C2#723211yescharacteristic
200C2#723611yescharacteristic
subgroupG/H
center, socleC22#1289
commutatorC22×C8C42
Frattini, cosocleC2×C8×D4C22
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C23×C16C4semidirect
7. C2×C22⋊C16C4semidirect
13. #1290C4semidirect

Maximal subgroups

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

subgroupindex
#129762
#86202
#86082