Finite Group Databasebeta

← #135949

Subgroups — #135949

Order 512.

301 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 86 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#33493C211yesmaximal, characteristic
2#12982C211yesmaximal, characteristic
3#12916C212yesmaximal, complemented
4#12633C211yesmaximal, characteristic
5#7208C212yesmaximal, complemented
6#3407C2211yescharacteristic
7#2931C2211yescharacteristic
8#2169—22no—
9D4×C16—22no—
10#2162—21no—
11#2153—22no—
12#2132C411yescharacteristic
13#2126C411yescharacteristic
14#2117—21no—
15#2115C2211yescharacteristic
16#2113—21no—
17#2113C2212yes—
18#1757C2212yes—
19#1395C412yes—
20#1379—22no—
21#1376—21no—
22C2×#131D411yescharacteristic
23C2×#131C2311yescharacteristic
24#532C2×C411yescharacteristic
25#504—42no—
26C2×M32—21no—
27C2×M32—21no—
28C2×M32—22no—
29C2×M32—21no—
30C2×M32D412yes—
31C22×C16—21no—
32C22×C16D412yes—
33#436—22no—
34C8×D4—22no—
35#423Q811yescharacteristic
36#413C2×C412yes—
37#410C2×C412yes—
38#408C2×C411yescharacteristic
39#408Q811yescharacteristic
40#405—22no—
41C2×C4×C8D411yescharacteristic
42#364—44no—
43C4⋊C16—21no—
44C4⋊C16—21no—
45#356C812yes—
46#355C812yes—
47#354—21no—
48#353—21no—
49#350—42no—
50#349—22no—
51C22⋊C16—22no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54C4×C16—22no—
55#344—22no—
56C2×#41C4⋊C411yescharacteristic
57C2×#41C22×C411yescharacteristic
58#131—41no—
59#131—41no—
60#131—41no—
61#131—41no—
62C2×M16—21no—
63C2×M16—22no—
64C2×M16C4⋊C411yescharacteristic
65C2×M16C2×Q811yescharacteristic
66C2×M16#4112yes—
67C22×C8—21no—
68C22×C8C4⋊C411yescharacteristic
69C22×C8C2×D412yes—
70C22×C8C2×D411yescharacteristic
71C4⋊Q8C2×C811yescharacteristic
72#127M1611yescharacteristic
73#124C2×C811yescharacteristic
74#124M1611yescharacteristic
75#121—22no—
76C4×D4—22no—
77#117—22no—
78C2×C42C4⋊C411yescharacteristic
79M32—22no—
80M32—24no—
81M32—41no—
82M32—42no—
83M32—41no—
84M32—41no—
85M32—81no—
86C2×C16—22no—
87C2×C16—22no—
88C2×C16—42no—
89C2×C16—41no—
90C2×C16—42no—
91C2×C16—41no—
92C2×C16—81no—
93C4⋊C8—22no—
94#101C2×C814yes—
95#100—22no—
96#99—22no—
97#99—22no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100#97—42no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—42no—
104#41—41no—
105#41—41no—
106#41—41no—
107#41—41no—
108C2×Q8C4⋊C811yescharacteristic
109C2×Q8C22×C811yescharacteristic
110C2×D4—22no—
111C2×D4—21no—
112C2×D4C4⋊C811yescharacteristic
113C2×D4C2×M1611yescharacteristic
114C2×D4#13112yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4C2×C4⋊C411yescharacteristic
118C22×C4C4×D412yes—
119M16—24no—
120M16—41no—
121M16—42no—
122M16—42no—
123M16—41no—
124M16—81no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8#12011yescharacteristic
136C2×C8C22⋊Q812yes—
137C2×C8#12311yescharacteristic
138C4⋊C4—22no—
139C4⋊C4—42no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—42no—
143C42—21no—
144C42—42no—
145C42C4⋊C811yescharacteristic
146C42C4⋊C811yescharacteristic
147C16—42no—
148C16—42no—
149C16—42no—
150C16—81no—
151C23—21no—
152C23#42411yescharacteristic
153C23#43512yes—
154Q8—41no—
155Q8—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—44no—
160D4—81no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—82no—
172C2×C4#38511yescharacteristic
173C2×C4C2×C4⋊C811yescharacteristic
174C2×C4C8×D412yes—
175C8—21no—
176C8—22no—
177C8—44no—
178C8—41no—
179C8—41no—
180C8B(4,𝔽2)11yescharacteristic
181C8#45811yescharacteristic
182C22—22no—
183C22—21no—
184C22—41no—
185C22—42no—
186C22—42no—
187C22—41no—
188C22#185011yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—162no—
195C4#186711yescharacteristic
196C4#186811yescharacteristic
197C2—21no—
198C2—42no—
199C2—41no—
200C2—81no—
201C2#885111yescharacteristic
subgroupG/H
centerC8B(4,𝔽2)
commutatorC2×Q8C22×C8
Frattini, cosocleC2×#131C23
socleC2#8851
prime radical#532C2×C4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G
LECSG ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12916C2semidirect
5. #7208C2semidirect

Maximal subgroups

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

subgroupindex
#334932
#129822
#129162
#126332
#72082