Finite Group Databasebeta

← #511560

Subgroups — #511560

Order 512.

424 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 146 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#28679C212yesmaximal
2#13515C211yesmaximal, characteristic
3#13514C211yesmaximal, characteristic
4#13453C212yesmaximal
5#13453C212yesmaximal
6#13263C211yesmaximal, characteristic
7#13166C212yesmaximal
8#13111C212yesmaximal
9#13110C212yesmaximal
10C2×#499C2212yes—
11C2×C8×Q8C2211yescharacteristic
12#2584—22no—
13C2×#367C2212yes—
14C2×#367C2212yes—
15C2×#366C2211yescharacteristic
16C2×#366C2211yescharacteristic
17C2×C4⋊C16C2211yescharacteristic
18C2×C4×C16C2211yescharacteristic
19#2088—22no—
20#2088—22no—
21#1989—22no—
22#1942C2212yes—
23#1942C2212yes—
24#1942C2212yes—
25#1942C2212yes—
26C4×#107C2212yes—
27#1380C2211yescharacteristic
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2212yes—
32#1380C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C2×#125—22no—
36C2×C4×Q8D411yescharacteristic
37C22×C16C2311yescharacteristic
38C22×C16C2311yescharacteristic
39C22×C16C2312yes—
40#499—42no—
41#499—42no—
42C8×Q8—41no—
43C8×Q8—41no—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#106—22no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D412yes—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2312yes—
53#398—42no—
54#395—42no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389—42no—
60#389—42no—
61C4×C4⋊C4—22no—
62#367—42no—
63#367Q818yescomplemented
64#366—41no—
65#366—41no—
66C4⋊C16—41no—
67C4×C16—41no—
68#340—22no—
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73C22×Q8D811yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D412yes—
76C22×C8C2411yescharacteristic
77#125—44no—
78#125—44no—
79C4×Q8—41no—
80C4×Q8—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4D811yescharacteristic
92C2×C4⋊C4D812yes—
93C2×C42—42no—
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96C2×C16—22no—
97C2×C16—22no—
98C2×C16—21no—
99C2×C16—21no—
100C2×C16—22no—
101C2×C16—22no—
102C2×C16C2×Q814yes—
103#107—22no—
104#107—22no—
105#107—22no—
106#107—22no—
107#107—22no—
108#107—22no—
109#107C2×Q814yes—
110#107C2×Q814yes—
111#106—44no—
112C4⋊C8—41no—
113C4⋊C8—42no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—22no—
117C4×C8—22no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123C2×Q8—22no—
124C2×Q8—41no—
125C2×Q8—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4C2×D811yescharacteristic
131C22×C4C2×D812yes—
132C22×C4C2×D812yes—
133C22×C4C2×D811yescharacteristic
134C22×C4C22×D411yescharacteristic
135C2×C8—22no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—24no—
139C2×C8C22×Q812yes—
140C2×C8H(5,𝔽2)11yescharacteristic
141C2×C8H(5,𝔽2)11yescharacteristic
142C2×C82−1+412yes—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—41no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—84no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C4⋊C4—84no—
157C4⋊C4—162no—
158C4⋊C4—162no—
159C42—21no—
160C42—22no—
161C42—21no—
162C42—22no—
163C42—162no—
164C16—24no—
165C16—41no—
166C16—41no—
167C23C22×D811yescharacteristic
168Q8—44no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#50812yes—
192C2×C4#53411yescharacteristic
193C2×C4D4×Q812yes—
194C2×C4#57511yescharacteristic
195C2×C4#57812yes—
196C8—42no—
197C8—41no—
198C8#54412yes—
199C8Q8211yescharacteristic
200C8#56311yescharacteristic
201C22#318611yescharacteristic
202C22Q8×D812yes—
203C22C2×#50811yescharacteristic
204C22#341411yescharacteristic
205C22#341612yes—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—41no—
210C4—84no—
211C4—84no—
212C4—162no—
213C4—162no—
214C4#273411yescharacteristic
215C4#320211yescharacteristic
216C4#339012yes—
217C2#3268211yescharacteristic
218C2#3375711yescharacteristic
219C2#3377011yescharacteristic
220C2#3394812yes—
221C2#3395512yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
63. #367Q8semidirect

Maximal subgroups

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

subgroupindex
#286792
#135152
#135142
#134532
#132632
#131662
#131112
#131102