Finite Group Databasebeta

← #511734

Subgroups — #511734

Order 512.

407 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 124 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#33273C212yesmaximal
2#13516C212yesmaximal
3#13515C211yesmaximal, characteristic
4#13477C212yesmaximal
5#13477C212yesmaximal
6#13475C212yesmaximal
7#13450C211yesmaximal, characteristic
8#13450C212yesmaximal
9#12899C211yesmaximal, characteristic
10C22×Q32C2211yescharacteristic
11#2645—22no—
12C2×#367C2212yes—
13C2×#366C2211yescharacteristic
14C2×C4⋊C16C2211yescharacteristic
15C2×C4⋊C16C2212yes—
16C2×C4⋊C16C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#358C2212yes—
21#1969C2212yes—
22#1969C2212yes—
23#1969C2212yes—
24#1942C2212yes—
25#1942C2212yes—
26#1942C2212yes—
27#1935—22no—
28#1935—22no—
29#1935—22no—
30#1726C2211yescharacteristic
31#1380C2212yes—
32#1380C2211yescharacteristic
33#1380C2212yes—
34#1380C2211yescharacteristic
35C22×Q16C2312yes—
36C2×Q32—41no—
37C2×Q32—42no—
38C22×C16C2311yescharacteristic
39C22×C16C2311yescharacteristic
40C22×C16C2312yes—
41C2×#107C2312yes—
42C2×#107C2312yes—
43C2×#107C2312yes—
44C2×#103—22no—
45C2×#103—22no—
46C2×#103—22no—
47C2×#97D411yescharacteristic
48C2×#97D412yes—
49C2×C4×C8C2311yescharacteristic
50C2×C4×C8C2312yes—
51#401—42no—
52#400—42no—
53#400—42no—
54#400—42no—
55#391—22no—
56#391—22no—
57#391—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#376D411yescharacteristic
62#367—42no—
63#366—41no—
64C4⋊C16—42no—
65C4⋊C16—41no—
66C4⋊C16—41no—
67C4⋊C16—42no—
68#358—42no—
69#358—42no—
70#358—42no—
71#340—22no—
72#340—22no—
73#340—22no—
74C22×Q8—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C22×C8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8C2411yescharacteristic
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C42C2×D411yescharacteristic
91C2×C42C2×D412yes—
92Q32—44no—
93C2×C16—22no—
94C2×C16—22no—
95C2×C16—22no—
96C2×C16—21no—
97C2×C16—22no—
98C2×C16—21no—
99C2×C16—22no—
100#107—22no—
101#107—22no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#103—44no—
107#103—44no—
108#103—44no—
109#97—42no—
110#97—41no—
111C4×C8—21no—
112C4×C8—22no—
113C4×C8—21no—
114C4×C8—22no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95D811yescharacteristic
125#95D811yescharacteristic
126#95D812yes—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4C2×D811yescharacteristic
138C22×C4C2×D812yes—
139C22×C4C2×D812yes—
140C22×C4C2×D811yescharacteristic
141C22×C4C22×D411yescharacteristic
142Q16—42no—
143Q16—42no—
144Q16—42no—
145Q16—42no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8H(5,𝔽2)11yescharacteristic
152C2×C8H(5,𝔽2)12yes—
153C2×C82−1+412yes—
154C2×C82−1+411yescharacteristic
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—162no—
162C4⋊C4—162no—
163C4⋊C4—162no—
164C42—22no—
165C42—21no—
166C42—22no—
167C42—21no—
168C16—41no—
169C16—42no—
170C16—41no—
171C23C22×D811yescharacteristic
172Q8—82no—
173Q8—82no—
174Q8—82no—
175Q8—82no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4#53712yes—
200C2×C4#53711yescharacteristic
201C2×C4#57512yes—
202C2×C4#57511yescharacteristic
203C8—42no—
204C8—41no—
205C8#56011yescharacteristic
206C8#56311yescharacteristic
207C8#56312yes—
208C22#324512yes—
209C22#324511yescharacteristic
210C22#341612yes—
211C22#341711yescharacteristic
212C22#341711yescharacteristic
213C4—42no—
214C4—42no—
215C4—41no—
216C4—41no—
217C4—162no—
218C4—162no—
219C4—162no—
220C4—162no—
221C4#282311yescharacteristic
222C4#325612yes—
223C4#325611yescharacteristic
224C2#3285811yescharacteristic
225C2#3382412yes—
226C2#3382412yes—
227C2#3382711yescharacteristic
228C2#3382911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332732
#135162
#135152
#134772
#134752
#134502
#128992