Finite Group Databasebeta

← #511423

Subgroups — #511423

Order 512.

457 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 128 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#28268C212yesmaximal
2#13539C212yesmaximal
3#13515C211yesmaximal, characteristic
4#13483C212yesmaximal
5#13450C211yesmaximal, characteristic
6#13264C211yesmaximal, characteristic
7#13150C212yesmaximal
8#13015C212yesmaximal
9#12936C212yesmaximal
10C22×Q32C2211yescharacteristic
11C2×C8×Q8C2211yescharacteristic
12C2×C4×Q16C2212yes—
13#2581—22no—
14C2×#366C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic
16C2×#358C2212yes—
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#346C2211yescharacteristic
20#2066—22no—
21#2066—22no—
22#1969C2212yes—
23#1969C2212yes—
24#1942C2212yes—
25#1942C2212yes—
26#1846C2212yes—
27#1805—22no—
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2211yescharacteristic
32#1380C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C22×Q16C2312yes—
36C2×C4×Q8—22no—
37C2×C4×Q8D411yescharacteristic
38C2×Q32—24no—
39C2×Q32—41no—
40C22×C16C2312yes—
41C22×C16C2311yescharacteristic
42C22×C16C2311yescharacteristic
43C8×Q8—41no—
44C8×Q8—41no—
45C4×Q16—42no—
46C4×Q16—42no—
47C2×#107C2312yes—
48C2×#107C2312yes—
49C2×#107C2312yes—
50C2×#106—22no—
51C2×C4⋊C8D412yes—
52C2×C4⋊C8D411yescharacteristic
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—22no—
56C2×C4×C8C2312yes—
57C2×C4×C8C2311yescharacteristic
58#400—42no—
59#395—42no—
60#391—22no—
61#391—22no—
62#389—22no—
63#389—22no—
64#387—42no—
65#387—42no—
66#382—22no—
67#366—41no—
68C4⋊C16—41no—
69#358—24no—
70#358—24no—
71#358—42no—
72#346—41no—
73#340—22no—
74#340—22no—
75C22×Q8—22no—
76C22×Q8D811yescharacteristic
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16#4114yes—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8C2411yescharacteristic
86C4×Q8—41no—
87C4×Q8—41no—
88C4×Q8—44no—
89C4×Q8—44no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4D811yescharacteristic
97C2×C4⋊C4D812yes—
98C2×C42—42no—
99C2×C42C2×D412yes—
100C2×C42C2×D411yescharacteristic
101Q32—44no—
102C2×C16—22no—
103C2×C16—22no—
104C2×C16—22no—
105C2×C16—22no—
106C2×C16—21no—
107C2×C16—21no—
108C2×C16—22no—
109#107—22no—
110#107—22no—
111#107—22no—
112#107—22no—
113#107—22no—
114#107—22no—
115#106—82no—
116C4⋊C8—41no—
117C4⋊C8—42no—
118#103—44no—
119#103—82no—
120#103—82no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—22no—
124C4×C8—22no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133C2×Q8—24no—
134C2×Q8—41no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4C2×D811yescharacteristic
146C22×C4C2×D812yes—
147C22×C4C2×D811yescharacteristic
148C22×C4C2×D812yes—
149C22×C4C22×D411yescharacteristic
150Q16—24no—
151Q16—24no—
152Q16—42no—
153Q16—42no—
154C2×C8—22no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8C2×#4112yes—
160C2×C8H(5,𝔽2)11yescharacteristic
161C2×C8H(5,𝔽2)11yescharacteristic
162C2×C82−1+412yes—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—84no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C42—21no—
175C42—22no—
176C42—21no—
177C42—22no—
178C42—84no—
179C16—42no—
180C16—41no—
181C16—41no—
182C23C22×D811yescharacteristic
183Q8—44no—
184Q8—44no—
185Q8—44no—
186Q8—82no—
187Q8—82no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—44no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#53411yescharacteristic
212C2×C4#54812yes—
213C2×C4#57511yescharacteristic
214C2×C4#57712yes—
215C8—42no—
216C8—41no—
217C8#53912yes—
218C8#55111yescharacteristic
219C8#56311yescharacteristic
220C22#318611yescharacteristic
221C22#335812yes—
222C22#341412yes—
223C22#341711yescharacteristic
224C22#341711yescharacteristic
225C4—41no—
226C4—42no—
227C4—42no—
228C4—41no—
229C4—84no—
230C4—162no—
231C4—162no—
232C4—162no—
233C4#272811yescharacteristic
234C4#320411yescharacteristic
235C4#337612yes—
236C2#3254411yescharacteristic
237C2#3376311yescharacteristic
238C2#3376811yescharacteristic
239C2#3394412yes—
240C2#3394412yes—
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

Maximal subgroups

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

subgroupindex
#282682
#135392
#135152
#134832
#134502
#132642
#131502
#130152
#129362