Finite Group Databasebeta

← #511409

Subgroups — #511409

Order 512.

468 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 138 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#28265C212yesmaximal
2#13538C212yesmaximal
3#13515C211yesmaximal, characteristic
4#13481C212yesmaximal
5#13450C211yesmaximal, characteristic
6#13263C211yesmaximal, characteristic
7#13148C212yesmaximal
8#13015C212yesmaximal
9#12936C212yesmaximal
10C22×Q32C2211yescharacteristic
11C2×C8×Q8C2211yescharacteristic
12C2×C4×Q16C2212yes—
13#2582—22no—
14C2×#366C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic
16C2×#358C2212yes—
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×C4×C16C2211yescharacteristic
20#2068—22no—
21#2068—22no—
22#1969C2212yes—
23#1969C2212yes—
24#1942C2212yes—
25#1942C2212yes—
26#1845C2212yes—
27#1803—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×#107—22no—
48C2×#107C2312yes—
49C2×#107C2312yes—
50C2×#107C2312yes—
51C2×C4⋊C8D412yes—
52C2×C4⋊C8D411yescharacteristic
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—22no—
56C2×C4×C8C2312yes—
57C2×C4×C8C2311yescharacteristic
58#398—42no—
59#398—42no—
60#391—22no—
61#391—22no—
62#391—42no—
63#389—22no—
64#389—22no—
65#389—42no—
66#384—22no—
67#366—41no—
68C4⋊C16—41no—
69#358—24no—
70#358—24no—
71#358—42no—
72C4×C16—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⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4D812yes—
99C2×C4⋊C4D811yescharacteristic
100C2×C42—42no—
101C2×C42C2×D412yes—
102C2×C42C2×D411yescharacteristic
103Q32—44no—
104C2×C16—22no—
105C2×C16—22no—
106C2×C16—22no—
107C2×C16—22no—
108C2×C16—21no—
109C2×C16—21no—
110C2×C16—22no—
111#107—22no—
112#107—22no—
113#107—22no—
114#107—22no—
115#107—22no—
116#107—22no—
117#107—82no—
118C4⋊C8—41no—
119C4⋊C8—42no—
120#103—44no—
121#103—82no—
122#103—82no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—22no—
126C4×C8—22no—
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—41no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8Q3214yescomplemented
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4C2×D811yescharacteristic
147C22×C4C2×D812yes—
148C22×C4C2×D811yescharacteristic
149C22×C4C2×D812yes—
150C22×C4C22×D411yescharacteristic
151Q16—24no—
152Q16—24no—
153Q16—42no—
154Q16—42no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8C2×#4112yes—
161C2×C8H(5,𝔽2)11yescharacteristic
162C2×C8H(5,𝔽2)11yescharacteristic
163C2×C82−1+412yes—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C4⋊C4—162no—
175C4⋊C4—162no—
176C4⋊C4—162no—
177C42—21no—
178C42—22no—
179C42—21no—
180C42—22no—
181C42—84no—
182C16—42no—
183C16—41no—
184C16—41no—
185C23C22×D811yescharacteristic
186Q8—44no—
187Q8—44no—
188Q8—44no—
189Q8—82no—
190Q8—82no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—44no—
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—82no—
212C2×C4C2×Q3212yes—
213C2×C4C2×Q3214yes—
214C2×C4#53411yescharacteristic
215C2×C4#54812yes—
216C2×C4#57511yescharacteristic
217C2×C4#57712yes—
218C8—42no—
219C8—41no—
220C8#53912yes—
221C8#55111yescharacteristic
222C8#56311yescharacteristic
223C22#318611yescharacteristic
224C22#335812yes—
225C22C22×Q3211yescharacteristic
226C22#341411yescharacteristic
227C22#341612yes—
228C4—24no—
229C4—22no—
230C4—42no—
231C4—41no—
232C4—84no—
233C4—162no—
234C4—162no—
235C4—162no—
236C4#272811yescharacteristic
237C4#320411yescharacteristic
238C4#337612yes—
239C2#3254411yescharacteristic
240C2#3375511yescharacteristic
241C2#3377011yescharacteristic
242C2#3393012yes—
243C2#3393812yes—
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
141. C2×Q8Q32semidirect

Maximal subgroups

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

subgroupindex
#282652
#135382
#135152
#134812
#134502
#132632
#131482
#130152
#129362