Finite Group Databasebeta

← #341695

Subgroups — #341695

Order 512.

522 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 150 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#28817C212yesmaximal
2#26512C212yesmaximal
3#10685C211yesmaximal, characteristic
4#10653C212yesmaximal
5#10580C212yesmaximal
6#10349C211yesmaximal, characteristic
7#9575C212yesmaximal
8#9537C212yesmaximal
9#8929C211yesmaximal, characteristic
10C2×#501—22no—
11#2552C2212yes—
12#2551C2211yescharacteristic
13#2369C2211yescharacteristic
14#2100—22no—
15#2087—22no—
16#2059—22no—
17#2058—22no—
18#1990C2212yes—
19#1987C2212yes—
20#1956C2211yescharacteristic
21#1948C2212yes—
22#1947—22no—
23#1947C2212yes—
24#1947C2212yes—
25#1943—22no—
26#1939C2212yes—
27#1921—22no—
28#1844C2212yes—
29#1844C2211yescharacteristic
30C4×#107C2212yes—
31#1770C2212yes—
32C2×#333C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2311yescharacteristic
41C2×#125—22no—
42#501—48no—
43C2×#107—22no—
44C2×#107—22no—
45C2×#107D412yes—
46C2×#107D412yes—
47C2×#106—22no—
48C2×#106—22no—
49C2×#106—22no—
50C2×#106—41no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—41no—
55C2×#97—22no—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2312yes—
58#398—42no—
59#395—41no—
60#391—41no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#384—22no—
66#384C2311yescharacteristic
67#384C2312yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2312yes—
70C4×C4⋊C4C2312yes—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340D412yes—
79#340D412yes—
80#336—22no—
81#336—22no—
82#333Q814yescomplemented
83C22×Q8—21no—
84C22×Q8—41no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8C2×D412yes—
89C22×C8C2×D412yes—
90C4⋊Q8—22no—
91C4⋊Q8—21no—
92C4⋊Q8—44no—
93C4⋊Q8—82no—
94C4⋊Q8C2×Q812yes—
95#125—44no—
96#125—82no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—41no—
115C2×C42C2411yescharacteristic
116#107—24no—
117#107—24no—
118#107—44no—
119#107—44no—
120#106—44no—
121#106—44no—
122#106—82no—
123#106—82no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—44no—
128C4⋊C8C2×Q814yes—
129#103—82no—
130#97—44no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134#95—22no—
135#95—22no—
136#95—42no—
137#95—41no—
138#95—42no—
139C2×Q8—22no—
140C2×Q8—41no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—84no—
144C2×Q8—161no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4C22×D412yes—
153C22×C4H(5,𝔽2)11yescharacteristic
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—24no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8AGL(1,ℤ/8)14yes—
169C4⋊C4—22no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—82no—
186C4⋊C4—84no—
187C4⋊C4—84no—
188C4⋊C4—84no—
189C4⋊C4—162no—
190C4⋊C4—161no—
191C4⋊C4#13714yes—
192C42—42no—
193C42—42no—
194C42—42no—
195C42—42no—
196C42—161no—
197C42C22×Q811yescharacteristic
198C422−1+411yescharacteristic
199C422−1+412yes—
200C23D4211yescharacteristic
201Q8—82no—
202Q8—161no—
203Q8—164no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4#53612yes—
227C2×C4#54411yescharacteristic
228C2×C4D4×Q812yes—
229C2×C4#56311yescharacteristic
230C2×C4C2×AGL(1,ℤ/8)12yes—
231C2×C4C2×#13712yes—
232C2×C4#57712yes—
233C2×C4#57812yes—
234C8—24no—
235C8—42no—
236C8—44no—
237C8—42no—
238C8—82no—
239C22#267711yescharacteristic
240C22#327012yes—
241C22#327212yes—
242C22#328811yescharacteristic
243C22#329111yescharacteristic
244C4—21no—
245C4—21no—
246C4—44no—
247C4—82no—
248C4—82no—
249C4—82no—
250C4—82no—
251C4—161no—
252C4—162no—
253C4#323812yes—
254C4#324212yes—
255C4#338612yes—
256C4#338712yes—
257C2#2009711yescharacteristic
258C2#2021112yes—
259C2#2031411yescharacteristic
260C2#3097312yes—
261C2#3097911yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
82. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#288172
#265122
#106852
#106532
#105802
#103492
#95752
#95372
#89292