Finite Group Databasebeta

← #345677

Subgroups — #345677

Order 512.

543 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 152 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#26506C212yesmaximal
2#26465C212yesmaximal
3#10594C212yesmaximal
4#10592C212yesmaximal
5#10581C212yesmaximal
6#9831C211yesmaximal, characteristic
7#9598C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#9058C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#479—22no—
12#2552C2212yes—
13#2365C2212yes—
14#2099—22no—
15#2086—22no—
16#2066—22no—
17#2026—22no—
18#1990C2212yes—
19#1988C2212yes—
20#1981C2212yes—
21#1956C2212yes—
22#1947—22no—
23#1947C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1918—22no—
28#1863—22no—
29#1834C2211yescharacteristic
30C4×#107C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#332C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8C2312yes—
40C2×#125—22no—
41C2×C4×Q8—22no—
42#500—41no—
43#500Q814yes—
44#479—48no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#107—41no—
48C2×#107D412yes—
49C2×#106D411yescharacteristic
50C2×#106D411yescharacteristic
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—22no—
58C2×#103—22no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#398—42no—
63#391—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#387—42no—
68#384—22no—
69#384C2312yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2312yes—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—41no—
80#340D411yescharacteristic
81#340D412yes—
82#340D411yescharacteristic
83#336—22no—
84#336—22no—
85#335—41no—
86#335—41no—
87#332—42no—
88#322—41no—
89C22×Q8—22no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8C2×Q814yes—
101#125—44no—
102#125—82no—
103C4×Q8—44no—
104C4×Q8—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—42no—
121C2×C42C2411yescharacteristic
122#107—44no—
123#107—42no—
124#107—44no—
125#107—44no—
126#106—41no—
127#106D814yes—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—44no—
133C4⋊C8—44no—
134#103—44no—
135#103—44no—
136#103—44no—
137#103—82no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8C2×Q812yes—
144#95—22no—
145#95—22no—
146#95—42no—
147#95—42no—
148#95—42no—
149C2×Q8—24no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C22×D411yescharacteristic
162C22×C4C22×D411yescharacteristic
163C22×C4H(5,𝔽2)11yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—21no—
172C2×C8—22no—
173C2×C8—21no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8C2×D812yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C4⋊C4—84no—
200C4⋊C4—82no—
201C4⋊C4—162no—
202C4⋊C4C2×D814yes—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42—84no—
208C42C22×Q811yescharacteristic
209C42H(5,𝔽2)11yescharacteristic
210C422−1+411yescharacteristic
211C422−1+411yescharacteristic
212C23D4211yescharacteristic
213Q8—44no—
214Q8—44no—
215Q8—82no—
216Q8—82no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#53711yescharacteristic
237C2×C4#53711yescharacteristic
238C2×C4#54311yescharacteristic
239C2×C4D4×Q811yescharacteristic
240C2×C4D4×Q811yescharacteristic
241C2×C4#56311yescharacteristic
242C2×C4C22×D811yescharacteristic
243C2×C4#57511yescharacteristic
244C2×C4#57512yes—
245C2×C4#57611yescharacteristic
246C2×C4#57811yescharacteristic
247C2×C4#57812yes—
248C8—22no—
249C8—42no—
250C8—41no—
251C8—41no—
252C8—41no—
253C8—82no—
254C8—82no—
255C22#268311yescharacteristic
256C22#326912yes—
257C22#327711yescharacteristic
258C22D4×D811yescharacteristic
259C22#328912yes—
260C4—21no—
261C4—21no—
262C4—44no—
263C4—84no—
264C4—82no—
265C4—82no—
266C4—82no—
267C4—162no—
268C4#324511yescharacteristic
269C4#325611yescharacteristic
270C4#326111yescharacteristic
271C4#326211yescharacteristic
272C4Q8×D811yescharacteristic
273C4#338911yescharacteristic
274C4#339012yes—
275C2#1976012yes—
276C2#2023712yes—
277C2#3123411yescharacteristic
278C2#3124911yescharacteristic
279C2#3125911yescharacteristic
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

Maximal subgroups

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

subgroupindex
#265062
#264652
#105942
#105922
#105812
#98312
#95982
#95962
#90582