Finite Group Databasebeta

← #341670

Subgroups — #341670

Order 512.

480 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 146 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#28836C212yesmaximal
2#26441C212yesmaximal
3#10685C211yesmaximal, characteristic
4#10654C212yesmaximal
5#10580C212yesmaximal
6#10352C211yesmaximal, characteristic
7#9576C212yesmaximal
8#9540C212yesmaximal
9#8928C211yesmaximal, characteristic
10#2571C2211yescharacteristic
11#2552C2212yes—
12#2369C2211yescharacteristic
13#2088—22no—
14#2085—22no—
15#2084—22no—
16#2059—22no—
17#1990C2212yes—
18#1989—22no—
19#1987C2212yes—
20#1956C2211yescharacteristic
21#1948C2212yes—
22#1948C2212yes—
23#1947C2212yes—
24#1944—22no—
25#1939C2212yes—
26#1918—22no—
27#1847—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⋊Q8C2311yescharacteristic
40C2×#125—22no—
41C2×#107—22no—
42C2×#107D412yes—
43C2×#107D412yes—
44C2×#106—22no—
45C2×#106—22no—
46C2×#106—22no—
47C2×#106—22no—
48C2×#106—41no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#103—41no—
53C2×#97—22no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2312yes—
56#400—41no—
57#398—42no—
58#391—41no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#384—22no—
64#384C2311yescharacteristic
65#384C2312yes—
66#382—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2312yes—
69C4×C4⋊C4C2312yes—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340D412yes—
78#340D412yes—
79#336—22no—
80#336—22no—
81#333Q814yes—
82C22×Q8—21no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8C2×D412yes—
87C22×C8C2×D412yes—
88C4⋊Q8—22no—
89C4⋊Q8—21no—
90C4⋊Q8C2×Q812yes—
91#125—44no—
92#125—82no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C2×D412yes—
105C2×C4⋊C4C2×D412yes—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—41no—
111C2×C42C2411yescharacteristic
112#107—24no—
113#107—42no—
114#107—82no—
115#106—44no—
116#106—82no—
117#106—82no—
118#106—82no—
119#106—82no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—82no—
124C4⋊C8C2×Q814yes—
125#103—82no—
126#97—44no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130#95—22no—
131#95—22no—
132#95—42no—
133#95—41no—
134#95—42no—
135#95—42no—
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—81no—
139C2×Q8—81no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4C22×D412yes—
148C22×C4H(5,𝔽2)11yescharacteristic
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—24no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8#13714yes—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C4⋊C4—162no—
185C4⋊C4—161no—
186C4⋊C4—161no—
187C42—42no—
188C42—42no—
189C42—42no—
190C42—42no—
191C42—161no—
192C42C22×Q811yescharacteristic
193C422−1+411yescharacteristic
194C422−1+412yes—
195C23D4211yescharacteristic
196Q8—82no—
197Q8—161no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4#53612yes—
221C2×C4#54411yescharacteristic
222C2×C4D4×Q812yes—
223C2×C4#56311yescharacteristic
224C2×C4C2×#13712yes—
225C2×C4#57512yes—
226C2×C4#57712yes—
227C2×C4#57712yes—
228C8—24no—
229C8—42no—
230C8—44no—
231C8—42no—
232C8—82no—
233C22#267711yescharacteristic
234C22#326712yes—
235C22#327812yes—
236C22#328811yescharacteristic
237C22#329111yescharacteristic
238C4—21no—
239C4—21no—
240C4—82no—
241C4—82no—
242C4—82no—
243C4—82no—
244C4—82no—
245C4—162no—
246C4—161no—
247C4#323612yes—
248C4#324412yes—
249C4#338712yes—
250C4#338812yes—
251C2#2014311yescharacteristic
252C2#2024912yes—
253C2#2032311yescharacteristic
254C2#3096812yes—
255C2#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

Maximal subgroups

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

subgroupindex
#288362
#264412
#106852
#106542
#105802
#103522
#95762
#95402
#89282