Finite Group Databasebeta

← #353073

Subgroups — #353073

Order 512.

586 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 218 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#33291C211yesmaximal, characteristic
2#33263C212yesmaximal
3#33256C211yesmaximal, characteristic
4#30307C212yesmaximal
5#8821C212yesmaximal
6#8821C212yesmaximal
7#8821C212yesmaximal
8#8819C212yesmaximal
9#7842C211yesmaximal, characteristic
10#2859C2212yes—
11#2859C2212yes—
12#2085—24no—
13#2059—24no—
14#1991—24no—
15#1944—24no—
16#1943—24no—
17#1941—24no—
18#1845C2212yes—
19#1845C2212yes—
20#1844C2212yes—
21#1844C2212yes—
22#1844C2212yes—
23#1834C2211yescharacteristic
24#1834C2211yescharacteristic
25#1834C2212yes—
26#1834C2212yes—
27#1834C2212yes—
28#1834C2212yes—
29#1834C2211yescharacteristic
30#1834C2211yescharacteristic
31#1834C2212yes—
32#1744C2211yescharacteristic
33C4×#97C2211yescharacteristic
34C42×C8C2211yescharacteristic
35C2×#322C2212yes—
36C2×#322C2212yes—
37C2×#125—22no—
38C2×#125—22no—
39C2×#107—24no—
40C2×#107—24no—
41C2×#106—24no—
42C2×#106—24no—
43C2×#106—24no—
44C2×#97D412yes—
45C2×#97D412yes—
46C2×C4×C8D414yes—
47C2×C4×C8C2311yescharacteristic
48C2×C4×C8C2311yescharacteristic
49C2×C4×C8C2312yes—
50C2×C4×C8C2312yes—
51#389—22no—
52#389—22no—
53#389—22no—
54#389—22no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#384C2312yes—
60#384C2312yes—
61#384C2312yes—
62#384C2312yes—
63#382—22no—
64#382—22no—
65C43C2311yescharacteristic
66#340—24no—
67#340—24no—
68#340—22no—
69#340—22no—
70#340—24no—
71#340—24no—
72#340—24no—
73#340—22no—
74#340—22no—
75#336D414yes—
76#322Q818yes—
77#322Q818yes—
78C22×C8C2×D414yes—
79C22×C8C2×D412yes—
80C22×C8C2×D412yes—
81C22×C8C2×D414yes—
82#125—48no—
83#125—84no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C42C2×D412yes—
101C2×C42C2×D412yes—
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2411yescharacteristic
105#107—48no—
106#107—84no—
107#106—48no—
108#106—48no—
109#106—84no—
110#97—42no—
111#97—42no—
112C4×C8—44no—
113C4×C8C2×Q814yes—
114C4×C8C2×Q814yes—
115C4×C8C2×Q814yes—
116C4×C8C2×Q814yes—
117C4×C8C2×Q814yes—
118C4×C8C2×Q814yes—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4#12712yes—
128C22×C4#12712yes—
129C22×C4C22×D411yescharacteristic
130C22×C4C22×D411yescharacteristic
131C22×C4C22×D411yescharacteristic
132C2×C8—24no—
133C2×C8—24no—
134C2×C8—24no—
135C2×C8—24no—
136C2×C8—24no—
137C2×C8—24no—
138C2×C8C4⋊Q814yes—
139C2×C8C4⋊Q818yes—
140C2×C8C4⋊Q814yes—
141C2×C8C4⋊Q818yes—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C4⋊C4—84no—
157C4⋊C4—84no—
158C42—24no—
159C42—22no—
160C42—22no—
161C42—24no—
162C42C22×Q812yes—
163C42C22×Q812yes—
164C23C2×#12711yescharacteristic
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4C2×C4⋊Q811yescharacteristic
184C2×C4C2×C4⋊Q812yes—
185C2×C4C2×C4⋊Q812yes—
186C2×C4C2×C4⋊Q811yescharacteristic
187C2×C4#57514yes—
188C2×C4#57512yes—
189C2×C4#57512yes—
190C2×C4#57714yes—
191C8—24no—
192C8—28no—
193C8—28no—
194C8—24no—
195C22#286811yescharacteristic
196C22#315011yescharacteristic
197C22#315011yescharacteristic
198C22#315312yes—
199C22#315412yes—
200C4—42no—
201C4—42no—
202C4—162no—
203C4—162no—
204C4—162no—
205C4—162no—
206C4#316212yes—
207C4#316314yes—
208C4#316614yes—
209C4#316612yes—
210C2#1969112yes—
211C2#1969212yes—
212C2#3331511yescharacteristic
213C2#3331711yescharacteristic
214C2#3331811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#332912
#332632
#332562
#303072
#88212
#88192
#78422