Finite Group Databasebeta

← #352769

Subgroups — #352769

Order 512.

638 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 186 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#33225C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic
4#30228C211yesmaximal, characteristic
5#30228C211yesmaximal, characteristic
6#30220C211yesmaximal, characteristic
7#9840C214yesmaximal
8#9821C214yesmaximal
9#7841C211yesmaximal, characteristic
10C2×#487C2212yes—
11C2×#487C2212yes—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2859C2211yescharacteristic
16#2100—24no—
17#2099—24no—
18#2099—24no—
19#2043—24no—
20#1991—22no—
21#1991—22no—
22#1990C2214yes—
23#1990C2214yes—
24#1990C2214yes—
25#1956—22no—
26#1956—22no—
27#1956C2214yes—
28#1834C2212yes—
29#1834C2212yes—
30#1744C2211yescharacteristic
31#1744C2211yescharacteristic
32C42×C8C2211yescharacteristic
33C2×#322C2212yes—
34C2×C82C2212yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8C2312yes—
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×#125—21no—
42#487—28no—
43#487—28no—
44C2×#103—24no—
45C2×#103—24no—
46C2×#103—28no—
47C2×C4×C8D412yes—
48C2×C4×C8D412yes—
49C2×C4×C8C2312yes—
50C2×C4×C8C2312yes—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#389—22no—
54#389—22no—
55#389—22no—
56#389—22no—
57#384—22no—
58#384—22no—
59#384—22no—
60#384—21no—
61#384—21no—
62#384C2312yes—
63#384C2312yes—
64#382—21no—
65#382—21no—
66C43C2311yescharacteristic
67#340—24no—
68#340—24no—
69#340—28no—
70#340—28no—
71#340—28no—
72#336D414yes—
73#336D414yes—
74#322—24no—
75C82—24no—
76C22×Q8—22no—
77C22×Q8—22no—
78C22×C8C2×D414yes—
79C22×C8C2×D414yes—
80C22×C8C2×D412yes—
81C22×C8C2×D412yes—
82C4⋊Q8—24no—
83C4⋊Q8—24no—
84C4⋊Q8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88C4⋊Q8—84no—
89#125—44no—
90#125—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C42C2×D412yes—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2411yescharacteristic
112#103—48no—
113#103—48no—
114#103—88no—
115C4×C8—24no—
116C4×C8—24no—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8#4114yes—
122C4×C8#4114yes—
123#95—42no—
124#95—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—48no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4#12712yes—
136C22×C4#12712yes—
137C22×C4C22×D411yescharacteristic
138C22×C4C22×D411yescharacteristic
139C22×C4C22×D411yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—28no—
147C2×C8—28no—
148C2×C8#12414yes—
149C2×C8#12414yes—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—84no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C42—24no—
168C42—24no—
169C42#13514yes—
170C42#13514yes—
171C42C2×#4112yes—
172C42C2×#4112yes—
173C23C2×#12711yescharacteristic
174Q8—84no—
175Q8—84no—
176Q8—88no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—84no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#49514yes—
193C2×C4#49514yes—
194C2×C4C2×#12411yescharacteristic
195C2×C4C2×#12411yescharacteristic
196C2×C4#53212yes—
197C2×C4#53212yes—
198C2×C4C2×#13512yes—
199C2×C4C2×#13512yes—
200C2×C4#57714yes—
201C2×C4#57814yes—
202C8—24no—
203C8—24no—
204C8—44no—
205C8—44no—
206C22#285811yescharacteristic
207C22C2×#49511yescharacteristic
208C22C2×#49511yescharacteristic
209C22#315412yes—
210C22#315612yes—
211C4—24no—
212C4—24no—
213C4—162no—
214C4—162no—
215C4—162no—
216C4—162no—
217C4#313112yes—
218C4#313112yes—
219C4#317414yes—
220C4#317714yes—
221C2#1965712yes—
222C2#1965912yes—
223C2#3322711yescharacteristic
224C2#3324611yescharacteristic
225C2#3324911yescharacteristic
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
#332252
#302302
#302282
#302202
#98402
#98212
#78412