Finite Group Databasebeta

← #325583

Subgroups — #325583

Order 512.

578 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 162 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#33242C211yesmaximal, characteristic
2#29349C212yesmaximal
3C8×#125C211yesmaximal, characteristic
4#10542C214yesmaximal
5#9733C212yesmaximal
6#9731C212yesmaximal
7#9700C212yesmaximal
8C4×#335C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#494C2211yescharacteristic
11C2×#487C2212yes—
12#2862C2212yes—
13C4×#125C2211yescharacteristic
14#2074—22no—
15#2026—22no—
16#1988C2214yes—
17#1981C2214yes—
18#1969—22no—
19#1969—22no—
20#1939C2212yes—
21#1933—24no—
22#1916—22no—
23#1831C2211yescharacteristic
24C8×C4⋊C4—22no—
25C8×C4⋊C4C2212yes—
26C4×C4⋊C8C2212yes—
27C42×C8C2211yescharacteristic
28C2×#335C2212yes—
29C2×#332C2214yes—
30C2×#332C2214yes—
31#1277C2212yes—
32C22×Q16—21no—
33C22×Q16—21no—
34C2×C4⋊Q8C2312yes—
35C2×C4⋊Q8C2312yes—
36C2×#125D411yescharacteristic
37#500—24no—
38#494—24no—
39#487—44no—
40C2×#107—21no—
41C2×#107—21no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8C2312yes—
44C2×C4⋊C8C2314yes—
45C2×#103—24no—
46C2×#103—24no—
47C2×#103—24no—
48C2×C4×C8—22no—
49C2×C4×C8—22no—
50C2×C4×C8D412yes—
51C2×C4×C8D412yes—
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#391—22no—
55#391—22no—
56#391—24no—
57#389D412yes—
58#387—22no—
59#382—24no—
60#382D411yescharacteristic
61C4×C4⋊C4—22no—
62C4×C4⋊C4C2312yes—
63C43C2311yescharacteristic
64#340—24no—
65#340—24no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336—22no—
70#335—24no—
71#332—44no—
72#332—44no—
73C22×Q8—22no—
74C22×Q8—22no—
75C2×Q16—44no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×Q16—44no—
79C22×C8—22no—
80C22×C8—24no—
81C22×C8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4⋊Q8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88#125—24no—
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⋊C4C2×D412yes—
96C2×C4⋊C4C2×D412yes—
97C2×C42—22no—
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D412yes—
100C2×C42C2×D411yescharacteristic
101C2×C42#4112yes—
102C2×C42C2411yescharacteristic
103#107—42no—
104#107—42no—
105C4⋊C8—24no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108C4⋊C8—24no—
109C4⋊C8—44no—
110#103—48no—
111#103—48no—
112#103—84no—
113C4×C8—24no—
114C4×C8—22no—
115C4×C8—42no—
116C4×C8—44no—
117C4×C8—82no—
118C4×C8#4114yes—
119#95—22no—
120#95—22no—
121#95—22no—
122#95—44no—
123#95—44no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—42no—
131C2×Q8—42no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4#12112yes—
137C22×C4#12112yes—
138C22×C4C22×D411yescharacteristic
139C22×C4C22×D411yescharacteristic
140C22×C4C2×#4111yescharacteristic
141Q16—84no—
142Q16—84no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—42no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—44no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8#12412yes—
154C2×C8#12412yes—
155C2×C8#13514yes—
156C4⋊C4—28no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—44no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4#13514yes—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—21no—
171C42—21no—
172C42—24no—
173C42—42no—
174C42—42no—
175C42C2×#4112yes—
176C42H(5,𝔽2)12yes—
177C23C2×#12111yescharacteristic
178Q8—82no—
179Q8—84no—
180Q8—82no—
181Q8—84no—
182Q8—84no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—84no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#47314yes—
201C2×C4C2×#12411yescharacteristic
202C2×C4#53511yescharacteristic
203C2×C4#55012yes—
204C2×C4#55912yes—
205C2×C4C2×#13512yes—
206C2×C4C2×#13512yes—
207C2×C4#57512yes—
208C2×C4#57611yescharacteristic
209C2×C4#57811yescharacteristic
210C8—22no—
211C8—42no—
212C8—84no—
213C8—82no—
214C8#48814yes—
215C22#254811yescharacteristic
216C22#303512yes—
217C22#305011yescharacteristic
218C22#306111yescharacteristic
219C22C2×#47312yes—
220C4—24no—
221C4—22no—
222C4—42no—
223C4—44no—
224C4—82no—
225C4—162no—
226C4—162no—
227C4C2×#48811yescharacteristic
228C4#314011yescharacteristic
229C4#321212yes—
230C4#339312yes—
231C4#339612yes—
232C2#1912912yes—
233C2#1922312yes—
234C2#2874011yescharacteristic
235C2#2875211yescharacteristic
236C2#2876211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
#332422
#293492
C8×#1252
#105422
#97332
#97312
#97002
C4×#3352