Finite Group Databasebeta

← #475721

Subgroups — #475721

Order 512.

657 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 194 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#21024C211yesmaximal, characteristic
2#20807C211yesmaximal, characteristic
3#11343C212yesmaximal
4#11338C212yesmaximal
5#11332C212yesmaximal
6#11295C212yesmaximal
7#11233C212yesmaximal
8#11223C212yesmaximal
9#10815C211yesmaximal, characteristic
10C2×#430C2212yes—
11C2×#430C2214yes—
12C22×#108C2212yes—
13#2905C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#2296C2211yescharacteristic
17#1917—22no—
18#1916—22no—
19#1849—22no—
20#1848—22no—
21C2×#344C2214yes—
22#1742C2212yes—
23C2×#341C2212yes—
24C2×#341C2214yes—
25C2×#338C2212yes—
26#1729C2211yescharacteristic
27#1729C2211yescharacteristic
28#1729C2211yescharacteristic
29#1728C2211yescharacteristic
30#1728C2214yes—
31C22×M16C2314yes—
32C22×M16C2311yescharacteristic
33C22×M16C2314yes—
34C22×M16C2311yescharacteristic
35C23×C8C2311yescharacteristic
36C23×C8C2311yescharacteristic
37C2×#117C2311yescharacteristic
38C2×#117C2311yescharacteristic
39C22×C42C2311yescharacteristic
40#430—44no—
41#430—44no—
42#430—44no—
43C2×#108—22no—
44C2×#108—24no—
45C2×#108—22no—
46C2×#108—44no—
47C2×#108—44no—
48#424—44no—
49#423—24no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—24no—
53#405—24no—
54C2×#97D412yes—
55C2×C4×C8D412yes—
56#389—21no—
57#389—21no—
58#385Q812yes—
59#385Q812yes—
60C4×C4⋊C4—22no—
61#344—44no—
62#343—22no—
63#343—22no—
64#341—24no—
65#341—44no—
66#338Q818yes—
67#336—21no—
68#336—22no—
69#336—22no—
70#336—21no—
71#336—22no—
72#336—44no—
73C23×C4C2×Q811yescharacteristic
74C23×C4C2×Q811yescharacteristic
75C23×C4C2411yescharacteristic
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—24no—
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—22no—
88C2×M16—24no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—24no—
92C2×M16C2×Q818yes—
93C2×M16C2×Q814yes—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—24no—
102C22×C8—22no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D412yes—
106C22×C8#4114yes—
107#117—22no—
108#117—22no—
109#117—22no—
110#117—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4#4111yescharacteristic
115C2×C4⋊C4#4111yescharacteristic
116C2×C22⋊C4C2×Q812yes—
117C2×C22⋊C4C2×Q812yes—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42C2×D412yes—
124#108—44no—
125#108—44no—
126#108—48no—
127#108—416no—
128C4⋊C8—48no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131#97—24no—
132C4×C8—24no—
133#95—22no—
134#95—22no—
135C24C22×Q811yescharacteristic
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—81no—
143C22×C4C22⋊Q812yes—
144C22×C4C22⋊Q812yes—
145C22×C4C22×D411yescharacteristic
146C22×C4C22×Q812yes—
147C22×C4C2×#4112yes—
148C22×C4C2×#4111yescharacteristic
149C22×C4H(5,𝔽2)12yes—
150C22×C4H(5,𝔽2)12yes—
151C22×C4H(5,𝔽2)12yes—
152C22×C42−1+412yes—
153M16—28no—
154M16—28no—
155M16—28no—
156M16—44no—
157M16—44no—
158M16—44no—
159M16—44no—
160M16—44no—
161M16—44no—
162M16—44no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—28no—
170C2×C8—28no—
171C2×C8—44no—
172C2×C8—44no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—44no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8#12312yes—
184C2×C8#12312yes—
185C2×C8C4⋊Q814yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C42—22no—
194C42—22no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42—41no—
199C42—42no—
200C23—41no—
201C23—41no—
202C23C2×C22⋊Q811yescharacteristic
203C23#54412yes—
204C23#55412yes—
205C23Q8212yes—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4C2×#12311yescharacteristic
223C2×C4C2×C4⋊Q811yescharacteristic
224C2×C4#53511yescharacteristic
225C2×C4#53511yescharacteristic
226C2×C4#53511yescharacteristic
227C2×C4#53611yescharacteristic
228C2×C4#53914yes—
229C2×C4#54212yes—
230C2×C4#54414yes—
231C2×C4#55312yes—
232C2×C4#55414yes—
233C2×C4#55412yes—
234C2×C4#56014yes—
235C8—22no—
236C8—22no—
237C8—24no—
238C8—48no—
239C8—81no—
240C8—84no—
241C8—81no—
242C22—21no—
243C22—22no—
244C22—21no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22#248711yescharacteristic
249C22#256711yescharacteristic
250C22#262811yescharacteristic
251C22#279214yes—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—81no—
257C4#250211yescharacteristic
258C4#262311yescharacteristic
259C4#266011yescharacteristic
260C4#279014yes—
261C4#281811yescharacteristic
262C2—21no—
263C2—21no—
264C2—24no—
265C2#1706711yescharacteristic
266C2#2769111yescharacteristic
267C2#2769211yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC23×C4C24
socleC22#2487
prime radicalC24C22×Q8
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#210242
#208072
#113432
#113382
#113322
#112952
#112332
#112232
#108152