Finite Group Databasebeta

← #305335

Subgroups — #305335

Order 512.

620 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 170 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#26109C212yesmaximal
2Q8×#103C212yesmaximal
3#10676C211yesmaximal, characteristic
4#10589C212yesmaximal
5#10352C211yesmaximal, characteristic
6#8884C212yesmaximal
7#8846C212yesmaximal
8#8647C212yesmaximal
9#8451C211yesmaximal, characteristic
10C2×Q82C2211yescharacteristic
11C2×C8×Q8C2212yes—
12#2571C2211yescharacteristic
13Q8×C4⋊C4C2212yes—
14#2101—22no—
15#2086—22no—
16#1990—22no—
17#1990C2212yes—
18#1987—22no—
19#1987C2212yes—
20#1956C2211yescharacteristic
21#1947C2212yes—
22#1939C2212yes—
23#1863—22no—
24#1845C2211yescharacteristic
25#1834C2212yes—
26#1805—22no—
27#1803—22no—
28#1803C2212yes—
29#1803C2212yes—
30C4×#103—22no—
31C4×#103C2212yes—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×Q8⋊C8C2212yes—
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1273C2211yescharacteristic
39Q82—24no—
40Q82—41no—
41Q82Q814yes—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8C2312yes—
48C8×Q8—24no—
49C8×Q8—42no—
50C2×#107—22no—
51C2×#107—41no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—22no—
58C2×#103—41no—
59C2×#103D412yes—
60C2×#103D412yes—
61C2×C4×C8—22no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2312yes—
64#400—41no—
65#389—22no—
66#389—22no—
67#389—42no—
68#384—22no—
69#384C2311yescharacteristic
70#384C2312yes—
71#382—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2312yes—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340D412yes—
84#340D412yes—
85#336—22no—
86#334—42no—
87#333—41no—
88Q8⋊C8—24no—
89C22×Q8—21no—
90C22×Q8—41no—
91C22×Q8C2×D412yes—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4⋊Q8—28no—
98C4⋊Q8—22no—
99C4⋊Q8—21no—
100C4⋊Q8—44no—
101C4⋊Q8—44no—
102C4⋊Q8—44no—
103C4⋊Q8C2×Q812yes—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—28no—
107C4×Q8—44no—
108C4×Q8—44no—
109C4×Q8—82no—
110C4×Q8C2×Q814yes—
111C4×Q8#4114yes—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D412yes—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—41no—
129C2×C42—42no—
130C2×C42C2411yescharacteristic
131#107—82no—
132#107—82no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—44no—
138C4⋊C8—82no—
139#103—24no—
140#103—24no—
141#103—44no—
142#103—44no—
143#103—82no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—44no—
148#95—22no—
149#95—42no—
150#95—41no—
151#95—42no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—44no—
157C2×Q8—41no—
158C2×Q8—81no—
159C2×Q8—81no—
160C2×Q8—161no—
161C2×Q8—161no—
162C2×Q8C22⋊Q814yes—
163C2×Q8#13514yes—
164C2×Q8#13714yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4C22×D412yes—
173C22×C4H(5,𝔽2)11yescharacteristic
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—48no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—84no—
206C4⋊C4—81no—
207C4⋊C4—82no—
208C4⋊C4—162no—
209C4⋊C4—161no—
210C4⋊C4C22⋊Q814yes—
211C42—24no—
212C42—24no—
213C42—44no—
214C42—42no—
215C42—42no—
216C42—84no—
217C42C22×Q811yescharacteristic
218C42C2×#4112yes—
219C42H(5,𝔽2)11yescharacteristic
220C23D4211yescharacteristic
221Q8—24no—
222Q8—24no—
223Q8—24no—
224Q8—82no—
225Q8—161no—
226Q8—164no—
227Q8#47918yes—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4C2×C22⋊Q812yes—
251C2×C4#54311yescharacteristic
252C2×C4#54612yes—
253C2×C4#55111yescharacteristic
254C2×C4C2×#13512yes—
255C2×C4C2×#13712yes—
256C2×C4#57712yes—
257C2×C4#57812yes—
258C8—42no—
259C8—42no—
260C8—42no—
261C8—82no—
262C8—82no—
263C22#243711yescharacteristic
264C22#327012yes—
265C22#328212yes—
266C22#328911yescharacteristic
267C22#329211yescharacteristic
268C4—28no—
269C4—21no—
270C4—21no—
271C4—44no—
272C4—82no—
273C4—82no—
274C4—82no—
275C4—162no—
276C4—161no—
277C4C2×#47912yes—
278C4#308212yes—
279C4#330012yes—
280C4#332012yes—
281C2#1983012yes—
282C2#2000311yescharacteristic
283C2#2011711yescharacteristic
284C2#2595712yes—
285C2#2598211yescharacteristic
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
#261092
Q8×#1032
#106762
#105892
#103522
#88842
#88462
#86472
#84512