Finite Group Databasebeta

← #305472

Subgroups — #305472

Order 512.

594 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 158 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#26089C212yesmaximal
2#22525C212yesmaximal
3#10685C211yesmaximal, characteristic
4#10593C212yesmaximal
5#10352C211yesmaximal, characteristic
6#8887C212yesmaximal
7#8848C212yesmaximal
8#8651C212yesmaximal
9#8451C211yesmaximal, characteristic
10C2×Q82C2211yescharacteristic
11C2×#446C2212yes—
12#2571C2211yescharacteristic
13#2355C2212yes—
14#2097—22no—
15#2066—22no—
16#1993—22no—
17#1990C2212yes—
18#1988—22no—
19#1987C2212yes—
20#1956C2211yescharacteristic
21#1949—22no—
22#1947C2212yes—
23#1939C2212yes—
24#1920—22no—
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1805—22no—
28#1803C2212yes—
29#1803C2212yes—
30#1763—22no—
31C4×#103C2212yes—
32C2×#335C2212yes—
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×Q8C2312yes—
47#446—24no—
48#446—42no—
49C2×#107—22no—
50C2×#106—22no—
51C2×#106—22no—
52C2×#106—41no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#103—22no—
58C2×#103—22no—
59C2×#103—41no—
60C2×#103D412yes—
61C2×#103D412yes—
62C2×#97—22no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2312yes—
65#400—41no—
66#391—22no—
67#389—22no—
68#387—42no—
69#384C2311yescharacteristic
70#384C2312yes—
71#382—22no—
72#382—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2312yes—
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#335—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×Q8C2×Q814yes—
109C4×Q8#4114yes—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—41no—
125C2×C42—42no—
126C2×C42C2411yescharacteristic
127#107—82no—
128#106—44no—
129#106—82no—
130#106—82no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—44no—
136C4⋊C8—44no—
137#103—24no—
138#103—42no—
139#103—44no—
140#103—82no—
141#103—82no—
142#97—44no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—22no—
146#95—22no—
147#95—42no—
148#95—42no—
149#95—41no—
150#95—42no—
151#95—42no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8—41no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×Q8—161no—
163C2×Q8—161no—
164C2×Q8C22⋊Q814yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—41no—
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—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8#13714yes—
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—81no—
206C4⋊C4—82no—
207C4⋊C4—161no—
208C4⋊C4C22⋊Q814yes—
209C42—24no—
210C42—24no—
211C42—44no—
212C42—42no—
213C42—42no—
214C42—162no—
215C42C22×Q811yescharacteristic
216C42C2×#4112yes—
217C42H(5,𝔽2)11yescharacteristic
218C23D4211yescharacteristic
219Q8—24no—
220Q8—24no—
221Q8—42no—
222Q8—42no—
223Q8—82no—
224Q8—161no—
225Q8—164no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4C2×C22⋊Q812yes—
249C2×C4#54311yescharacteristic
250C2×C4#54612yes—
251C2×C4#55111yescharacteristic
252C2×C4C2×#13712yes—
253C2×C4#57512yes—
254C2×C4#57712yes—
255C2×C4#57812yes—
256C8—24no—
257C8—42no—
258C8—42no—
259C8—82no—
260C8—82no—
261C22#243711yescharacteristic
262C22#326912yes—
263C22#327812yes—
264C22#328811yescharacteristic
265C22#329111yescharacteristic
266C4—21no—
267C4—21no—
268C4—44no—
269C4—44no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—162no—
274C4—161no—
275C4#307812yes—
276C4#308212yes—
277C4#331212yes—
278C4#332612yes—
279C2#2012511yescharacteristic
280C2#2023612yes—
281C2#2032311yescharacteristic
282C2#2597312yes—
283C2#2598311yescharacteristic
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
#260892
#225252
#106852
#105932
#103522
#88872
#88482
#86512
#84512