Finite Group Databasebeta

← #329592

Subgroups — #329592

Order 512.

708 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 154 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#32954C212yesmaximal
2#29808C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4#10624C212yesmaximal, complemented
5#10487C214yesmaximal, complemented
6#10276C212yesmaximal, complemented
7#9132C212yesmaximal
8#8599C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#494C2211yescharacteristic
11C2×#475C2214yes—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2056—22no—
16#2001—22no—
17#1984—22no—
18#1984C2212yes—
19#1984C2212yes—
20#1956C2212yes—
21#1865—24no—
22#1853—22no—
23#1834C2211yescharacteristic
24#1817—22no—
25#1803C2214yes—
26#1797C2212yes—
27#1787—22no—
28C2×#335C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C23×Q8D411yescharacteristic
33C22×Q16—22no—
34C2×C4⋊Q8C2312yes—
35C2×C22⋊Q8—22no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—42no—
39C2×C4×Q8—42no—
40C2×C4×Q8D412yes—
41C2×C4×Q8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#494—42no—
47#475—44no—
48#475—44no—
49#431—42no—
50#431—42no—
51C2×#106—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2314yes—
54C2×C4⋊C8C2312yes—
55C2×#103—24no—
56C2×#103—24no—
57C2×#103—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8D412yes—
61C2×C4×C8—21no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—22no—
65#389—22no—
66#389—22no—
67#384C2312yes—
68#380—22no—
69#380—22no—
70#340—22no—
71#340—24no—
72#336—24no—
73#336—24no—
74#336—22no—
75#335—42no—
76#332—44no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—42no—
81C22×Q8—42no—
82C22×Q8Q1614yescomplemented
83C22×Q8C2×D412yes—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C2×Q16—44no—
90C2×Q16—84no—
91C22×C8—22no—
92C22×C8—24no—
93C22×C8C2×D412yes—
94C22×C8C2×D412yes—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C22⋊Q8—84no—
98C4×Q8—24no—
99C4×Q8—24no—
100C4×Q8—44no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×Q8—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—42no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42C2411yescharacteristic
127#106—48no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—42no—
132#103—48no—
133#103—48no—
134#103—84no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—44no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—42no—
141#95—42no—
142#95—42no—
143C24#12111yescharacteristic
144C2×Q8—22no—
145C2×Q8—24no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—48no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—48no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8C2×Q1614yes—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—82no—
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C2×Q1612yes—
177C22×C4#13512yes—
178C22×C4C22×D411yescharacteristic
179C22×C4C22×D411yescharacteristic
180C22×C4C2×#4111yescharacteristic
181Q16—88no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—24no—
185C2×C8—24no—
186C2×C8—44no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C4⋊C4—24no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—82no—
200C4⋊C4—84no—
201C22⋊C4—84no—
202C42—21no—
203C42—22no—
204C42—21no—
205C42—22no—
206C42—44no—
207C42—42no—
208C42—42no—
209C42H(5,𝔽2)12yes—
210C42H(5,𝔽2)12yes—
211C23—21no—
212C23—22no—
213C23—21no—
214C23—22no—
215C23#46712yes—
216C23C2×#12111yescharacteristic
217Q8—28no—
218Q8—48no—
219Q8—44no—
220Q8—84no—
221Q8—84no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—48no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—82no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4#46712yes—
248C2×C4#53412yes—
249C2×C4#53411yescharacteristic
250C2×C4#53511yescharacteristic
251C2×C4#53812yes—
252C2×C4C22×Q1611yescharacteristic
253C2×C4C2×#13511yescharacteristic
254C2×C4#57512yes—
255C2×C4#57714yes—
256C8—42no—
257C8—42no—
258C8—84no—
259C8—82no—
260C22—22no—
261C22—24no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22#261711yescharacteristic
266C22#304312yes—
267C22C2×#46711yescharacteristic
268C22#305411yescharacteristic
269C22#306912yes—
270C4—22no—
271C4—22no—
272C4—41no—
273C4—44no—
274C4—41no—
275C4—82no—
276C4—162no—
277C4—162no—
278C4#318912yes—
279C4#319112yes—
280C4#320012yes—
281C4#322112yes—
282C2—42no—
283C2#1939912yes—
284C2#1943612yes—
285C2#2972211yescharacteristic
286C2#2973111yescharacteristic
287C2#2975211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #10624C2semidirect
5. #10487C2semidirect
6. #10276C2semidirect
82. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#329542
#298082
#218182
#106242
#104872
#102762
#91322
#85992