Finite Group Databasebeta

← #166117

Subgroups — #166117

Order 512.

873 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#2809C211yesmaximal, characteristic
2#24064C212yesmaximal
3#23455C211yesmaximal, characteristic
4#7771C218yesmaximal, complemented
5#7607C212yesmaximal
6#7595C211yesmaximal, characteristic
7C2×#533C2211yescharacteristic
8C2×#484—22no—
9C2×#419—21no—
10#2809—22no—
11#2809—24no—
12#2809C2214yes—
13C2×#400—21no—
14C2×#398—22no—
15C2×#391—21no—
16C2×#391C2211yescharacteristic
17C2×#389C2212yes—
18C2×#387—22no—
19#2057C2214yes—
20#2043C2214yes—
21#2026—22no—
22#2024C2214yes—
23#1872—22no—
24#1863—24no—
25#1817C2212yes—
26#1736—21no—
27#1728C2211yescharacteristic
28#1272C2214yes—
29#1271C2218yes—
30C23×Q8D411yescharacteristic
31C22×M16—21no—
32#533—44no—
33C2×C4⋊Q8C2312yes—
34C2×#125—21no—
35C2×C22⋊Q8—22no—
36C2×C4×Q8—22no—
37C2×#117—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#484—48no—
47C2×#107—24no—
48#419—44no—
49C2×#103—24no—
50C2×#103—24no—
51C2×#103—24no—
52C2×C22⋊C8—22no—
53#400—24no—
54#400—24no—
55#398—28no—
56#398—28no—
57#391—24no—
58#391—24no—
59#391—22no—
60#391D412yes—
61#391C2312yes—
62#389—22no—
63#389—22no—
64#389D414yes—
65#389C2314yes—
66#387—28no—
67#387—28no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—22no—
72C2×#95—22no—
73#340—24no—
74#340—24no—
75#336—22no—
76#336—21no—
77#336C2314yes—
78#336C2312yes—
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—42no—
83C22×Q8—42no—
84C22×Q8Q1614yes—
85C22×Q8C2×D412yes—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—42no—
95C22×C8—22no—
96C22×C8—24no—
97C4⋊Q8—44no—
98#125—44no—
99C22⋊Q8—28no—
100C22⋊Q8—44no—
101C4×Q8—44no—
102C4×Q8—44no—
103#117—24no—
104#117—42no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4—82no—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D414yes—
124C2×C22⋊C4—41no—
125C2×C42—24no—
126C2×C42—42no—
127C2×C42—41no—
128C2×C42—82no—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2411yescharacteristic
132#107—48no—
133#103—48no—
134#103—48no—
135#103—48no—
136C22⋊C8—44no—
137#95—24no—
138#95—28no—
139#95—28no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144C24#12011yescharacteristic
145C2×Q8—22no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—48no—
151C2×Q8—42no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—84no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×Q8C2×Q1614yes—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4#12012yes—
180C22×C4#12011yescharacteristic
181C22×C4C2×Q1612yes—
182C22×C4#13712yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D412yes—
185M16—84no—
186C2×C8—44no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—44no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—48no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—48no—
203C4⋊C4—48no—
204C4⋊C4—81no—
205C4⋊C4—82no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C22⋊C4—44no—
209C42—42no—
210C42—41no—
211C42—41no—
212C42—44no—
213C42—84no—
214C23—22no—
215C23—22no—
216C23#45112yes—
217C23#45512yes—
218C23B(4,𝔽2)12yes—
219C23C2×#12011yescharacteristic
220Q8—44no—
221Q8—48no—
222Q8—82no—
223Q8—84no—
224Q8—84no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—84no—
251C2×C4—88no—
252C2×C4—88no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#45112yes—
256C2×C4C22×Q1611yescharacteristic
257C2×C4C2×#13711yescharacteristic
258C2×C4#57514yes—
259C8—84no—
260C8—82no—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—24no—
265C22#160214yes—
266C22C2×#45111yescharacteristic
267C22#300512yes—
268C22C2×#45511yescharacteristic
269C22#302012yes—
270C22C2×B(4,𝔽2)11yescharacteristic
271C4—21no—
272C4—21no—
273C4—22no—
274C4—24no—
275C4—44no—
276C4—42no—
277C4—82no—
278C4—84no—
279C4—84no—
280C4—84no—
281C2—24no—
282C2C2×#160211yescharacteristic
283C2#1813412yes—
284C2#1855211yescharacteristic
285C2#1877912yes—
286C2#1899911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7771C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
#240642
#234552
#77712
#76072
#75952