Finite Group Databasebeta

← #238115

Subgroups — #238115

Order 512.

1032 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2571C211yesmaximal, characteristic, complemented
2#21395C212yesmaximal
3#21200C211yesmaximal, characteristic
4#10352C214yesmaximal, complemented
5#10351C214yesmaximal, complemented
6#8040C211yesmaximal, characteristic, complemented
7#7904C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#432C2212yes—
10C2×#417—22no—
11C2×C4×M16C2211yescharacteristic
12#2571—22no—
13#2571—24no—
14#2571C2214yes—
15C2×#400—22no—
16C2×#389—22no—
17C2×#384C2211yescharacteristic
18C2×#382—22no—
19C2×C4×C4⋊C4C2212yes—
20#1987C2214yes—
21#1982C2214yes—
22#1959—22no—
23#1957C2212yes—
24#1956C2212yes—
25#1874—22no—
26#1874—22no—
27#1762—24no—
28#1296C2214yes—
29#1279C2218yes—
30C22×#41—21no—
31C22×M16—21no—
32#532—24no—
33#532—22no—
34#532Q814yes—
35C2×C4⋊Q8C2311yescharacteristic
36C2×#127C2311yescharacteristic
37C2×#124—21no—
38C2×#121—22no—
39C2×C4×D4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C42—22no—
46C22×C42C2311yescharacteristic
47#432—24no—
48#432—24no—
49C2×C4⋊C8C2314yes—
50#417—48no—
51C2×#103—24no—
52C2×#102—24no—
53C2×#102—24no—
54C2×#102—24no—
55C2×C22⋊C8—22no—
56C4×M16—24no—
57C2×#97—21no—
58C2×#97—22no—
59C2×C4×C8C2312yes—
60#400—28no—
61#400—28no—
62#389—28no—
63#389—28no—
64#384—22no—
65#384D412yes—
66#384C2312yes—
67#382—28no—
68#382—28no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C4×C4⋊C4C2314yes—
73C2×#95—22no—
74C2×#95—24no—
75C2×#95—22no—
76#340—24no—
77#340—24no—
78C2×#41—24no—
79C2×#41—44no—
80C2×#41—44no—
81C2×#41—42no—
82C22×Q8—21no—
83C22×D4—22no—
84C22×D4—21no—
85C22×D4—41no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—41no—
90C23×C4C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C2×M16—24no—
93C2×M16—42no—
94C2×M16—42no—
95C22×C8—22no—
96C22×C8—24no—
97C4⋊Q8—22no—
98C4⋊Q8—21no—
99C4⋊Q8C2×Q812yes—
100#127—22no—
101#127—21no—
102#127C2×Q812yes—
103#124—24no—
104#124—42no—
105#121—28no—
106#121—44no—
107C4×D4—28no—
108C4×D4—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—84no—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C22⋊C4—42no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—48no—
128C2×C42C2×D412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C2×C42#4112yes—
133C2×C42C2411yescharacteristic
134C4⋊C8—24no—
135C4⋊C8—24no—
136#103—48no—
137#102—48no—
138#102—48no—
139#102—48no—
140C22⋊C8—44no—
141#97—42no—
142#97—44no—
143C4×C8—24no—
144#95—28no—
145#95—216no—
146#95—24no—
147#95—24no—
148C24—41no—
149C24#12011yescharacteristic
150#41—416no—
151#41—88no—
152C2×Q8—22no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—41no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—82no—
166C2×D4—82no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—41no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4#12011yescharacteristic
187C22×C4#12012yes—
188C22×C4C22⋊Q814yes—
189C22×C4C22⋊Q812yes—
190C22×C4#12414yes—
191C22×C4#12412yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194M16—48no—
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—44no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—42no—
211C4⋊C4—48no—
212C4⋊C4—82no—
213C4⋊C4—88no—
214C22⋊C4—48no—
215C42—48no—
216C42—48no—
217C42C22⋊Q814yes—
218C42C22⋊Q812yes—
219C42#12414yes—
220C42#12412yes—
221C42C22×Q811yescharacteristic
222C42C2×#4112yes—
223C42C2×#4111yescharacteristic
224C23—22no—
225C23—22no—
226C23—42no—
227C23—82no—
228C23—82no—
229C23—81no—
230C23—81no—
231C23#39312yes—
232C23#45312yes—
233C23#45412yes—
234C23C2×#12011yescharacteristic
235Q8—44no—
236Q8—82no—
237D4—44no—
238D4—48no—
239D4—82no—
240D4—88no—
241D4—84no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—28no—
251C2×C4—28no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—88no—
262C2×C4—88no—
263C2×C4—88no—
264C2×C4—88no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4#39314yes—
268C2×C4#39312yes—
269C2×C4C2×C22⋊Q811yescharacteristic
270C2×C4C2×C22⋊Q812yes—
271C2×C4C2×#12411yescharacteristic
272C2×C4C2×#12412yes—
273C2×C4#57514yes—
274C2×C4#57514yes—
275C8—44no—
276C8—84no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—84no—
282C22—82no—
283C22—161no—
284C22—162no—
285C22—161no—
286C22#202714yes—
287C22C2×#39311yescharacteristic
288C22#300512yes—
289C22#300512yes—
290C22C2×#45311yescharacteristic
291C22C2×#45411yescharacteristic
292C4—24no—
293C4—21no—
294C4—22no—
295C4—21no—
296C4—84no—
297C4—84no—
298C4—84no—
299C4—82no—
300C4#308014yes—
301C4#314214yes—
302C2—24no—
303C2—162no—
304C2#1852611yescharacteristic
305C2#1853612yes—
306C2#1854711yescharacteristic
307C2#2345312yes—
308C2C2×#202711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. C2×#2571C2semidirect
4. #10352C2semidirect
5. #10351C2semidirect
6. #8040C2semidirect
7. #7904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#213952
#212002
#103522
#103512
#80402
#79042