Finite Group Databasebeta

← #341589

Subgroups — #341589

Order 512.

534 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 142 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#28799C212yesmaximal, complemented
2#26529C212yesmaximal
3#10677C211yesmaximal, characteristic
4#10655C212yesmaximal, complemented
5#10577C212yesmaximal, complemented
6#10348C211yesmaximal, characteristic
7#9571C212yesmaximal
8#9538C212yesmaximal, complemented
9#8924C211yesmaximal, characteristic
10#2571C2212yes—
11#2551C2211yescharacteristic
12#2366C2211yescharacteristic
13#2092—22no—
14#2085—22no—
15#2067—22no—
16#2058—22no—
17#1993—22no—
18#1982C2212yes—
19#1957C2212yes—
20#1957C2211yescharacteristic
21#1948C2212yes—
22#1947C2212yes—
23#1947C2212yes—
24#1943—22no—
25#1939C2212yes—
26#1864—22no—
27#1844C2211yescharacteristic
28#1834C2212yes—
29#1804—22no—
30C4×#106C2212yes—
31#1770C2212yes—
32C2×#331C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8—22no—
40C2×#127C2311yescharacteristic
41C2×C4×D4—22no—
42C2×#107—22no—
43C2×#107—22no—
44C2×#106—22no—
45C2×#106—41no—
46C2×#106D412yes—
47C2×#106D412yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#102—22no—
52C2×#102—22no—
53C2×#102—22no—
54C2×#102—22no—
55C2×#102—41no—
56C2×#97—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2312yes—
59#400—42no—
60#395—41no—
61#390—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#386—41no—
66#384C2312yes—
67#384C2311yescharacteristic
68#382—22no—
69#382—22no—
70C4×C4⋊C4C2312yes—
71C4×C4⋊C4C2312yes—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340D412yes—
78#340D412yes—
79#336—22no—
80#336—22no—
81#331Q814yescomplemented
82C22×Q8—41no—
83C22×D4—22no—
84C22×D4—21no—
85C23×C4—41no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8C2×D412yes—
90C22×C8C2×D412yes—
91C4⋊Q8—44no—
92C4⋊Q8—82no—
93#127—21no—
94#127—22no—
95#127C2×Q812yes—
96C4×D4—44no—
97C4×D4—82no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C2×D412yes—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—41no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42C2411yescharacteristic
114#107—44no—
115#107—82no—
116#106—42no—
117#106—42no—
118#106—82no—
119#106—82no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—82no—
124C4⋊C8C2×Q814yes—
125#102—44no—
126#102—82no—
127#102—82no—
128#102—82no—
129#102—82no—
130#97—82no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134#95—22no—
135#95—22no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—41no—
140C24—41no—
141C2×Q8—84no—
142C2×Q8—161no—
143C2×D4—22no—
144C2×D4—24no—
145C2×D4—42no—
146C2×D4—41no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—81no—
152C2×D4—81no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—84no—
160C22×C4—161no—
161C22×C4C22×D412yes—
162C22×C4H(5,𝔽2)11yescharacteristic
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—24no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—82no—
194C4⋊C4—84no—
195C4⋊C4—162no—
196C22⋊C4—84no—
197C22⋊C4—161no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42C22×Q811yescharacteristic
203C422−1+411yescharacteristic
204C422−1+412yes—
205C23—42no—
206C23—82no—
207C23—82no—
208C23—81no—
209C23—81no—
210C23D4211yescharacteristic
211Q8—164no—
212D4—44no—
213D4—44no—
214D4—82no—
215D4—82no—
216D4—82no—
217D4—161no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4—164no—
238C2×C4#53612yes—
239C2×C4#54411yescharacteristic
240C2×C4D4×Q812yes—
241C2×C4#56311yescharacteristic
242C2×C4#57512yes—
243C2×C4#57512yes—
244C2×C4#57612yes—
245C2×C4#57712yes—
246C8—42no—
247C8—44no—
248C8—42no—
249C8—42no—
250C8—82no—
251C22—82no—
252C22—84no—
253C22—162no—
254C22—161no—
255C22—161no—
256C22#267711yescharacteristic
257C22#326812yes—
258C22#327512yes—
259C22#328811yescharacteristic
260C22#329111yescharacteristic
261C4—21no—
262C4—21no—
263C4—82no—
264C4—82no—
265C4—82no—
266C4—82no—
267C4—162no—
268C4—161no—
269C4#324012yes—
270C4#324112yes—
271C4#338812yes—
272C4#338912yes—
273C2—162no—
274C2#2015211yescharacteristic
275C2#2026012yes—
276C2#2028911yescharacteristic
277C2#3096312yes—
278C2#3097911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28799C2semidirect
4. #10655C2semidirect
5. #10577C2semidirect
8. #9538C2semidirect
81. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#287992
#265292
#106772
#106552
#105772
#103482
#95712
#95382
#89242