Finite Group Databasebeta

← #341661

Subgroups — #341661

Order 512.

576 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 146 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#28797C212yesmaximal, complemented
2#26460C212yesmaximal
3#10671C211yesmaximal, characteristic
4#10657C212yesmaximal, complemented
5#10573C212yesmaximal, complemented
6#10351C211yesmaximal, characteristic
7#9571C212yesmaximal
8#9540C212yesmaximal, complemented
9#8924C211yesmaximal, characteristic
10C2×#474—22no—
11#2571C2211yescharacteristic
12#2551C2212yes—
13#2366C2211yescharacteristic
14#2085—22no—
15#2067—22no—
16#2057—22no—
17#2023—22no—
18#1993—22no—
19#1982C2212yes—
20#1957C2212yes—
21#1957C2211yescharacteristic
22#1948C2212yes—
23#1947C2212yes—
24#1947C2212yes—
25#1944—22no—
26#1939C2212yes—
27#1862—22no—
28#1845C2211yescharacteristic
29#1834C2212yes—
30C4×#107C2212yes—
31#1770C2212yes—
32C2×#331C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×#127C2311yescharacteristic
42C2×C4×D4—22no—
43#474—48no—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107—41no—
47C2×#107D412yes—
48C2×#107D412yes—
49C2×#106—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#102—22no—
54C2×#102—22no—
55C2×#102—22no—
56C2×#102—22no—
57C2×#102—41no—
58C2×#97—22no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2312yes—
61#400—41no—
62#395—42no—
63#390—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#386—41no—
68#384C2311yescharacteristic
69#384C2312yes—
70#382—22no—
71C4×C4⋊C4C2312yes—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340D412yes—
79#340D412yes—
80#336—22no—
81#336—22no—
82#331Q814yescomplemented
83C22×Q8—42no—
84C22×D4—22no—
85C22×D4—21no—
86C23×C4—41no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8C2×D412yes—
91C22×C8C2×D412yes—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94C4⋊Q8—82no—
95C4⋊Q8—82no—
96#127—21no—
97#127—22no—
98#127C2×Q812yes—
99C4×D4—44no—
100C4×D4—82no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C2×D412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42C2411yescharacteristic
118#107—24no—
119#107—42no—
120#107—44no—
121#107—82no—
122#107—82no—
123#106—44no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—44no—
128C4⋊C8C2×Q814yes—
129#102—44no—
130#102—44no—
131#102—44no—
132#102—82no—
133#102—82no—
134#97—82no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138#95—22no—
139#95—22no—
140#95—41no—
141#95—42no—
142#95—42no—
143C24—41no—
144C2×Q8—84no—
145C2×Q8—84no—
146C2×Q8—162no—
147C2×D4—22no—
148C2×D4—24no—
149C2×D4—42no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—81no—
156C2×D4—81no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—84no—
164C22×C4—161no—
165C22×C4C22×D412yes—
166C22×C4H(5,𝔽2)11yescharacteristic
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—24no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—84no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—84no—
199C4⋊C4—161no—
200C4⋊C4AGL(1,ℤ/8)14yes—
201C22⋊C4—84no—
202C22⋊C4—161no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42C22×Q811yescharacteristic
208C422−1+411yescharacteristic
209C422−1+412yes—
210C23—42no—
211C23—82no—
212C23—82no—
213C23—81no—
214C23—81no—
215C23D4211yescharacteristic
216Q8—168no—
217D4—44no—
218D4—44no—
219D4—82no—
220D4—82no—
221D4—82no—
222D4—161no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—164no—
243C2×C4#53612yes—
244C2×C4#54411yescharacteristic
245C2×C4D4×Q812yes—
246C2×C4#56311yescharacteristic
247C2×C4C2×AGL(1,ℤ/8)12yes—
248C2×C4#57512yes—
249C2×C4#57612yes—
250C2×C4#57812yes—
251C8—42no—
252C8—44no—
253C8—42no—
254C8—42no—
255C8—82no—
256C22—82no—
257C22—84no—
258C22—161no—
259C22—161no—
260C22—162no—
261C22#267711yescharacteristic
262C22#326412yes—
263C22#327712yes—
264C22#328911yescharacteristic
265C22#329211yescharacteristic
266C4—21no—
267C4—21no—
268C4—44no—
269C4—82no—
270C4—82no—
271C4—82no—
272C4—162no—
273C4—161no—
274C4#323512yes—
275C4#324312yes—
276C4#338612yes—
277C4#338912yes—
278C2—162no—
279C2#2005311yescharacteristic
280C2#2016812yes—
281C2#2031911yescharacteristic
282C2#3096612yes—
283C2#3097811yescharacteristic
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. #28797C2semidirect
4. #10657C2semidirect
5. #10573C2semidirect
8. #9540C2semidirect
82. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#287972
#264602
#106712
#106572
#105732
#103512
#95712
#95402
#89242