Finite Group Databasebeta

← #321587

Subgroups — #321587

Order 512.

816 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 198 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#33225C211yesmaximal, characteristic, complemented
2#28509C211yesmaximal, characteristic
3#28508C211yesmaximal, characteristic
4#21681C211yesmaximal, characteristic
5C2×#1703C214yesmaximal, complemented
6#10275C214yesmaximal, complemented
7#9692C211yesmaximal, characteristic, complemented
8#9691C211yesmaximal, characteristic, complemented
9C4×#334C211yesmaximal, characteristic, complemented
10C2×#487C2212yes—
11C2×#487C2212yes—
12C2×#476C2214yes—
13C2×#474C2214yes—
14C2×#436—22no—
15C2×#436C2212yes—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18C4×#127C2211yescharacteristic
19#2094—22no—
20#2093—22no—
21#1957—21no—
22#1957—21no—
23#1956—22no—
24#1956—22no—
25#1954—22no—
26#1954C2212yes—
27#1845—21no—
28#1844—21no—
29#1842C2211yescharacteristic
30C4×C4⋊C8C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C42×C8C2211yescharacteristic
33#1703—216no—
34C2×#334C2212yes—
35C2×#333C2214yes—
36C2×#332C2214yes—
37#1277C2212yes—
38C22×M16—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2312yes—
43C2×#127D411yescharacteristic
44C2×C4×D4—22no—
45C2×C4×D4C2312yes—
46#487—28no—
47#487—44no—
48#476—28no—
49#476—44no—
50#474—28no—
51#474—44no—
52#436—28no—
53#436—48no—
54C2×#107—24no—
55C2×#106—24no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—24no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#102—24no—
64C2×#102—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#390—22no—
74#390D412yes—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—22no—
80#384—22no—
81#384D411yescharacteristic
82C43C2311yescharacteristic
83#336—21no—
84#336—21no—
85#334D418yescomplemented
86#333—28no—
87#332—28no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×D4—22no—
91C22×D4C2×D412yes—
92C23×C4—22no—
93C2×M16—48no—
94C2×M16—44no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8C2×D412yes—
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107C4⋊Q8—84no—
108#127—22no—
109#127—22no—
110C4×D4—24no—
111C4×D4—24no—
112C4×D4—44no—
113C4×D4—44no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4C2×D412yes—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131#107—48no—
132#106—48no—
133C4⋊C8—24no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—44no—
137C4⋊C8C2×D414yes—
138C4⋊C8C2×D414yes—
139#103—48no—
140#103—44no—
141#103—44no—
142#102—48no—
143#102—48no—
144C22⋊C8—48no—
145C22⋊C8—44no—
146C4×C8—24no—
147C4×C8—24no—
148C4×C8—22no—
149C4×C8—42no—
150C4×C8—42no—
151C4×C8C2×D414yes—
152C24—22no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—84no—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—44no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178M16—88no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—24no—
183C2×C8—24no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—44no—
192C2×C8#12712yes—
193C2×C8#12712yes—
194C4⋊C4—24no—
195C4⋊C4—44no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C22⋊C4—44no—
206C22⋊C4—48no—
207C42—22no—
208C42—24no—
209C42#13514yes—
210C42AGL(1,ℤ/8)14yes—
211C42#13712yes—
212C42#13712yes—
213C42C22×D412yes—
214C42H(5,𝔽2)12yes—
215C23—44no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—44no—
220C23C2×#12111yescharacteristic
221Q8—82no—
222Q8—82no—
223Q8—84no—
224Q8—164no—
225D4—44no—
226D4—44no—
227D4—88no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4#46314yes—
241C2×C4#46314yes—
242C2×C4#47014yes—
243C2×C4#47114yes—
244C2×C4C2×#12711yescharacteristic
245C2×C4#53511yescharacteristic
246C2×C4D4212yes—
247C2×C4#55012yes—
248C2×C4C2×#13512yes—
249C2×C4C2×AGL(1,ℤ/8)12yes—
250C2×C4C2×#13711yescharacteristic
251C2×C4C2×#13711yescharacteristic
252C2×C4#57812yes—
253C8—22no—
254C8—22no—
255C8—42no—
256C8—44no—
257C8—44no—
258C8—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—84no—
263C22—84no—
264C22#254111yescharacteristic
265C22C2×#46312yes—
266C22C2×#47011yescharacteristic
267C22#305911yescharacteristic
268C22C2×#47112yes—
269C4—22no—
270C4—24no—
271C4—82no—
272C4—162no—
273C4—162no—
274C4#170318yescomplemented
275C4#314711yescharacteristic
276C4#315111yescharacteristic
277C4#320712yes—
278C4#327612yes—
279C4#327612yes—
280C2—162no—
281C2C2×#170312yes—
282C2#1919912yes—
283C2#2844311yescharacteristic
284C2#2848911yescharacteristic
285C2#2849411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33225C2semidirect
5. C2×#1703C2semidirect
6. #10275C2semidirect
7. #9692C2semidirect
8. #9691C2semidirect
9. C4×#334C2semidirect
85. #334D4semidirect
274. C4#1703semidirect

Maximal subgroups

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

subgroupindex
#332252
#285092
#285082
#216812
C2×#17032
#102752
#96922
#96912
C4×#3342