Finite Group Databasebeta

← #314642

Subgroups — #314642

Order 512.

480 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 118 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#21927C211yesmaximal, characteristic, complemented
2#21858C211yesmaximal, characteristic
3#10668C211yesmaximal, characteristic, complemented
4#10667C211yesmaximal, characteristic, complemented
5#10630C211yesmaximal, characteristic, complemented
6#10629C211yesmaximal, characteristic, complemented
7#9584C212yesmaximal, complemented
8#9581C212yesmaximal, complemented
9#8861C212yesmaximal, complemented
10#8733C211yesmaximal, characteristic, complemented
11#8411C212yesmaximal
12#3455C2211yescharacteristic
13#2905C2211yescharacteristic
14#2298C2211yescharacteristic
15#1995C2211yescharacteristic
16#1994C2211yescharacteristic
17#1963C2211yescharacteristic
18#1962C2211yescharacteristic
19#1924—22no—
20#1917—22no—
21#1871—22no—
22#1843C2211yescharacteristic
23#1832C2211yescharacteristic
24#1806—22no—
25#1798C2212yes—
26#1767C2212yes—
27#1571C2212yes—
28#1571C2212yes—
29#1542C2212yes—
30#1540C2212yes—
31#1481C2212yes—
32#1480C2212yes—
33#1458C2212yes—
34#1301C2212yes—
35#1301C2212yes—
36#1276C2212yes—
37#1274C2211yescharacteristic
38#1274C2211yescharacteristic
39C22×M16—21no—
40#542—22no—
41#541—21no—
42#540—21no—
43#539—22no—
44#538—22no—
45#533C2311yescharacteristic
46#532C2311yescharacteristic
47#529—42no—
48#525—21no—
49C4×#41—21no—
50C2×#117C2311yescharacteristic
51#431C2312yes—
52#431C2312yes—
53#424—42no—
54#423—21no—
55#423C2311yescharacteristic
56#423C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#104—22no—
61C2×#104—22no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64#388—22no—
65#387—22no—
66C4×C4⋊C4D412yes—
67C4×C22⋊C4D412yes—
68#376C2311yescharacteristic
69C43C2311yescharacteristic
70#339—22no—
71#339—22no—
72#336—22no—
73#336—22no—
74#336—22no—
75#335—42no—
76#335—42no—
77#333—22no—
78#332—22no—
79#332—22no—
80#331—22no—
81Q8⋊C8—42no—
82D4⋊C8—42no—
83#324—22no—
84#323—22no—
85C2×#41—21no—
86C23×C4C2×D411yescharacteristic
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—41no—
90C2×M16—82no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C4⋊Q8#4111yescharacteristic
95C4⋊Q8#4112yes—
96#127#4111yescharacteristic
97#126—42no—
98#126—42no—
99#125—21no—
100#125—42no—
101#124—21no—
102#124—42no—
103#123—42no—
104#123—42no—
105#123—41no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8—42no—
109#121—22no—
110#121—22no—
111#121—42no—
112#120—41no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—42no—
118C4×D4—42no—
119#117—21no—
120#117—21no—
121#117—22no—
122#117—21no—
123#117—41no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4#4112yes—
126C2×C22⋊C4—22no—
127C2×C22⋊C4#4112yes—
128C2×C42—22no—
129C2×C42—41no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2411yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—42no—
143C4⋊C8—82no—
144#104—82no—
145#104—82no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—82no—
149#97—82no—
150C4×C8—22no—
151C4×C8—22no—
152#95—22no—
153#95—22no—
154#95—22no—
155C24#12311yescharacteristic
156#41—42no—
157#41—42no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—41no—
163C2×D4—42no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4#12311yescharacteristic
173C22×C4#12312yes—
174C22×C4C22×D411yescharacteristic
175C22×C4C2×#4112yes—
176M16—42no—
177M16—42no—
178M16—82no—
179C2×C8—42no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—82no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—41no—
205C22⋊C4—42no—
206C22⋊C4—82no—
207C42—21no—
208C42—21no—
209C42—22no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—42no—
214C42—41no—
215C42—82no—
216C42C2×#4111yescharacteristic
217C42C2×#4111yescharacteristic
218C42H(5,𝔽2)12yes—
219C23—21no—
220C23—22no—
221C23—42no—
222C23—41no—
223C23C2×#12311yescharacteristic
224Q8—42no—
225Q8—82no—
226D4—42no—
227D4—82no—
228D4—82no—
229D4—162no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—162no—
249C2×C4#52511yescharacteristic
250C2×C4#53511yescharacteristic
251C2×C4#55112yes—
252C2×C4#55112yes—
253C2×C4#57612yes—
254C2×C4#57812yes—
255C8—82no—
256C8—82no—
257C8—82no—
258C8—82no—
259C8—81no—
260C8—81no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—82no—
265C22—81no—
266C22—82no—
267C22#248211yescharacteristic
268C22#309812yes—
269C4—22no—
270C4—22no—
271C4—41no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—81no—
276C4—82no—
277C4#312412yes—
278C4#322011yescharacteristic
279C4#322311yescharacteristic
280C2—22no—
281C2—81no—
282C2—161no—
283C2#2759511yescharacteristic
284C2#2759912yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2482
prime radical#3455C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21927C2semidirect
3. #10668C2semidirect
4. #10667C2semidirect
5. #10630C2semidirect
6. #10629C2semidirect
7. #9584C2semidirect
8. #9581C2semidirect
9. #8861C2semidirect
10. #8733C2semidirect

Maximal subgroups

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

subgroupindex
#219272
#218582
#106682
#106672
#106302
#106292
#95842
#95812
#88612
#87332
#84112