Finite Group Databasebeta

← #150442

Subgroups — #150442

Order 512.

356 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 56 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#29114C211yesmaximal, characteristic
2#12581C211yesmaximal, characteristic
3#11507C211yesmaximal, characteristic
4#7428C212yesmaximal, complemented
5#7412C212yesmaximal, complemented
6C2×#410—21no—
7#2851C411yescharacteristic, complemented
8#2297C2211yescharacteristic
9#2074C411yescharacteristic, complemented
10#2073C2211yescharacteristic
11#2026C411yescharacteristic, complemented
12#2025C2211yescharacteristic
13#1936—21no—
14#1935—21no—
15#1825—21no—
16#1788—21no—
17#1749C411yescharacteristic, complemented
18#1464—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×#117—41no—
26C2×#117D411yescharacteristic
27C22×C42C2×C411yescharacteristic
28#432—41no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#102—21no—
32C2×#102—21no—
33#410—24no—
34#410—42no—
35C2×#99—22no—
36C2×#99—41no—
37C2×C22⋊C8—21no—
38C2×#97—21no—
39#391C2×C411yescharacteristic
40#391D411yescharacteristic
41#390D411yescharacteristic
42#390C2311yescharacteristic
43#389—21no—
44#387—41no—
45#386—41no—
46#385—41no—
47#382C2×C411yescharacteristic
48#380—21no—
49C4×C22⋊C4—41no—
50#340—21no—
51#340—21no—
52#340—21no—
53#340—21no—
54#336C2×C411yescharacteristic
55#336C2×C411yescharacteristic
56#324—22no—
57#323—41no—
58C2×#41—22no—
59C2×#41—21no—
60C2×#41—21no—
61C2×#41—41no—
62C2×#41—41no—
63C2×#41—81no—
64C22×Q8C22⋊C411yescharacteristic
65C22×D4—21no—
66C22×D4C2×D411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C22⋊C411yescharacteristic
70C22×C8—21no—
71C22×C8—21no—
72#125—81no—
73#117—41no—
74#117—41no—
75#117—44no—
76#117—82no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—81no—
82C2×C4⋊C4C22⋊C411yescharacteristic
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C22⋊C4—21no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—81no—
92C2×C42—81no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C22×C411yescharacteristic
95C4⋊C8—81no—
96#103—42no—
97#103—81no—
98#102—41no—
99#102—41no—
100#102—81no—
101#99—24no—
102#99—44no—
103#99—42no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106C22⋊C8—42no—
107#97—81no—
108#97—81no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C24—21no—
115C24#9911yescharacteristic
116#41—81no—
117#41—81no—
118#41—81no—
119#41—81no—
120#41—82no—
121#41—82no—
122C2×Q8—21no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8#10412yes—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—81no—
132C2×D4—81no—
133C2×D4AGL(1,ℤ/8)11yescharacteristic
134C2×D4#13711yescharacteristic
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—82no—
156C22×C4#9911yescharacteristic
157C22×C4C2×C22⋊C411yescharacteristic
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—82no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C22⋊C4—44no—
175C22⋊C4—81no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C22⋊C4—82no—
180C42—41no—
181C42—41no—
182C42—41no—
183C42—41no—
184C42—81no—
185C42—84no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—81no—
196C23—81no—
197C23C2×#9911yescharacteristic
198Q8—42no—
199Q8—81no—
200D4—41no—
201D4—41no—
202D4—81no—
203D4—82no—
204D4—82no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—84no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—84no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—162no—
239C2×C4—162no—
240C2×C4#35411yescharacteristic
241C2×C4#35411yescharacteristic
242C2×C4#41711yescharacteristic
243C2×C4C2×#10411yescharacteristic
244C8—81no—
245C8—81no—
246C8—162no—
247C22—21no—
248C22—21no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—161no—
261C22#150411yescharacteristic
262C22#212111yescharacteristic
263C22C2×#35411yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—161no—
274C4—161no—
275C4—164no—
276C2—21no—
277C2—21no—
278C2—81no—
279C2—81no—
280C2—81no—
281C2#1175911yescharacteristic
282C2#1359411yescharacteristic
283C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7428C2semidirect
5. #7412C2semidirect
7. #2851C4semidirect
9. #2074C4semidirect
11. #2026C4semidirect
17. #1749C4semidirect

Maximal subgroups

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

subgroupindex
#291142
#125812
#115072
#74282
#74122