Finite Group Databasebeta

← #280932

Subgroups — #280932

Order 512.

451 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 126 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#28295C211yesmaximal, characteristic
2#22816C211yesmaximal, characteristic
3#10637C212yesmaximal, complemented
4#10469C212yesmaximal, complemented
5#9668C211yesmaximal, characteristic
6#9630C211yesmaximal, characteristic, complemented
7#9177C212yesmaximal, complemented
8#8793C211yesmaximal, characteristic, complemented
9#8771C212yesmaximal, complemented
10#8675C211yesmaximal, characteristic, complemented
11#8256C211yesmaximal, characteristic, complemented
12#3077C2212yes—
13#2960C2211yescharacteristic
14C2×C4×D8C2211yescharacteristic
15#2357C2212yes—
16#2063—22no—
17#1978—21no—
18#1972—21no—
19#1952—21no—
20#1925—22no—
21#1848C2211yescharacteristic
22#1846C2211yescharacteristic
23#1804C2212yes—
24#1800C2212yes—
25#1776—21no—
26#1770C2211yescharacteristic
27#1571C2212yes—
28#1564C2211yescharacteristic
29#1542C2211yescharacteristic
30#1518C2212yes—
31#1503C2211yescharacteristic
32#1478C2212yes—
33#1460C2212yes—
34#1450C2211yescharacteristic
35#1303C2211yescharacteristic
36#1303C2212yes—
37#1301C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41C22×D8D411yescharacteristic
42#533C2311yescharacteristic
43C2×C4×D4C2312yes—
44#478—42no—
45#476—22no—
46#474—22no—
47#446—21no—
48#446—21no—
49C8×Q8—21no—
50C8×Q8—21no—
51C4×D8—21no—
52C4×D8—21no—
53C4×D8—21no—
54C4×D8—21no—
55#431C2311yescharacteristic
56#431C2311yescharacteristic
57#431C2312yes—
58C2×#108—21no—
59C2×#108—21no—
60#428—22no—
61C2×#107—21no—
62C2×#107D411yescharacteristic
63C2×#106—21no—
64#424—21no—
65#423C2312yes—
66#423C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68#418—22no—
69C2×#102—22no—
70C2×#102D412yes—
71C2×#100—22no—
72#405—21no—
73C2×#97—21no—
74C2×C4×C8C2311yescharacteristic
75#390—22no—
76#388—42no—
77#382C2312yes—
78#381—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2311yescharacteristic
81C4×C22⋊C4—22no—
82#343—22no—
83#340—22no—
84#336—21no—
85#335—21no—
86#335—42no—
87#334—21no—
88#333—21no—
89#332—21no—
90#328—22no—
91#328—42no—
92D4⋊C8—42no—
93#323—42no—
94#322—21no—
95C82—21no—
96C22×D4C2×D412yes—
97C23×C4—22no—
98C2×D8—21no—
99C2×D8—21no—
100C2×D8—22no—
101C2×D8—22no—
102C2×M16—21no—
103C2×M16—22no—
104C22×C8—21no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8#4111yescharacteristic
108C4⋊Q8#4111yescharacteristic
109#125—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118#117—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C2×D412yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C42—21no—
126C2×C42—42no—
127C2×C42C2411yescharacteristic
128#108—42no—
129#108—81no—
130#107—21no—
131#107—21no—
132#107—42no—
133#107—81no—
134#106—42no—
135#106—42no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—22no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—22no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—22no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147#102—22no—
148#102—22no—
149#102—42no—
150#102—42no—
151#102—82no—
152#100—82no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—22no—
156#97—41no—
157#97—42no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—41no—
162C4×C8#4111yescharacteristic
163C4×C8#4111yescharacteristic
164C4×C8#4111yescharacteristic
165C4×C8#4111yescharacteristic
166#95—22no—
167#95—22no—
168#95—42no—
169#95#4112yes—
170C24—22no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—162no—
185C22×C4C22×D411yescharacteristic
186C22×C4C2×#4111yescharacteristic
187C22×C4H(5,𝔽2)11yescharacteristic
188D8—42no—
189D8—42no—
190M16—42no—
191M16—41no—
192M16—42no—
193M16—41no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8#12311yescharacteristic
205C2×C8#12311yescharacteristic
206C2×C8#12311yescharacteristic
207C2×C8#12311yescharacteristic
208C2×C8#13512yes—
209C2×C8#13712yes—
210C4⋊C4—22no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—84no—
225C22⋊C4—162no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—162no—
230C42C2×#4111yescharacteristic
231C42C2×#4111yescharacteristic
232C42C2×#4111yescharacteristic
233C42H(5,𝔽2)11yescharacteristic
234C23—42no—
235C23—42no—
236C23—42no—
237C23—82no—
238C23—82no—
239C23#54611yescharacteristic
240Q8—81no—
241Q8—81no—
242D4—42no—
243D4—42no—
244D4—42no—
245D4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—162no—
259C2×C4—162no—
260C2×C4C2×#12311yescharacteristic
261C2×C4#52911yescharacteristic
262C2×C4#54011yescharacteristic
263C2×C4#54611yescharacteristic
264C2×C4#55311yescharacteristic
265C2×C4#55511yescharacteristic
266C2×C4C2×#13511yescharacteristic
267C2×C4C2×#13711yescharacteristic
268C2×C4#57611yescharacteristic
269C2×C4#57811yescharacteristic
270C8—21no—
271C8—21no—
272C8—82no—
273C8—81no—
274C8—81no—
275C8—81no—
276C8—81no—
277C8—82no—
278C8#48512yes—
279C8#48512yes—
280C22—21no—
281C22—21no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—162no—
286C22—162no—
287C22#242011yescharacteristic
288C22#330211yescharacteristic
289C22#333311yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—82no—
295C4—81no—
296C4—82no—
297C4—81no—
298C4—162no—
299C4C2×#48511yescharacteristic
300C4#309811yescharacteristic
301C4#330511yescharacteristic
302C4#332811yescharacteristic
303C2—21no—
304C2—21no—
305C2—162no—
306C2#2497011yescharacteristic
307C2#2498611yescharacteristic
308C2#2498811yescharacteristic
subgroupG/H
center, socleC22#2420
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10637C2semidirect
4. #10469C2semidirect
6. #9630C2semidirect
7. #9177C2semidirect
8. #8793C2semidirect
9. #8771C2semidirect
10. #8675C2semidirect
11. #8256C2semidirect

Maximal subgroups

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

subgroupindex
#282952
#228162
#106372
#104692
#96682
#96302
#91772
#87932
#87712
#86752
#82562