Finite Group Databasebeta

← #280866

Subgroups — #280866

Order 512.

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

normal subgroups only

This group has 454 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#22810C211yesmaximal, characteristic
3#10641C212yesmaximal, complemented
4#10468C212yesmaximal, complemented
5#9668C211yesmaximal, characteristic
6#9628C211yesmaximal, characteristic, complemented
7#9177C212yesmaximal, complemented
8#8793C211yesmaximal, characteristic, complemented
9#8770C212yesmaximal, complemented
10#8677C211yesmaximal, characteristic, complemented
11#8252C211yesmaximal, characteristic, complemented
12#3077C2212yes—
13#2960C2211yescharacteristic
14C2×C4×D8C2211yescharacteristic
15#2357C2212yes—
16#2063—22no—
17#1978—21no—
18#1970—21no—
19#1952—21no—
20#1926—22no—
21#1848C2211yescharacteristic
22#1846C2211yescharacteristic
23#1804C2212yes—
24#1800C2212yes—
25C4×#107—21no—
26C8×C4⋊C4C2211yescharacteristic
27#1570C2212yes—
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×C4×C8—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#334—21no—
87#334—42no—
88#333—21no—
89#332—21no—
90#328—21no—
91#328—21no—
92#328—42no—
93D4⋊C8—42no—
94#323—42no—
95#322—21no—
96C82—21no—
97C22×D4C2×D412yes—
98C23×C4—22no—
99C2×D8—21no—
100C2×D8—21no—
101C2×D8—22no—
102C2×D8—22no—
103C2×M16—21no—
104C2×M16—22no—
105C22×C8—21no—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8#4111yescharacteristic
109C4⋊Q8#4111yescharacteristic
110#125—21no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119#117—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C42—21no—
127C2×C42—42no—
128C2×C42C2411yescharacteristic
129#108—42no—
130#108—81no—
131#107—21no—
132#107—21no—
133#107—42no—
134#107—42no—
135#106—42no—
136#106—81no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—22no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148#102—22no—
149#102—22no—
150#102—42no—
151#102—42no—
152#102—82no—
153#100—82no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—22no—
157#97—41no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—41no—
162C4×C8—42no—
163C4×C8#4111yescharacteristic
164C4×C8#4111yescharacteristic
165C4×C8#4111yescharacteristic
166C4×C8#4111yescharacteristic
167#95—22no—
168#95—22no—
169#95—42no—
170#95#4111yescharacteristic
171#95#4111yescharacteristic
172C24—22no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—162no—
188C22×C4C22×D411yescharacteristic
189C22×C4C2×#4111yescharacteristic
190C22×C4H(5,𝔽2)11yescharacteristic
191D8—42no—
192D8—42no—
193M16—42no—
194M16—41no—
195M16—42no—
196M16—41no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8#12311yescharacteristic
208C2×C8#12311yescharacteristic
209C2×C8#12311yescharacteristic
210C2×C8#12311yescharacteristic
211C2×C8#13512yes—
212C2×C8#13712yes—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—84no—
228C22⋊C4—162no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—162no—
233C42C2×#4111yescharacteristic
234C42C2×#4111yescharacteristic
235C42C2×#4111yescharacteristic
236C42H(5,𝔽2)11yescharacteristic
237C23—42no—
238C23—42no—
239C23—42no—
240C23—82no—
241C23—82no—
242C23#54611yescharacteristic
243Q8—81no—
244Q8—81no—
245D4—42no—
246D4—42no—
247D4—42no—
248D4—42no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—162no—
265C2×C4—162no—
266C2×C4C2×#12311yescharacteristic
267C2×C4#52911yescharacteristic
268C2×C4#54011yescharacteristic
269C2×C4#54611yescharacteristic
270C2×C4#55311yescharacteristic
271C2×C4#55511yescharacteristic
272C2×C4C2×#13511yescharacteristic
273C2×C4C2×#13711yescharacteristic
274C2×C4#57711yescharacteristic
275C2×C4#57711yescharacteristic
276C8—21no—
277C8—21no—
278C8—82no—
279C8—81no—
280C8—81no—
281C8—81no—
282C8—81no—
283C8—82no—
284C8#48512yes—
285C8#48512yes—
286C22—21no—
287C22—21no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—162no—
292C22—162no—
293C22#242011yescharacteristic
294C22#330411yescharacteristic
295C22#333111yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—81no—
302C4—81no—
303C4—82no—
304C4—81no—
305C4—162no—
306C4C2×#48511yescharacteristic
307C4#309911yescharacteristic
308C4#330411yescharacteristic
309C4#332911yescharacteristic
310C2—21no—
311C2—21no—
312C2—162no—
313C2#2497211yescharacteristic
314C2#2498511yescharacteristic
315C2#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. #10641C2semidirect
4. #10468C2semidirect
6. #9628C2semidirect
7. #9177C2semidirect
8. #8793C2semidirect
9. #8770C2semidirect
10. #8677C2semidirect
11. #8252C2semidirect

Maximal subgroups

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

subgroupindex
#282952
#228102
#106412
#104682
#96682
#96282
#91772
#87932
#87702
#86772
#82522