Finite Group Databasebeta

← #141140

Subgroups — #141140

Order 512.

468 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 96 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#33180C211yesmaximal, characteristic
2#11700C212yesmaximal
3#11694C212yesmaximal
4#10711C211yesmaximal, characteristic
5#8601C211yesmaximal, characteristic
6C2×#494C411yescharacteristic, complemented
7C2×#470—21no—
8C2×#467—21no—
9C2×#405—21no—
10C2×#405C2211yescharacteristic
11#2621C2212yes—
12#1971—21no—
13#1969—21no—
14#1960C412yescomplemented
15#1846C411yescharacteristic, complemented
16C2×#344—21no—
17#1742C2211yescharacteristic
18#1740C2211yescharacteristic
19#1564—21no—
20#1458C2212yes—
21#1346—24no—
22#1344—24no—
23#1275—21no—
24C22×Q16—21no—
25C22×Q16—21no—
26C22×M16—21no—
27C22×M16D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4⋊Q8C2×C412yes—
30C2×C22⋊Q8—22no—
31C2×#117—21no—
32C2×#117C2311yescharacteristic
33#494—41no—
34#494—41no—
35#470—42no—
36#470—42no—
37#467—42no—
38#467—42no—
39#431—22no—
40C2×#107—21no—
41C2×#106—21no—
42C2×#103—22no—
43C2×#103—22no—
44#405—24no—
45#405—41no—
46#405—41no—
47#405—42no—
48C2×#97—41no—
49C2×#97C2×C411yescharacteristic
50C2×#97D411yescharacteristic
51C2×C4×C8—41no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#398—22no—
55#398—22no—
56#397—22no—
57#397—22no—
58#391—22no—
59#382C2×C412yes—
60#344—24no—
61#343—21no—
62#343D412yes—
63#342—21no—
64#342D412yes—
65#335—41no—
66#334—41no—
67#324—22no—
68#323—22no—
69C22×Q8—22no—
70C23×C4C2×D411yescharacteristic
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16—42no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×M16—21no—
79C2×M16—21no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—41no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8D812yes—
97C22×C8SD1612yes—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104#117—22no—
105#117—22no—
106#117—41no—
107#117—41no—
108#117—41no—
109#117—41no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D411yescharacteristic
116C2×C4⋊C4#4111yescharacteristic
117C2×C22⋊C4—22no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4C2×D411yescharacteristic
120C2×C22⋊C4#4111yescharacteristic
121C2×C42—41no—
122C2×C42C22×C411yescharacteristic
123C2×C42C2×D411yescharacteristic
124#107—42no—
125#106—42no—
126C4⋊C8—42no—
127#103—44no—
128#103—44no—
129C22⋊C8—42no—
130#97—41no—
131#97—41no—
132#97—44no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—42no—
138C4×C8—44no—
139#95—22no—
140#95—22no—
141#95—42no—
142#95—42no—
143C24#12111yescharacteristic
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C4×D411yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4#12111yescharacteristic
161C22×C4#12411yescharacteristic
162C22×C4#12711yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4C2×SD1611yescharacteristic
165C22×C4AGL(1,ℤ/8)11yescharacteristic
166C22×C4AGL(1,ℤ/8)11yescharacteristic
167Q16—84no—
168Q16—82no—
169Q16—82no—
170M16—24no—
171M16—42no—
172M16—42no—
173M16—42no—
174M16—42no—
175M16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—82no—
191C2×C8#9911yescharacteristic
192C2×C8#9911yescharacteristic
193C2×C8#10212yes—
194C2×C8#10212yes—
195C2×C8#10412yes—
196C2×C8#10412yes—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C22⋊C4—24no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—42no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—42no—
218C42—42no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23#39011yescharacteristic
224C23#45911yescharacteristic
225C23#46011yescharacteristic
226Q8—82no—
227Q8—82no—
228Q8—82no—
229Q8—82no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4C2×#9911yescharacteristic
258C2×C4#41011yescharacteristic
259C2×C4C2×#10211yescharacteristic
260C2×C4#41811yescharacteristic
261C2×C4C2×#10411yescharacteristic
262C2×C4C2×#10411yescharacteristic
263C2×C4#45911yescharacteristic
264C2×C4#46011yescharacteristic
265C8—22no—
266C8—22no—
267C8—44no—
268C8—41no—
269C8—162no—
270C8#32712yes—
271C8#32712yes—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#196211yescharacteristic
282C22#196411yescharacteristic
283C22#198211yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—84no—
291C4—162no—
292C4—162no—
293C4C2×#32711yescharacteristic
294C4#150211yescharacteristic
295C4#150511yescharacteristic
296C4#150711yescharacteristic
297C2—21no—
298C2—21no—
299C2—41no—
300C2—41no—
301C2#971211yescharacteristic
302C2#1323011yescharacteristic
303C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#494C4semidirect
14. #1960C4semidirect
15. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331802
#117002
#116942
#107112
#86012