Finite Group Databasebeta

← #227106

Subgroups — #227106

Order 512.

774 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 198 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#22969C212yesmaximal, complemented
2#22503C212yesmaximal
3C2×#1634C211yesmaximal, characteristic
4#9825C211yesmaximal, characteristic, complemented
5#9576C212yesmaximal, complemented
6#9223C212yesmaximal, complemented
7#9066C211yesmaximal, characteristic, complemented
8#8882C212yesmaximal, complemented
9#8206C212yesmaximal, complemented
10C2×#555C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#476C2212yes—
13C2×#436C2212yes—
14#2369C2211yescharacteristic
15D4×C4⋊C4C2212yes—
16#1990C2212yes—
17#1990C2212yes—
18#1987C2212yes—
19#1947C2212yes—
20#1944—22no—
21#1920—22no—
22#1895—22no—
23#1874—22no—
24#1862—22no—
25#1845C2211yescharacteristic
26#1825—22no—
27#1802C2212yes—
28#1770C2212yes—
29#1762—22no—
30C4×#102C2212yes—
31#1736—22no—
32#1634C418yescomplemented
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35C2×D4⋊C8C2212yes—
36C2×C82C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39#1277C2212yes—
40C22×Q16—41no—
41C22×M16—22no—
42#555—24no—
43#555—41no—
44#555C2×C414yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#124—41no—
48C2×#121—22no—
49C2×C4×D4—22no—
50C2×C4×D4C2312yes—
51C2×#117—22no—
52C22×C4⋊C4—22no—
53#494—41no—
54#494C2×C414yescomplemented
55#476—24no—
56#476C2×C418yescomplemented
57#436—24no—
58#436—42no—
59C2×#107—22no—
60C2×#107—41no—
61C2×#106—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#102—22no—
65C2×#102—22no—
66C2×#102D412yes—
67C2×#102D412yes—
68C2×C22⋊C8—22no—
69C2×#97—22no—
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72#391—41no—
73#389—22no—
74#389—22no—
75#385—42no—
76#384C2312yes—
77#380—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4C2312yes—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340D412yes—
89#340D412yes—
90#336—22no—
91#333—41no—
92#333C2×C414yescomplemented
93D4⋊C8C2×C418yescomplemented
94C82—41no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×D4—41no—
98C22×D4C2×D412yes—
99C23×C4—22no—
100C23×C4—42no—
101C2×Q16—44no—
102C2×Q16—82no—
103C2×Q16—82no—
104C2×M16—44no—
105C2×M16—44no—
106C2×M16—82no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8C22×C412yes—
117C4⋊Q8C22×C412yes—
118#124—44no—
119#124—44no—
120#121—28no—
121#121—44no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—44no—
125C4×D4—82no—
126C4×D4C22×C414yes—
127C4×D4#4114yes—
128#117—28no—
129#117—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—82no—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C42—22no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42C2411yescharacteristic
149#107—82no—
150#107—82no—
151#106—28no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8C22×C414yes—
156#102—28no—
157#102—24no—
158#102—42no—
159#102—82no—
160C22⋊C8—44no—
161#97—82no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—21no—
165C4×C8C22×C412yes—
166#95—22no—
167#95—42no—
168#95—42no—
169C24—22no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—44no—
184C2×D4—161no—
185C2×D4—161no—
186C2×D4C4×D414yes—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—162no—
202C22×C4C22×D412yes—
203C22×C4H(5,𝔽2)11yescharacteristic
204Q16—88no—
205M16—88no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8#13714yes—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4—161no—
237C4⋊C4C4×D414yes—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—48no—
241C22⋊C4—84no—
242C42—44no—
243C42—42no—
244C42—42no—
245C42—161no—
246C42C23×C411yescharacteristic
247C42C2×#4112yes—
248C42H(5,𝔽2)11yescharacteristic
249C23—24no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23D4211yescharacteristic
257Q8—82no—
258Q8—82no—
259Q8—161no—
260Q8—161no—
261D4—24no—
262D4—24no—
263D4—42no—
264D4—42no—
265D4—164no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—168no—
291C2×C4C2×C4×D412yes—
292C2×C4#51811yescharacteristic
293C2×C4#54612yes—
294C2×C4#55111yescharacteristic
295C2×C4C2×#13712yes—
296C2×C4#57512yes—
297C2×C4#57612yes—
298C2×C4#57812yes—
299C8—24no—
300C8—24no—
301C8—44no—
302C8—42no—
303C8—82no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22#235811yescharacteristic
312C22#327312yes—
313C22#327612yes—
314C22#329111yescharacteristic
315C22#329211yescharacteristic
316C4—21no—
317C4—21no—
318C4—44no—
319C4—82no—
320C4—82no—
321C4—82no—
322C4—82no—
323C4—162no—
324C4C4×#13712yes—
325C4#295112yes—
326C4#331412yes—
327C4#331512yes—
328C2—84no—
329C2#2030211yescharacteristic
330C2#2030512yes—
331C2#2032411yescharacteristic
332C2#2286212yes—
333C2#2289711yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22969C2semidirect
4. #9825C2semidirect
5. #9576C2semidirect
6. #9223C2semidirect
7. #9066C2semidirect
8. #8882C2semidirect
9. #8206C2semidirect
32. #1634C4semidirect
54. #494C2×C4semidirect
56. #476C2×C4semidirect
92. #333C2×C4semidirect
93. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#229692
#225032
C2×#16342
#98252
#95762
#92232
#90662
#88822
#82062