Finite Group Databasebeta

← #227025

Subgroups — #227025

Order 512.

764 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 202 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#23061C212yesmaximal, complemented
2#22489C212yesmaximal
3C2×#1634C211yesmaximal, characteristic
4#9825C211yesmaximal, characteristic, complemented
5#9567C212yesmaximal, complemented
6#9215C212yesmaximal, complemented
7#9066C211yesmaximal, characteristic, complemented
8#8885C212yesmaximal, complemented
9C8×#102C212yesmaximal, complemented
10C2×#555C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#476C2212yes—
13C2×C8×D4C2212yes—
14#2369C2211yescharacteristic
15#2354C2212yes—
16#1990C2212yes—
17#1990C2212yes—
18#1987C2212yes—
19#1948C2212yes—
20#1934—22no—
21#1926—22no—
22#1920—22no—
23#1893—22no—
24#1875—22no—
25#1845C2211yescharacteristic
26#1825—22no—
27#1802C2212yes—
28C4×#106—22no—
29C8×C4⋊C4C2212yes—
30C4×#102C2212yes—
31#1735—22no—
32#1634C418yescomplemented
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35C2×D4⋊C8C2212yes—
36C2×C82C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39#1277C2212yes—
40C22×Q16—41no—
41C23×C8—22no—
42#555—24no—
43#555—41no—
44#555C2×C414yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#124—41no—
48C2×#121—22no—
49C2×C4×D4C2312yes—
50C2×#117—22no—
51C2×#117—22no—
52#494—41no—
53#494C2×C414yes—
54#476—24no—
55#476C2×C418yes—
56C8×D4—24no—
57C8×D4—42no—
58C2×#107—41no—
59C2×#106—22no—
60C2×#106—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#102—22no—
64C2×#102—22no—
65C2×#102D412yes—
66C2×#102D412yes—
67C2×C22⋊C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8C2312yes—
70C2×C4×C8C2311yescharacteristic
71#391—41no—
72#390—22no—
73#389—22no—
74#388—42no—
75#384C2312yes—
76#380—22no—
77C4×C4⋊C4—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×C414yes—
93D4⋊C8C2×C418yes—
94C82—41no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×D4—41no—
98C22×D4C2×D412yes—
99C23×C4—22no—
100C2×Q16—44no—
101C2×Q16—82no—
102C2×Q16—82no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—44no—
107C22×C8—44no—
108C22×C8—82no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C22×C412yes—
116C4⋊Q8C22×C412yes—
117#124—44no—
118#124—44no—
119#121—28no—
120#121—44no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4C22×C414yes—
124C4×D4#4114yes—
125#117—28no—
126#117—44no—
127#117—44no—
128#117—82no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C42—22no—
143C2×C42—41no—
144C2×C42—42no—
145C2×C42—41no—
146C2×C42C2411yescharacteristic
147#107—82no—
148#106—28no—
149#106—44no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8C22×C414yes—
154#102—28no—
155#102—24no—
156#102—24no—
157#102—82no—
158C22⋊C8—44no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—21no—
162C4×C8—44no—
163C4×C8C22×C412yes—
164#95—22no—
165#95—42no—
166#95—42no—
167C24—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—44no—
182C2×D4—161no—
183C2×D4—161no—
184C2×D4C4×D414yes—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4C22×D412yes—
198C22×C4H(5,𝔽2)11yescharacteristic
199Q16—88no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—24no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—88no—
215C2×C8#13514yes—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—161no—
230C4⋊C4C4×D414yes—
231C4⋊C4#13514yes—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—48no—
235C22⋊C4—84no—
236C22⋊C4—162no—
237C42—44no—
238C42—42no—
239C42—42no—
240C42—84no—
241C42—161no—
242C42C23×C411yescharacteristic
243C42C2×#4112yes—
244C42H(5,𝔽2)11yescharacteristic
245C23—24no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23D4211yescharacteristic
253Q8—82no—
254Q8—82no—
255Q8—161no—
256Q8—161no—
257D4—24no—
258D4—24no—
259D4—42no—
260D4—42no—
261D4—164no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4C2×C4×D412yes—
287C2×C4#51811yescharacteristic
288C2×C4#54612yes—
289C2×C4#55111yescharacteristic
290C2×C4C2×#13512yes—
291C2×C4C2×#13512yes—
292C2×C4#57712yes—
293C2×C4#57812yes—
294C8—24no—
295C8—24no—
296C8—44no—
297C8—42no—
298C8—82no—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22#235811yescharacteristic
307C22#328212yes—
308C22#328312yes—
309C22#329111yescharacteristic
310C22#329211yescharacteristic
311C4—21no—
312C4—21no—
313C4—44no—
314C4—44no—
315C4—82no—
316C4—82no—
317C4—82no—
318C4—162no—
319C4C4×#13512yes—
320C4#294712yes—
321C4#329612yes—
322C4#330212yes—
323C2—84no—
324C2#1999712yes—
325C2#2000311yescharacteristic
326C2#2001911yescharacteristic
327C2#2287612yes—
328C2#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. #23061C2semidirect
4. #9825C2semidirect
5. #9567C2semidirect
6. #9215C2semidirect
7. #9066C2semidirect
8. #8885C2semidirect
9. C8×#102C2semidirect
32. #1634C4semidirect

Maximal subgroups

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

subgroupindex
#230612
#224892
C2×#16342
#98252
#95672
#92152
#90662
#88852
C8×#1022