Finite Group Databasebeta

← #227105

Subgroups — #227105

Order 512.

716 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 194 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#23069C212yesmaximal, complemented
2#22543C212yesmaximal
3C2×#1630C211yesmaximal, characteristic
4#9842C211yesmaximal, characteristic
5#9572C212yesmaximal
6#9216C212yesmaximal, complemented
7#9063C211yesmaximal, characteristic
8#8883C212yesmaximal, complemented
9#8205C212yesmaximal, complemented
10C2×#559C2211yescharacteristic
11C2×#492C2211yescharacteristic
12C2×#477C2212yes—
13C2×C8×Q8C2212yes—
14#2369C2211yescharacteristic
15#2355C2212yes—
16#1990C2212yes—
17#1983—22no—
18#1982C2212yes—
19#1958—22no—
20#1957C2212yes—
21#1948C2212yes—
22#1935—22no—
23#1927—22no—
24#1918—22no—
25#1845C2211yescharacteristic
26#1804—22no—
27#1803C2212yes—
28#1776—22no—
29#1770C2212yes—
30C4×#103C2212yes—
31C4×#102—22no—
32#1630C418yescomplemented
33C2×#333C2211yescharacteristic
34C2×#331C2211yescharacteristic
35C2×Q8⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39#1277C2212yes—
40C22×SD16—41no—
41#559—24no—
42#559—41no—
43#559C2×C414yes—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#124—22no—
47C2×#123—41no—
48C2×C4×Q8C2312yes—
49C2×C4×D4—22no—
50#492—41no—
51#492C2×C414yes—
52#477—24no—
53#477C2×C418yes—
54C8×Q8—24no—
55C8×Q8—42no—
56C2×#107—22no—
57C2×#107—41no—
58C2×#106—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103D412yes—
65C2×#103D412yes—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#391—41no—
76#389—22no—
77#387—42no—
78#384C2312yes—
79#382—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4C2312yes—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340D412yes—
89#340D412yes—
90#336—22no—
91#333—41no—
92#331C2×C414yes—
93Q8⋊C8C2×C418yes—
94#322—41no—
95C22×Q8—21no—
96C22×Q8C2×D412yes—
97C22×D4—22no—
98C22×D4—21no—
99C23×C4—41no—
100C2×SD16—44no—
101C2×SD16—82no—
102C2×SD16—82no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8C2×D412yes—
107C22×C8C2×D412yes—
108C4⋊Q8—22no—
109C4⋊Q8—21no—
110C4⋊Q8C22×C412yes—
111#127—22no—
112#127—21no—
113#127C22×C412yes—
114#124—28no—
115#124—44no—
116#123—44no—
117#123—44no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8C22×C414yes—
121C4×Q8#4114yes—
122C4×D4—28no—
123C4×D4—44no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—41no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—42no—
138C2×C42—41no—
139C2×C42C2411yescharacteristic
140#107—28no—
141#107—82no—
142#106—44no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—44no—
147C4⋊C8C22×C414yes—
148#103—28no—
149#103—42no—
150#103—42no—
151#103—82no—
152#102—44no—
153#102—44no—
154#102—82no—
155#102—82no—
156#97—44no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—44no—
161C4×C8C22×C412yes—
162#95—22no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167C24—41no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8C4×D414yes—
178C2×D4—24no—
179C2×D4—22no—
180C2×D4—42no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4—81no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—161no—
196C22×C4—161no—
197C22×C4C22×D412yes—
198C22×C4H(5,𝔽2)11yescharacteristic
199SD16—88no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—161no—
230C4⋊C4C4×D414yes—
231C22⋊C4—48no—
232C22⋊C4—44no—
233C42—24no—
234C42—24no—
235C42—42no—
236C42—42no—
237C42—162no—
238C42—161no—
239C42C23×C411yescharacteristic
240C42C2×#4112yes—
241C42H(5,𝔽2)11yescharacteristic
242C23—42no—
243C23—82no—
244C23—82no—
245C23—81no—
246C23—81no—
247C23D4211yescharacteristic
248Q8—24no—
249Q8—24no—
250Q8—42no—
251Q8—42no—
252Q8—82no—
253Q8—161no—
254D4—44no—
255D4—44no—
256D4—82no—
257D4—82no—
258D4—82no—
259D4—161no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—164no—
281C2×C4C2×C4×D412yes—
282C2×C4#51811yescharacteristic
283C2×C4#54612yes—
284C2×C4#55111yescharacteristic
285C2×C4#57512yes—
286C2×C4#57512yes—
287C2×C4#57712yes—
288C2×C4#57812yes—
289C8—24no—
290C8—44no—
291C8—42no—
292C8—42no—
293C8—82no—
294C22—84no—
295C22—82no—
296C22—162no—
297C22—161no—
298C22—161no—
299C22#235811yescharacteristic
300C22#326912yes—
301C22#327112yes—
302C22#329111yescharacteristic
303C22#329211yescharacteristic
304C4—21no—
305C4—21no—
306C4—44no—
307C4—44no—
308C4—82no—
309C4—82no—
310C4—82no—
311C4—162no—
312C4#294312yes—
313C4#294712yes—
314C4#331312yes—
315C4#331712yes—
316C2—162no—
317C2#2014912yes—
318C2#2015011yescharacteristic
319C2#2015711yescharacteristic
320C2#2286112yes—
321C2#2289711yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #23069C2semidirect
6. #9216C2semidirect
8. #8883C2semidirect
9. #8205C2semidirect
32. #1630C4semidirect

Maximal subgroups

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

subgroupindex
#230692
#225432
C2×#16302
#98422
#95722
#92162
#90632
#88832
#82052