Finite Group Databasebeta

← #227124

Subgroups — #227124

Order 512.

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

normal subgroups only

This group has 732 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#22971C212yesmaximal, complemented
2#22546C212yesmaximal
3C2×#1630C211yesmaximal, characteristic
4#9820C211yesmaximal, characteristic
5#9572C212yesmaximal
6#9224C212yesmaximal, complemented
7#9063C211yesmaximal, characteristic
8#8888C212yesmaximal, complemented
9#8207C212yesmaximal, complemented
10C2×#559C2211yescharacteristic
11C2×#492C2211yescharacteristic
12C2×#477C2212yes—
13C2×#446C2212yes—
14#2369C2211yescharacteristic
15Q8×C4⋊C4C2212yes—
16#1990C2212yes—
17#1983—22no—
18#1982C2212yes—
19#1959—22no—
20#1957C2212yes—
21#1948C2212yes—
22#1943—22no—
23#1918—22no—
24#1865—22no—
25#1844C2211yescharacteristic
26#1803C2212yes—
27#1802—22no—
28#1770C2212yes—
29#1763—22no—
30#1762—22no—
31C4×#103C2212yes—
32#1630C418yescomplemented
33C2×#333C2211yescharacteristic
34C2×#331C2211yescharacteristic
35C2×Q8⋊C8C2212yes—
36C2×C82C2211yescharacteristic
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×Q8—22no—
49C2×C4×Q8C2312yes—
50C2×C4×D4—22no—
51#492—41no—
52#492C2×C414yes—
53#477—24no—
54#477C2×C418yes—
55#446—24no—
56#446—42no—
57C2×#107—22no—
58C2×#106—22no—
59C2×#106—41no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103D412yes—
66C2×#103D412yes—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#391—41no—
76#389—22no—
77#389—22no—
78#389—42no—
79#384—22no—
80#384C2312yes—
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—
94C82—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×Q8—44no—
121C4×Q8—82no—
122C4×Q8C22×C414yes—
123C4×Q8#4114yes—
124C4×D4—28no—
125C4×D4—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—41no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—42no—
142C2×C42—41no—
143C2×C42C2411yescharacteristic
144#107—28no—
145#106—82no—
146#106—82no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—44no—
151C4⋊C8C22×C414yes—
152#103—28no—
153#103—24no—
154#103—42no—
155#103—82no—
156#102—44no—
157#102—44no—
158#102—82no—
159#102—82no—
160#97—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—41no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×Q8C4×D414yes—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—81no—
189C2×D4—81no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—161no—
198C22×C4—161no—
199C22×C4C22×D412yes—
200C22×C4H(5,𝔽2)11yescharacteristic
201SD16—88no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8#13714yes—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—162no—
233C4⋊C4—161no—
234C4⋊C4C4×D414yes—
235C22⋊C4—48no—
236C22⋊C4—44no—
237C42—24no—
238C42—24no—
239C42—42no—
240C42—42no—
241C42—84no—
242C42—161no—
243C42C23×C411yescharacteristic
244C42C2×#4112yes—
245C42H(5,𝔽2)11yescharacteristic
246C23—42no—
247C23—82no—
248C23—82no—
249C23—81no—
250C23—81no—
251C23D4211yescharacteristic
252Q8—24no—
253Q8—24no—
254Q8—42no—
255Q8—42no—
256Q8—82no—
257Q8—161no—
258D4—44no—
259D4—44no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4—161no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—164no—
285C2×C4C2×C4×D412yes—
286C2×C4#51811yescharacteristic
287C2×C4#54612yes—
288C2×C4#55111yescharacteristic
289C2×C4C2×#13712yes—
290C2×C4#57512yes—
291C2×C4#57712yes—
292C2×C4#57712yes—
293C8—24no—
294C8—24no—
295C8—44no—
296C8—42no—
297C8—82no—
298C22—84no—
299C22—82no—
300C22—161no—
301C22—162no—
302C22—161no—
303C22#235811yescharacteristic
304C22#327512yes—
305C22#327812yes—
306C22#329111yescharacteristic
307C22#329211yescharacteristic
308C4—21no—
309C4—21no—
310C4—44no—
311C4—44no—
312C4—82no—
313C4—82no—
314C4—82no—
315C4—162no—
316C4C4×#13712yes—
317C4#295212yes—
318C4#331612yes—
319C4#331712yes—
320C2—162no—
321C2#2029411yescharacteristic
322C2#2029912yes—
323C2#2032311yescharacteristic
324C2#2286712yes—
325C2#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. #22971C2semidirect
6. #9224C2semidirect
8. #8888C2semidirect
9. #8207C2semidirect
32. #1630C4semidirect

Maximal subgroups

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

subgroupindex
#229712
#225462
C2×#16302
#98202
#95722
#92242
#90632
#88882
#82072