Finite Group Databasebeta

← #323594

Subgroups — #323594

Order 512.

694 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 174 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#33263C211yesmaximal, characteristic, complemented
2#29220C212yesmaximal, complemented
3#29190C212yesmaximal
4#21692C211yesmaximal, characteristic
5#10685C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic
7#10676C211yesmaximal, characteristic, complemented
8#10671C211yesmaximal, characteristic
9#9829C212yesmaximal
10#8821C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#440—22no—
14#2859C2212yes—
15C4×C4⋊Q8C2211yescharacteristic
16C4×#127C2211yescharacteristic
17#2101—24no—
18#2065—24no—
19#1990C2212yes—
20#1990C2212yes—
21#1989—24no—
22#1973—22no—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1956C2211yescharacteristic
27#1944—22no—
28#1943—22no—
29#1864—24no—
30#1845C2211yescharacteristic
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1844C2211yescharacteristic
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#125—22no—
47C2×C4×Q8—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50#492—22no—
51#492Q814yes—
52#440—48no—
53C2×#107—22no—
54C2×#107—22no—
55C2×#106—22no—
56C2×#106—24no—
57C2×C4⋊C8D414yes—
58C2×#103—22no—
59C2×#102—24no—
60C2×#102—22no—
61C2×#102—24no—
62C2×#97D412yes—
63C2×#97D412yes—
64C2×C4×C8C2312yes—
65C2×C4×C8C2312yes—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#390—22no—
70#390—21no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2312yes—
79#384C2312yes—
80#382—22no—
81C4×C4⋊C4—22no—
82C43C2311yescharacteristic
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—24no—
91C22×Q8—21no—
92C22×D4—21no—
93C22×D4—22no—
94C23×C4—41no—
95C2×SD16—28no—
96C2×SD16—44no—
97C2×SD16—44no—
98C22×C8C2×D414yes—
99C22×C8C2×D414yes—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—21no—
103C4⋊Q8—22no—
104C4⋊Q8C2×Q812yes—
105#127—21no—
106#127—22no—
107#127C2×Q812yes—
108#125—48no—
109C4×Q8—44no—
110C4×D4—44no—
111C4×D4—84no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—42no—
128C2×C42—41no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2411yescharacteristic
134#107—44no—
135#107—44no—
136#106—44no—
137#106—84no—
138C4⋊C8—44no—
139#103—44no—
140#102—48no—
141#102—44no—
142#102—84no—
143#97—24no—
144#97—42no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8C2×Q812yes—
151C4×C8#4112yes—
152#95—42no—
153#95—42no—
154#95—42no—
155C24—41no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—44no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—82no—
174C22×C4—84no—
175C22×C4#12712yes—
176C22×C4#12711yescharacteristic
177C22×C4#12711yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C22×D412yes—
180SD16—416no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8C22⋊Q814yes—
192C2×C8AGL(1,ℤ/8)14yes—
193C4⋊C4—24no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—82no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C22⋊C4—82no—
209C22⋊C4—84no—
210C42—22no—
211C42—22no—
212C42—24no—
213C42—24no—
214C42—82no—
215C42C22×Q811yescharacteristic
216C42C2×#4111yescharacteristic
217C422−1+412yes—
218C23—42no—
219C23—81no—
220C23—81no—
221C23—82no—
222C23—82no—
223C23C2×#12711yescharacteristic
224Q8—44no—
225Q8—82no—
226D4—44no—
227D4—48no—
228D4—84no—
229D4—82no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—164no—
251C2×C4C2×C22⋊Q811yescharacteristic
252C2×C4#53611yescharacteristic
253C2×C4#54712yes—
254C2×C4D4×Q812yes—
255C2×C4C2×AGL(1,ℤ/8)12yes—
256C2×C4#57512yes—
257C2×C4#57514yes—
258C2×C4#57714yes—
259C8—24no—
260C8—24no—
261C8—44no—
262C8—44no—
263C22—82no—
264C22—84no—
265C22—161no—
266C22—161no—
267C22—162no—
268C22#254211yescharacteristic
269C22#315011yescharacteristic
270C22#315111yescharacteristic
271C22#315312yes—
272C22#315412yes—
273C4—41no—
274C4—41no—
275C4—42no—
276C4—82no—
277C4—161no—
278C4—161no—
279C4—162no—
280C4—162no—
281C4#307712yes—
282C4#323412yes—
283C4#335014yes—
284C4#338814yes—
285C2—162no—
286C2#1969112yes—
287C2#1971311yescharacteristic
288C2#1975011yescharacteristic
289C2#2851611yescharacteristic
290C2#2854712yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33263C2semidirect
2. #29220C2semidirect
5. #10685C2semidirect
7. #10676C2semidirect
10. #8821C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332632
#292202
#291902
#216922
#106852
#106772
#106762
#106712
#98292
#88212
#82632