Finite Group Databasebeta

← #323589

Subgroups — #323589

Order 512.

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

normal subgroups only

This group has 752 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#29192C212yesmaximal, complemented
3#29143C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic
5#10677C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic, complemented
7#10673C211yesmaximal, characteristic, complemented
8#10671C211yesmaximal, characteristic, complemented
9#9832C212yesmaximal
10#8821C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#442—22no—
14#2859C2212yes—
15C4×#127C2211yescharacteristic
16C4×#127C2211yescharacteristic
17#2067—24no—
18#2025—24no—
19#1972—22no—
20#1957C2212yes—
21#1957C2212yes—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1944—22no—
27#1943—22no—
28#1862—24no—
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1844C2211yescharacteristic
32#1844C2211yescharacteristic
33#1834C2212yes—
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1804—24no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×D8—22no—
44C2×#127C2311yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#125—22no—
47C2×C4×D4—22no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—22no—
51#493—22no—
52#493Q814yes—
53#442—48no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#106—22no—
57C2×#106—22no—
58C2×C4⋊C8D414yes—
59C2×#102—24no—
60C2×#102—22no—
61C2×#102—22no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#97D412yes—
66C2×#97D412yes—
67C2×C4×C8C2312yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#390—22no—
72#390—21no—
73#390—21no—
74#390—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#384C2311yescharacteristic
80#384C2311yescharacteristic
81#384C2312yes—
82#384C2312yes—
83#382—22no—
84C43C2311yescharacteristic
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—24no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—22no—
94C22×D4—22no—
95C23×C4—41no—
96C23×C4—41no—
97C2×D8—28no—
98C2×D8—44no—
99C2×D8—44no—
100C22×C8C2×D414yes—
101C22×C8C2×D414yes—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104#127—21no—
105#127—21no—
106#127—22no—
107#127—22no—
108#127C2×Q812yes—
109#127C2×Q812yes—
110#125—84no—
111C4×D4—48no—
112C4×D4—44no—
113C4×D4—44no—
114C4×D4—84no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—42no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2411yescharacteristic
136#107—44no—
137#107—84no—
138#106—44no—
139#106—44no—
140C4⋊C8—44no—
141#102—48no—
142#102—44no—
143#102—44no—
144#102—84no—
145#102—84no—
146#102—84no—
147#97—24no—
148#97—42no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8C2×Q812yes—
155C4×C8#4112yes—
156#95—42no—
157#95—42no—
158C24—41no—
159C24—41no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—44no—
175C2×D4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4#12711yescharacteristic
185C22×C4#12712yes—
186C22×C4#12711yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D412yes—
189D8—416no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8C22⋊Q814yes—
201C2×C8AGL(1,ℤ/8)14yes—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—84no—
213C4⋊C4—84no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C42—22no—
219C42—22no—
220C42—24no—
221C42—24no—
222C42C22×Q811yescharacteristic
223C42C2×#4111yescharacteristic
224C422−1+412yes—
225C23—42no—
226C23—42no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23C2×#12711yescharacteristic
236D4—44no—
237D4—44no—
238D4—48no—
239D4—48no—
240D4—84no—
241D4—84no—
242D4—82no—
243D4—82no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—164no—
262C2×C4—164no—
263C2×C4C2×C22⋊Q811yescharacteristic
264C2×C4#53611yescharacteristic
265C2×C4#54712yes—
266C2×C4D4×Q812yes—
267C2×C4C2×AGL(1,ℤ/8)12yes—
268C2×C4#57512yes—
269C2×C4#57514yes—
270C2×C4#57614yes—
271C8—24no—
272C8—24no—
273C8—44no—
274C8—44no—
275C22—82no—
276C22—82no—
277C22—84no—
278C22—84no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22—162no—
284C22—162no—
285C22#254211yescharacteristic
286C22#315011yescharacteristic
287C22#315111yescharacteristic
288C22#315312yes—
289C22#315512yes—
290C4—41no—
291C4—41no—
292C4—42no—
293C4—162no—
294C4—161no—
295C4—161no—
296C4—162no—
297C4#307712yes—
298C4#323412yes—
299C4#334714yes—
300C4#338914yes—
301C2—162no—
302C2—162no—
303C2#1969512yes—
304C2#1972711yescharacteristic
305C2#1974911yescharacteristic
306C2#2851611yescharacteristic
307C2#2854412yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #29192C2semidirect
3. #29143C2semidirect
5. #10677C2semidirect
6. #10677C2semidirect
7. #10673C2semidirect
8. #10671C2semidirect
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
#291922
#291432
#216902
#106772
#106732
#106712
#98322
#88212
#82632