Finite Group Databasebeta

← #331481

Subgroups — #331481

Order 512.

703 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 154 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#30278C211yesmaximal, characteristic
2#30276C211yesmaximal, characteristic
3#30262C211yesmaximal, characteristic, complemented
4#30230C211yesmaximal, characteristic, complemented
5#30219C211yesmaximal, characteristic
6#30206C211yesmaximal, characteristic, complemented
7#10314C212yesmaximal, complemented
8#10311C212yesmaximal
9#10309C212yesmaximal, complemented
10#10306C212yesmaximal
11#8135C211yesmaximal, characteristic, complemented
12C2×#500C2212yes—
13C2×#500C2211yescharacteristic
14C2×#498C2211yescharacteristic
15C2×#494C2212yes—
16C2×#487C2212yes—
17C2×#486C2211yescharacteristic
18C2×#486C2212yes—
19C2×#486C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2858C2211yescharacteristic
24#2100—24no—
25#2093—22no—
26#2093—22no—
27#2092—24no—
28#2074—22no—
29#2042—24no—
30#2026—22no—
31#2016—24no—
32#1990C2212yes—
33#1990C2212yes—
34#1990C2212yes—
35#1957C2212yes—
36#1956C2212yes—
37#1956C2212yes—
38#1744C2211yescharacteristic
39#1744C2211yescharacteristic
40#1744C2211yescharacteristic
41#1273C2212yes—
42#1273C2212yes—
43C22×Q16—24no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×#127C2311yescharacteristic
50C2×#125—21no—
51C2×#124—22no—
52#500—42no—
53#500—42no—
54#500—42no—
55#498—42no—
56#494—42no—
57#494—42no—
58#487—42no—
59#487—42no—
60#486—42no—
61#486—42no—
62#486—42no—
63#486—42no—
64C2×#107—22no—
65C2×#107—24no—
66C2×#106—22no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—24no—
73C2×#102—24no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2312yes—
78#391—21no—
79#391—22no—
80#391—21no—
81#390—22no—
82#390—21no—
83#390—21no—
84#389—22no—
85#389—22no—
86#384—21no—
87#384—21no—
88#384—22no—
89#384C2312yes—
90#382—21no—
91#382—22no—
92C43C2311yescharacteristic
93#340—24no—
94#340—24no—
95#340—24no—
96#336D412yes—
97#336D414yes—
98#336D412yes—
99#336D414yes—
100C22×Q8—21no—
101C22×Q8—22no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C2×Q16—48no—
108C2×Q16—84no—
109C2×Q16—84no—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C22×C8C2×D414yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—84no—
125#127—22no—
126#127—22no—
127#125—82no—
128#124—84no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2411yescharacteristic
150#107—44no—
151#107—48no—
152#106—44no—
153#103—48no—
154#103—84no—
155#103—84no—
156#102—44no—
157#102—44no—
158#102—48no—
159#102—84no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166#95—42no—
167#95—42no—
168#95—42no—
169C24—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4#12712yes—
195C22×C4#12712yes—
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D411yescharacteristic
199Q16—816no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—24no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C22⋊C4—84no—
230C22⋊C4—84no—
231C42—24no—
232C42—22no—
233C42—24no—
234C42—22no—
235C42H(5,𝔽2)12yes—
236C42H(5,𝔽2)12yes—
237C23—81no—
238C23—82no—
239C23—82no—
240C23—81no—
241C23—81no—
242C23C2×#12711yescharacteristic
243Q8—84no—
244Q8—84no—
245Q8—84no—
246Q8—84no—
247Q8—84no—
248D4—84no—
249D4—82no—
250D4—82no—
251D4—84no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4#53512yes—
272C2×C4#53511yescharacteristic
273C2×C4#53511yescharacteristic
274C2×C4#53512yes—
275C2×C4#57614yes—
276C2×C4#57712yes—
277C2×C4#57814yes—
278C2×C4#57812yes—
279C8—44no—
280C8—42no—
281C8—42no—
282C8—44no—
283C22—162no—
284C22—162no—
285C22—161no—
286C22—161no—
287C22—161no—
288C22#265711yescharacteristic
289C22#315411yescharacteristic
290C22#315512yes—
291C22#315612yes—
292C22#315611yescharacteristic
293C4—42no—
294C4—42no—
295C4—162no—
296C4—161no—
297C4—162no—
298C4—161no—
299C4—161no—
300C4#322014yes—
301C4#322312yes—
302C4#322312yes—
303C4#322514yes—
304C2—321no—
305C2#1974112yes—
306C2#1974412yes—
307C2#3027011yescharacteristic
308C2#3027211yescharacteristic
309C2#3027511yescharacteristic
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
3. #30262C2semidirect
4. #30230C2semidirect
6. #30206C2semidirect
7. #10314C2semidirect
9. #10309C2semidirect
11. #8135C2semidirect

Maximal subgroups

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

subgroupindex
#302782
#302762
#302622
#302302
#302192
#302062
#103142
#103112
#103092
#103062
#81352