Finite Group Databasebeta

← #320416

Subgroups — #320416

Order 512.

715 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 182 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#33226C211yesmaximal, characteristic
2#29191C212yesmaximal, complemented
3#28654C212yesmaximal
4C4×#498C211yesmaximal, characteristic, complemented
5#10685C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic
7#10316C211yesmaximal, characteristic, complemented
8#10312C211yesmaximal, characteristic
9#9828C212yesmaximal, complemented
10#9827C212yesmaximal
11#8260C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#498C2212yes—
14C2×#488—22no—
15C2×#486C2212yes—
16#2859C2211yescharacteristic
17#2858C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19#2066—24no—
20#2065—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1958—22no—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1957C2212yes—
27#1956C2212yes—
28#1956C2212yes—
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1941—22no—
32#1926—24no—
33#1865—24no—
34#1844C2212yes—
35#1844C2211yescharacteristic
36#1834C2212yes—
37C4×#106—22no—
38C4×C4⋊C8C2212yes—
39C42×C8C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C2×C4⋊Q8C2311yescharacteristic
44C2×C4⋊Q8C2312yes—
45C2×#127C2311yescharacteristic
46C2×#125—21no—
47C2×#124—21no—
48C2×#124—21no—
49C2×C4×Q8—22no—
50#498—24no—
51#498—24no—
52#498—42no—
53#488—48no—
54#486—24no—
55#486—42no—
56C2×#106—22no—
57C2×#106—24no—
58C2×#106—22no—
59C2×#106—24no—
60C2×C4⋊C8D414yes—
61C2×#103—24no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—22no—
68C2×#102—22no—
69C2×C4×C8D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#390—22no—
77#390—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384C2312yes—
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85#382—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C43C2311yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—42no—
94C22×Q8—22no—
95C22×Q8—21no—
96C22×D4—21no—
97C22×D4—22no—
98C22×C8C2×D414yes—
99C22×C8C2×D414yes—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—21no—
106C4⋊Q8—22no—
107C4⋊Q8—21no—
108C4⋊Q8#4114yes—
109#127—22no—
110#127—21no—
111#127—21no—
112#125—44no—
113#124—44no—
114#124—44no—
115C4×Q8—84no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C42—42no—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#106—28no—
139#106—48no—
140#106—44no—
141#106—84no—
142C4⋊C8—44no—
143#103—48no—
144#103—44no—
145#103—44no—
146#103—84no—
147#102—48no—
148#102—48no—
149#102—44no—
150#102—44no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—21no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—42no—
159C4×C8SD1618yes—
160#95—42no—
161#95—42no—
162#95—42no—
163C24—41no—
164C2×Q8—24no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—44no—
178C2×D4—42no—
179C2×D4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4#12712yes—
186C22×C4#12711yescharacteristic
187C22×C4#12711yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D412yes—
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×C8C2×SD1614yes—
201C2×C8C2×SD1614yes—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—84no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C22⋊C4—84no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—22no—
222C42—24no—
223C42—24no—
224C42—84no—
225C42C2×SD1614yes—
226C42C2×#4112yes—
227C42H(5,𝔽2)11yescharacteristic
228C422−1+411yescharacteristic
229C23—81no—
230C23—82no—
231C23—81no—
232C23—82no—
233C23—81no—
234C23C2×#12711yescharacteristic
235Q8—48no—
236Q8—84no—
237Q8—82no—
238Q8—82no—
239D4—82no—
240D4—84no—
241D4—82no—
242D4—84no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4#49212yes—
262C2×C4#49212yes—
263C2×C4#52511yescharacteristic
264C2×C4#53711yescharacteristic
265C2×C4#54612yes—
266C2×C4#54812yes—
267C2×C4C22×SD1612yes—
268C2×C4#57512yes—
269C2×C4#57514yes—
270C2×C4#57714yes—
271C8—24no—
272C8—24no—
273C8—44no—
274C8—44no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—162no—
279C22—162no—
280C22#254011yescharacteristic
281C22C2×#49211yescharacteristic
282C22#315012yes—
283C22#315311yescharacteristic
284C22#315412yes—
285C4—22no—
286C4—22no—
287C4—42no—
288C4—84no—
289C4—161no—
290C4—161no—
291C4—161no—
292C4—162no—
293C4#310312yes—
294C4#324612yes—
295C4#332614yes—
296C4#337614yes—
297C2—321no—
298C2#1960811yescharacteristic
299C2#1965411yescharacteristic
300C2#1969212yes—
301C2#2837411yescharacteristic
302C2#2841012yes—
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
2. #29191C2semidirect
4. C4×#498C2semidirect
5. #10685C2semidirect
7. #10316C2semidirect
9. #9828C2semidirect
11. #8260C2semidirect

Maximal subgroups

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

subgroupindex
#332262
#291912
#286542
C4×#4982
#106852
#106772
#103162
#103122
#98282
#98272
#82602