Finite Group Databasebeta

← #331268

Subgroups — #331268

Order 512.

687 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 162 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#33210C212yesmaximal
2#29876C211yesmaximal, characteristic
3#29845C212yesmaximal
4#29808C211yesmaximal, characteristic
5#10680C212yesmaximal, complemented
6#10675C212yesmaximal, complemented
7#10318C212yesmaximal, complemented
8#10314C212yesmaximal, complemented
9#7850C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#498C2212yes—
12C2×#494C2211yescharacteristic
13C2×#487C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2857C2212yes—
18#2857C2212yes—
19#2086—22no—
20#2075—22no—
21#2066—24no—
22#2056—22no—
23#2035—24no—
24#2001—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1956C2212yes—
30#1956C2212yes—
31#1943—24no—
32#1845C2212yes—
33#1844C2212yes—
34#1834C2211yescharacteristic
35#1834C2211yescharacteristic
36#1809—24no—
37#1273C2212yes—
38#1273C2212yes—
39C22×Q16—22no—
40C22×M16D412yes—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2312yes—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C22×C42C2311yescharacteristic
46#500—41no—
47#500—41no—
48#498—42no—
49#498—42no—
50#494—41no—
51#494—41no—
52#487—42no—
53#487—42no—
54#431—41no—
55#431—41no—
56#431—41no—
57#431—41no—
58C2×#107—24no—
59C2×#107—22no—
60C2×#106—24no—
61C2×#106—24no—
62C2×C4⋊C8D412yes—
63C2×C4⋊C8D412yes—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8D412yes—
69C4×M16—41no—
70C4×M16—41no—
71C2×#97D412yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76#391—22no—
77#391—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384C2312yes—
83#384C2312yes—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#340—22no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—22no—
93#340—24no—
94C22×Q8—22no—
95C22×Q8—22no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×Q16—44no—
100C2×Q16—84no—
101C2×M16—28no—
102C2×M16—42no—
103C22×C8C2×D412yes—
104C22×C8C2×D414yes—
105C22×C8C2×D412yes—
106C22×C8C2×D414yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C22⋊Q8—84no—
116C22⋊Q8—84no—
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—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2411yescharacteristic
139#107—44no—
140#107—48no—
141#106—48no—
142#106—84no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145#103—48no—
146#103—84no—
147#103—84no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150#97—42no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—21no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163C24#12711yescharacteristic
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12711yescharacteristic
187C22×C4#12711yescharacteristic
188C22×C4#12711yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192Q16—88no—
193M16—48no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8#13718yes—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C22⋊C4—84no—
221C22⋊C4—84no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42H(5,𝔽2)11yescharacteristic
229C42H(5,𝔽2)11yescharacteristic
230C42H(5,𝔽2)11yescharacteristic
231C422−1+411yescharacteristic
232C23—22no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23C2×#12711yescharacteristic
238Q8—84no—
239Q8—84no—
240Q8—84no—
241Q8—84no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4#53411yescharacteristic
268C2×C4#53411yescharacteristic
269C2×C4#53411yescharacteristic
270C2×C4#53711yescharacteristic
271C2×C4#53711yescharacteristic
272C2×C4#53711yescharacteristic
273C2×C4C2×#13714yes—
274C2×C4#57612yes—
275C2×C4#57714yes—
276C2×C4#57812yes—
277C8—28no—
278C8—42no—
279C8—42no—
280C8—44no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#265611yescharacteristic
287C22#315212yes—
288C22#315412yes—
289C22#315511yescharacteristic
290C22#315611yescharacteristic
291C4—41no—
292C4—41no—
293C4—41no—
294C4—162no—
295C4—162no—
296C4—162no—
297C4—162no—
298C4#319512yes—
299C4#320012yes—
300C4#320312yes—
301C4#325512yes—
302C4#326012yes—
303C4#326212yes—
304C2—81no—
305C2#1972012yes—
306C2#1972612yes—
307C2#3025011yescharacteristic
308C2#3025411yescharacteristic
309C2#3025511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10680C2semidirect
6. #10675C2semidirect
7. #10318C2semidirect
8. #10314C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#298762
#298452
#298082
#106802
#106752
#103182
#103142
#78502