Finite Group Databasebeta

← #319460

Subgroups — #319460

Order 512.

796 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 190 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#33210C211yesmaximal, characteristic
2#27950C212yesmaximal
3#25600C212yesmaximal
4#21322C211yesmaximal, characteristic
5#10674C212yesmaximal, complemented
6#10318C212yesmaximal, complemented
7#9833C212yesmaximal, complemented
8#9825C212yesmaximal, complemented
9#8146C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#498C2211yescharacteristic
12C2×#494C2212yes—
13C2×#487C2212yes—
14C2×C4×M16C2211yescharacteristic
15#2857C2212yes—
16C2×#384C2211yescharacteristic
17#2083—24no—
18#2001—24no—
19#1992—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1956C2212yes—
27#1943—22no—
28#1878—24no—
29#1852—24no—
30#1845C2212yes—
31#1845C2212yes—
32#1845C2211yescharacteristic
33#1809—22no—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2211yescharacteristic
37C2×C82C2211yescharacteristic
38#1273C2212yes—
39C22×Q16—24no—
40C22×M16D412yes—
41#533—24no—
42#533—41no—
43#533—41no—
44C2×C4⋊Q8C2312yes—
45C2×C4⋊Q8C2312yes—
46C2×#125—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C42C2311yescharacteristic
54#498—41no—
55#498—41no—
56#494—24no—
57#494—42no—
58#487—24no—
59#487—42no—
60C2×#107—22no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#103—24no—
65C2×#103—24no—
66C2×C22⋊C8D414yes—
67C4×M16—41no—
68C4×M16—41no—
69C2×#97D412yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#389—22no—
76#389—22no—
77#384—22no—
78#384—21no—
79#384C2312yes—
80#384C2312yes—
81#380—22no—
82#380—22no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—41no—
91C82—41no—
92C22×Q8—22no—
93C22×Q8—22no—
94C23×C4—41no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C2×Q16—48no—
98C2×Q16—48no—
99C2×Q16—84no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16Q1618yescomplemented
103C22×C8C2×D414yes—
104C22×C8C2×D412yes—
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#4114yes—
115#125—24no—
116#125—42no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C22⋊Q8—84no—
120C4×Q8—44no—
121C4×Q8—44no—
122#117—24no—
123#117—42no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—84no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—42no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—41no—
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42C2411yescharacteristic
149#107—44no—
150#107—84no—
151#107—84no—
152#106—44no—
153#103—48no—
154#103—84no—
155C22⋊C8—28no—
156#97—42no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—21no—
162C4×C8—22no—
163C4×C8—21no—
164C4×C8—21no—
165#95—42no—
166#95—42no—
167C24#12711yescharacteristic
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—84no—
189C22×C4#12712yes—
190C22×C4#12711yescharacteristic
191C22×C4C2×Q1614yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194Q16—816no—
195M16—48no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—24no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8C2×Q1614yes—
206C2×C8C2×Q1614yes—
207C2×C8#13718yes—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—48no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—82no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C22⋊C4—44no—
227C22⋊C4—84no—
228C42—21no—
229C42—21no—
230C42—22no—
231C42—22no—
232C42—44no—
233C42C2×#4112yes—
234C42H(5,𝔽2)11yescharacteristic
235C42H(5,𝔽2)11yescharacteristic
236C23—22no—
237C23—21no—
238C23—22no—
239C23—21no—
240C23#49412yes—
241C23C2×#12711yescharacteristic
242Q8—48no—
243Q8—84no—
244Q8—84no—
245Q8—84no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
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—164no—
268C2×C4#49412yes—
269C2×C4#52511yescharacteristic
270C2×C4#53411yescharacteristic
271C2×C4#54612yes—
272C2×C4#54612yes—
273C2×C4C22×Q1612yes—
274C2×C4C2×#13714yes—
275C2×C4#57512yes—
276C2×C4#57614yes—
277C8—24no—
278C8—24no—
279C8—28no—
280C8—44no—
281C22—22no—
282C22—24no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22#253911yescharacteristic
287C22C2×#49411yescharacteristic
288C22#315212yes—
289C22#315311yescharacteristic
290C22#315512yes—
291C4—22no—
292C4—42no—
293C4—84no—
294C4—162no—
295C4—162no—
296C4—162no—
297C4#310412yes—
298C4#318912yes—
299C4#331414yes—
300C4#333314yes—
301C2—42no—
302C2#1963312yes—
303C2#1972911yescharacteristic
304C2#1975111yescharacteristic
305C2#2830411yescharacteristic
306C2#2836012yes—
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. #10674C2semidirect
6. #10318C2semidirect
7. #9833C2semidirect
8. #9825C2semidirect
102. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
#332102
#279502
#256002
#213222
#106742
#103182
#98332
#98252
#81462