Finite Group Databasebeta

← #305439

Subgroups — #305439

Order 512.

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

normal subgroups only

This group has 732 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#26040C212yesmaximal, complemented
2#22503C212yesmaximal
3#10674C211yesmaximal, characteristic, complemented
4#10589C212yesmaximal, complemented
5#10349C211yesmaximal, characteristic, complemented
6#8882C212yesmaximal, complemented
7#8849C212yesmaximal, complemented
8#8652C212yesmaximal, complemented
9#8452C211yesmaximal, characteristic
10C2×#555C2211yescharacteristic
11C2×#474—22no—
12C2×#436C2212yes—
13#2551C2211yescharacteristic
14D4×C4⋊C4C2212yes—
15#2086—22no—
16#2057—22no—
17#1990C2212yes—
18#1987C2212yes—
19#1956C2211yescharacteristic
20#1947C2212yes—
21#1939C2212yes—
22#1895—22no—
23#1874—22no—
24#1862—22no—
25#1845C2212yes—
26#1845C2211yescharacteristic
27#1825—22no—
28#1802C2212yes—
29#1802C2212yes—
30C4×#102C2212yes—
31#1736—22no—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×D4⋊C8C2212yes—
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1273C2211yescharacteristic
39C22×M16—22no—
40#555—24no—
41#555—41no—
42#555Q814yescomplemented
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#124—41no—
46C2×#121—22no—
47C2×C4×D4—22no—
48C2×C4×D4C2312yes—
49C2×#117—22no—
50C22×C4⋊C4—22no—
51#474—48no—
52#436—24no—
53#436—42no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—41no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×#103—41no—
62C2×#102—22no—
63C2×#102—22no—
64C2×#102D412yes—
65C2×#102D412yes—
66C2×C22⋊C8—22no—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2312yes—
69#395—41no—
70#389—22no—
71#389—22no—
72#385—42no—
73#384C2311yescharacteristic
74#384C2312yes—
75#380—22no—
76C4×C4⋊C4C2312yes—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340D412yes—
85#340D412yes—
86#336—22no—
87#334—42no—
88#333—41no—
89D4⋊C8—24no—
90C22×Q8—21no—
91C22×Q8—41no—
92C22×D4—41no—
93C22×D4C2×D412yes—
94C23×C4—22no—
95C23×C4—42no—
96C2×M16—44no—
97C2×M16—44no—
98C2×M16—82no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C4⋊Q8—22no—
105C4⋊Q8—21no—
106C4⋊Q8—44no—
107C4⋊Q8—82no—
108C4⋊Q8C2×Q812yes—
109#124—44no—
110#124—44no—
111#121—28no—
112#121—44no—
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—44no—
116C4×D4—82no—
117C4×D4C2×Q814yes—
118C4×D4#4114yes—
119#117—28no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42—22no—
138C2×C42—41no—
139C2×C42C2411yescharacteristic
140#107—44no—
141#107—44no—
142#107—82no—
143#107—82no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—44no—
149#103—82no—
150#102—24no—
151#102—24no—
152#102—44no—
153#102—82no—
154C22⋊C8—44no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158#95—22no—
159#95—42no—
160#95—41no—
161C24—22no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—84no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×Q8—161no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4—161no—
175C2×D4—161no—
176C2×D4C22⋊Q814yes—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—162no—
192C22×C4C22×D412yes—
193C22×C4H(5,𝔽2)11yescharacteristic
194M16—88no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8#13714yes—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4C22⋊Q814yes—
227C4⋊C4AGL(1,ℤ/8)14yes—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—48no—
231C22⋊C4—84no—
232C42—44no—
233C42—42no—
234C42—42no—
235C42C22×Q811yescharacteristic
236C42C2×#4112yes—
237C42H(5,𝔽2)11yescharacteristic
238C23—24no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23D4211yescharacteristic
246Q8—82no—
247Q8—161no—
248Q8—164no—
249D4—24no—
250D4—24no—
251D4—42no—
252D4—42no—
253D4—164no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—168no—
278C2×C4C2×C22⋊Q812yes—
279C2×C4#54311yescharacteristic
280C2×C4#54612yes—
281C2×C4#55111yescharacteristic
282C2×C4C2×AGL(1,ℤ/8)12yes—
283C2×C4C2×#13712yes—
284C2×C4#57612yes—
285C2×C4#57812yes—
286C8—24no—
287C8—42no—
288C8—42no—
289C8—82no—
290C8—82no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22#243711yescharacteristic
299C22#326412yes—
300C22#327612yes—
301C22#328911yescharacteristic
302C22#329211yescharacteristic
303C4—21no—
304C4—21no—
305C4—44no—
306C4—44no—
307C4—82no—
308C4—82no—
309C4—162no—
310C4—161no—
311C4#307912yes—
312C4#308312yes—
313C4#331412yes—
314C4#333012yes—
315C2—84no—
316C2#2005311yescharacteristic
317C2#2016712yes—
318C2#2032411yescharacteristic
319C2#2597012yes—
320C2#2598211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#555C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26040C2semidirect
3. #10674C2semidirect
4. #10589C2semidirect
5. #10349C2semidirect
6. #8882C2semidirect
7. #8849C2semidirect
8. #8652C2semidirect
42. #555Q8semidirect

Maximal subgroups

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

subgroupindex
#260402
#225032
#106742
#105892
#103492
#88822
#88492
#86522
#84522