Finite Group Databasebeta

← #319486

Subgroups — #319486

Order 512.

916 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 178 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#33176C211yesmaximal, characteristic, complemented
2#27948C212yesmaximal, complemented
3#25673C212yesmaximal
4#21200C211yesmaximal, characteristic
5#10685C211yesmaximal, characteristic, complemented
6#10683C211yesmaximal, characteristic, complemented
7#10309C211yesmaximal, characteristic, complemented
8#10306C211yesmaximal, characteristic, complemented
9#9836C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#500C2212yes—
14C2×#494C2211yescharacteristic
15C2×#486C2212yes—
16C2×#470—22no—
17C2×#417—22no—
18C2×C4×M16C2211yescharacteristic
19#2857C2212yes—
20C2×#384C2211yescharacteristic
21#2084—24no—
22#2033—24no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1971—22no—
27#1959—22no—
28#1957C2211yescharacteristic
29#1957C2212yes—
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1862—24no—
35#1853—24no—
36#1844C2212yes—
37#1844C2212yes—
38#1844C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×#41—21no—
44C22×Q16—22no—
45C22×M16D412yes—
46#532—22no—
47#532—22no—
48#532—42no—
49C2×C4⋊Q8C2312yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×#127C2311yescharacteristic
52C2×#124—21no—
53C2×C22⋊Q8—22no—
54C2×#121—22no—
55C2×C4×D4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C42C2311yescharacteristic
59#500—24no—
60#500—42no—
61#494—42no—
62#486—24no—
63#486—42no—
64#470—48no—
65C2×#107—24no—
66C2×#106—22no—
67C2×#106—24no—
68C2×#106—24no—
69#417—48no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—22no—
76C2×#102—22no—
77C2×C22⋊C8D414yes—
78C4×M16—42no—
79C2×#97D412yes—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2312yes—
82C2×C4×C8C2312yes—
83C2×C4×C8C2311yescharacteristic
84#391—22no—
85#389—22no—
86#389—22no—
87#384—22no—
88#384—21no—
89#384C2312yes—
90#384C2311yescharacteristic
91#384C2311yescharacteristic
92#380—22no—
93#380—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#336D414yes—
98#322—42no—
99C2×#41—24no—
100C2×#41—42no—
101C2×#41—44no—
102C2×#41—44no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×D4—22no—
106C22×D4—21no—
107C22×D4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C2×Q16—44no—
113C2×Q16—44no—
114C2×Q16—44no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—42no—
118C22×C8C2×D412yes—
119C22×C8C2×D414yes—
120C22×C8C2×D412yes—
121C22×C8C2×D414yes—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8#4112yes—
129#127—21no—
130#127—22no—
131#127#4112yes—
132#124—24no—
133#124—42no—
134C22⋊Q8—48no—
135#121—44no—
136#121—44no—
137C4×D4—44no—
138C4×D4—44no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—84no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42C2×D412yes—
160C2×C42C2×D411yescharacteristic
161C2×C42C2411yescharacteristic
162#107—48no—
163#106—44no—
164#106—84no—
165#106—84no—
166#103—44no—
167#103—48no—
168#103—44no—
169#102—48no—
170#102—44no—
171#102—44no—
172#102—48no—
173C22⋊C8—28no—
174#97—42no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181#95—42no—
182#95—42no—
183C24—41no—
184C24#12711yescharacteristic
185#41—416no—
186#41—88no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—44no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—82no—
205C2×D4—82no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—44no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—84no—
223C22×C4#12712yes—
224C22×C4#12711yescharacteristic
225C22×C4AGL(1,ℤ/8)14yes—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D412yes—
228Q16—88no—
229M16—48no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8—22no—
235C2×C8—24no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8AGL(1,ℤ/8)14yes—
241C2×C8#13718yes—
242C4⋊C4—24no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—82no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C22⋊C4—48no—
259C22⋊C4—84no—
260C42—21no—
261C42—21no—
262C42—22no—
263C42—22no—
264C42C2×#4111yescharacteristic
265C42C2×#4111yescharacteristic
266C42H(5,𝔽2)12yes—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—21no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—82no—
275C23—82no—
276C23#49612yes—
277C23C2×#12711yescharacteristic
278Q8—44no—
279Q8—82no—
280Q8—84no—
281Q8—84no—
282D4—48no—
283D4—44no—
284D4—88no—
285D4—82no—
286D4—84no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—88no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—164no—
311C2×C4#49612yes—
312C2×C4#52511yescharacteristic
313C2×C4#53411yescharacteristic
314C2×C4#54612yes—
315C2×C4#54612yes—
316C2×C4C2×AGL(1,ℤ/8)12yes—
317C2×C4C2×AGL(1,ℤ/8)12yes—
318C2×C4C2×#13714yes—
319C2×C4#57714yes—
320C8—24no—
321C8—28no—
322C8—44no—
323C8—42no—
324C22—22no—
325C22—24no—
326C22—41no—
327C22—41no—
328C22—42no—
329C22—82no—
330C22—84no—
331C22—161no—
332C22—161no—
333C22—162no—
334C22#253911yescharacteristic
335C22C2×#49611yescharacteristic
336C22#315111yescharacteristic
337C22#315212yes—
338C22#315412yes—
339C4—22no—
340C4—42no—
341C4—82no—
342C4—161no—
343C4—161no—
344C4—162no—
345C4—162no—
346C4#310912yes—
347C4#319212yes—
348C4#331614yes—
349C4#333114yes—
350C2—42no—
351C2—162no—
352C2#1967212yes—
353C2#1971311yescharacteristic
354C2#1974611yescharacteristic
355C2#2831011yescharacteristic
356C2#2836212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #33176C2semidirect
2. #27948C2semidirect
5. #10685C2semidirect
6. #10683C2semidirect
7. #10309C2semidirect
8. #10306C2semidirect
9. #9836C2semidirect
10. #9827C2semidirect
11. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279482
#256732
#212002
#106852
#106832
#103092
#103062
#98362
#98272
#81472