Finite Group Databasebeta

← #319602

Subgroups — #319602

Order 512.

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

normal subgroups only

This group has 928 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#33210C211yesmaximal, characteristic, complemented
2#27910C212yesmaximal, complemented
3#25671C212yesmaximal
4#21200C211yesmaximal, characteristic
5#10674C211yesmaximal, characteristic, complemented
6#10673C211yesmaximal, characteristic, complemented
7#10316C211yesmaximal, characteristic, complemented
8#10312C211yesmaximal, characteristic, complemented
9#9841C212yesmaximal, complemented
10#9830C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#498C2212yes—
14C2×#498C2211yescharacteristic
15C2×#486C2212yes—
16C2×#484—24no—
17C2×#417—22no—
18C2×C4×M16C2211yescharacteristic
19#2857C2212yes—
20C2×#384C2211yescharacteristic
21#2095—24no—
22#2034—24no—
23#1990C2212yes—
24#1990C2212yes—
25#1959—22no—
26#1957C2212yes—
27#1957C2211yescharacteristic
28#1957C2211yescharacteristic
29#1956C2212yes—
30#1956C2212yes—
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1943—22no—
34#1864—24no—
35#1845C2212yes—
36#1845C2212yes—
37#1845C2211yescharacteristic
38#1809—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×#41—21no—
44C22×M16D412yes—
45#532—22no—
46#532—22no—
47#532—42no—
48C2×C4⋊Q8C2312yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#124—21no—
52C2×C22⋊Q8—22no—
53C2×#121—22no—
54C2×C4×D4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C42C2311yescharacteristic
58#498—24no—
59#498—42no—
60#498—42no—
61#486—24no—
62#486—42no—
63#484—416no—
64C2×#107—22no—
65C2×#107—24no—
66C2×#107—24no—
67C2×#106—22no—
68C2×#106—24no—
69#417—48no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—22no—
77C2×#102—22no—
78C2×C22⋊C8D414yes—
79C4×M16—42no—
80C2×#97D412yes—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2312yes—
83C2×C4×C8C2312yes—
84C2×C4×C8C2311yescharacteristic
85#391—22no—
86#389—22no—
87#389—22no—
88#384—22no—
89#384—21no—
90#384C2312yes—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#380—22no—
94#380—22no—
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×M16—24no—
113C2×M16—24no—
114C2×M16—42no—
115C22×C8C2×D412yes—
116C22×C8C2×D414yes—
117C22×C8C2×D412yes—
118C22×C8C2×D414yes—
119C4⋊Q8—21no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8#4112yes—
126#127—21no—
127#127—22no—
128#127#4112yes—
129#124—24no—
130#124—42no—
131C22⋊Q8—48no—
132#121—44no—
133#121—44no—
134C4×D4—44no—
135C4×D4—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2411yescharacteristic
159#107—48no—
160#107—44no—
161#107—48no—
162#106—48no—
163#106—44no—
164#103—44no—
165#103—48no—
166#103—44no—
167#103—48no—
168#102—48no—
169#102—44no—
170#102—48no—
171#102—44no—
172C22⋊C8—28no—
173#97—42no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—22no—
180#95—42no—
181#95—42no—
182C24—41no—
183C24#12711yescharacteristic
184#41—416no—
185#41—88no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—44no—
192C2×Q8—42no—
193C2×Q8—44no—
194C2×D4—22no—
195C2×D4—24no—
196C2×D4—44no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—44no—
203C2×D4—82no—
204C2×D4—82no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—44no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—84no—
221C22×C4—82no—
222C22×C4#12712yes—
223C22×C4#12711yescharacteristic
224C22×C4#13714yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D412yes—
227M16—48no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—24no—
238C2×C8AGL(1,ℤ/8)18yes—
239C2×C8AGL(1,ℤ/8)14yes—
240C4⋊C4—24no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C22⋊C4—48no—
257C22⋊C4—84no—
258C42—21no—
259C42—21no—
260C42—22no—
261C42—22no—
262C42C2×#4111yescharacteristic
263C42C2×#4111yescharacteristic
264C42H(5,𝔽2)12yes—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—82no—
273C23—82no—
274C23#49712yes—
275C23C2×#12711yescharacteristic
276Q8—44no—
277Q8—82no—
278Q8—84no—
279Q8—84no—
280D4—48no—
281D4—44no—
282D4—88no—
283D4—82no—
284D4—84no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—88no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—164no—
309C2×C4#49712yes—
310C2×C4#52511yescharacteristic
311C2×C4#53411yescharacteristic
312C2×C4#54612yes—
313C2×C4#54612yes—
314C2×C4C2×AGL(1,ℤ/8)14yes—
315C2×C4C2×AGL(1,ℤ/8)12yes—
316C2×C4C2×#13712yes—
317C2×C4#57814yes—
318C8—24no—
319C8—28no—
320C8—44no—
321C8—42no—
322C22—22no—
323C22—24no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—161no—
330C22—161no—
331C22—162no—
332C22#253911yescharacteristic
333C22C2×#49711yescharacteristic
334C22#315111yescharacteristic
335C22#315112yes—
336C22#315612yes—
337C4—22no—
338C4—42no—
339C4—82no—
340C4—162no—
341C4—161no—
342C4—161no—
343C4—162no—
344C4#311012yes—
345C4#319312yes—
346C4#330814yes—
347C4#333014yes—
348C2—42no—
349C2—162no—
350C2#1968412yes—
351C2#1973111yescharacteristic
352C2#1974511yescharacteristic
353C2#2831111yescharacteristic
354C2#2835812yes—
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. #33210C2semidirect
2. #27910C2semidirect
5. #10674C2semidirect
6. #10673C2semidirect
7. #10316C2semidirect
8. #10312C2semidirect
9. #9841C2semidirect
10. #9830C2semidirect
11. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#279102
#256712
#212002
#106742
#106732
#103162
#103122
#98412
#98302
#81472