Finite Group Databasebeta

← #319612

Subgroups — #319612

Order 512.

888 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 170 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#27961C212yesmaximal, complemented
3#25597C212yesmaximal
4#21200C211yesmaximal, characteristic
5#10676C211yesmaximal, characteristic, complemented
6#10671C211yesmaximal, characteristic, complemented
7#10317C211yesmaximal, characteristic, complemented
8#10315C211yesmaximal, characteristic, complemented
9#9840C212yesmaximal, complemented
10#9829C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#498C2211yescharacteristic
14C2×#492C2212yes—
15C2×#487C2212yes—
16C2×#417—22no—
17C2×C4×M16C2211yescharacteristic
18#2857C2212yes—
19C2×#384C2211yescharacteristic
20#2100—24no—
21#2000—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1959—22no—
27#1957C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2212yes—
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1943—22no—
33#1877—24no—
34#1845C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1824—24no—
38#1809—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×#41—21no—
44C22×SD16—24no—
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—22no—
57C22×C4⋊C4—21no—
58C22×C42C2311yescharacteristic
59#498—42no—
60#492—24no—
61#492—42no—
62#487—24no—
63#487—42no—
64C2×#107—22no—
65C2×#106—22no—
66#417—48no—
67C2×#103—24no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—24no—
73C2×C22⋊C8D414yes—
74C4×M16—42no—
75C2×#97D412yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391—22no—
81#389—22no—
82#389—22no—
83#384—22no—
84#384—21no—
85#384C2312yes—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#380—22no—
89#380—22no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#336D414yes—
97#322—42no—
98C2×#41—24no—
99C2×#41—42no—
100C2×#41—44no—
101C2×#41—44no—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C2×SD16—48no—
112C2×SD16—48no—
113C2×SD16—84no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—42no—
117C22×C8C2×D412yes—
118C22×C8C2×D414yes—
119C22×C8C2×D414yes—
120C22×C8C2×D412yes—
121C4⋊Q8—21no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8#4112yes—
128#127—21no—
129#127—22no—
130#127#4112yes—
131#124—24no—
132#124—42no—
133C22⋊Q8—84no—
134#121—44no—
135#121—44no—
136C4×D4—44no—
137C4×D4—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42C2×D412yes—
159C2×C42C2×D411yescharacteristic
160C2×C42C2411yescharacteristic
161#107—44no—
162#106—44no—
163#103—48no—
164#103—44no—
165#103—44no—
166#102—44no—
167#102—44no—
168#102—84no—
169C22⋊C8—44no—
170#97—42no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177#95—42no—
178#95—42no—
179C24—41no—
180C24#12711yescharacteristic
181#41—416no—
182#41—88no—
183C2×Q8—22no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—44no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—82no—
201C2×D4—82no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—44no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—84no—
219C22×C4#12712yes—
220C22×C4#12711yescharacteristic
221C22×C4#13714yes—
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D412yes—
224SD16—816no—
225M16—48no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8AGL(1,ℤ/8)14yes—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C22⋊C4—48no—
255C22⋊C4—84no—
256C42—21no—
257C42—21no—
258C42—22no—
259C42—22no—
260C42C2×#4111yescharacteristic
261C42C2×#4111yescharacteristic
262C42H(5,𝔽2)12yes—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—42no—
268C23—81no—
269C23—81no—
270C23—82no—
271C23—82no—
272C23#49712yes—
273C23C2×#12711yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—84no—
277Q8—84no—
278D4—44no—
279D4—48no—
280D4—88no—
281D4—82no—
282D4—84no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—88no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—164no—
307C2×C4#49712yes—
308C2×C4#52511yescharacteristic
309C2×C4#53411yescharacteristic
310C2×C4#54612yes—
311C2×C4#54612yes—
312C2×C4C2×AGL(1,ℤ/8)12yes—
313C2×C4C2×#13712yes—
314C2×C4#57514yes—
315C2×C4#57714yes—
316C8—24no—
317C8—42no—
318C8—44no—
319C8—44no—
320C22—22no—
321C22—24no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—82no—
326C22—84no—
327C22—161no—
328C22—161no—
329C22—162no—
330C22#253911yescharacteristic
331C22C2×#49711yescharacteristic
332C22#315111yescharacteristic
333C22#315312yes—
334C22#315412yes—
335C4—22no—
336C4—42no—
337C4—82no—
338C4—162no—
339C4—161no—
340C4—161no—
341C4—162no—
342C4#311012yes—
343C4#319312yes—
344C4#331714yes—
345C4#332314yes—
346C2—42no—
347C2—162no—
348C2#1968512yes—
349C2#1972111yescharacteristic
350C2#1974911yescharacteristic
351C2#2831111yescharacteristic
352C2#2836612yes—
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. #27961C2semidirect
5. #10676C2semidirect
6. #10671C2semidirect
7. #10317C2semidirect
8. #10315C2semidirect
9. #9840C2semidirect
10. #9829C2semidirect
11. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#279612
#255972
#212002
#106762
#106712
#103172
#103152
#98402
#98292
#81472