Finite Group Databasebeta

← #315786

Subgroups — #315786

Order 512.

738 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 186 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#33174C211yesmaximal, characteristic
2#27949C212yesmaximal, complemented
3#27891C212yesmaximal
4#20824C211yesmaximal, characteristic, complemented
5#10679C212yesmaximal, complemented
6#10672C212yesmaximal, complemented
7#9842C211yesmaximal, characteristic, complemented
8#9820C211yesmaximal, characteristic, complemented
9#8819C211yesmaximal, characteristic, complemented
10#8818C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#468—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2211yescharacteristic
16#2856C2211yescharacteristic
17#2285C2212yes—
18#2085—24no—
19#2033—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1973—22no—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1956C2211yescharacteristic
27#1925—24no—
28#1854—24no—
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1845C2211yescharacteristic
32#1845C2212yes—
33#1844C2212yes—
34#1844C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1809—22no—
38#1776—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×SD16—22no—
44C22×M16D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×#121—21no—
49C2×#121—22no—
50C2×#117—22no—
51C22×C42C2311yescharacteristic
52#492—22no—
53#492—22no—
54#468—48no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—24no—
58C2×#107—24no—
59C2×#106—22no—
60C2×#106—24no—
61C2×#103—22no—
62C2×#102—24no—
63C2×#102—22no—
64C2×#102—24no—
65C2×C22⋊C8D414yes—
66C4×M16—22no—
67C4×M16—22no—
68C2×#97D412yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73#391—21no—
74#390—22no—
75#390—21no—
76#389—22no—
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#384C2312yes—
80#384C2312yes—
81#382—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#336D414yes—
92#322—22no—
93C82—22no—
94C22×Q8—21no—
95C22×D4—21no—
96C22×D4—22no—
97C22×D4—41no—
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C2×SD16—44no—
101C2×SD16—44no—
102C2×SD16—44no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—24no—
106C22×C8C2×D412yes—
107C22×C8C2×D414yes—
108C22×C8C2×D414yes—
109C22×C8C2×D412yes—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—22no—
113#127—21no—
114#127—21no—
115#127—22no—
116C22⋊Q8—44no—
117#121—44no—
118#121—84no—
119#117—84no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
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—41no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—42no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—42no—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2411yescharacteristic
143#107—44no—
144#107—44no—
145#107—84no—
146#107—84no—
147#106—44no—
148#106—84no—
149#103—44no—
150#102—44no—
151#102—48no—
152#102—84no—
153C22⋊C8—28no—
154#97—24no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8#4112yes—
162C4×C8#4112yes—
163#95—42no—
164#95—42no—
165#95—42no—
166C24—41no—
167C24#12711yescharacteristic
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—82no—
181C2×D4—84no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#12712yes—
196C22×C4#12711yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D412yes—
199SD16—88no—
200M16—48no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8#12314yes—
210C2×C8#13514yes—
211C2×C8AGL(1,ℤ/8)14yes—
212C2×C8#13718yes—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—84no—
235C42C2×#4111yescharacteristic
236C42C2×#4111yescharacteristic
237C422−1+412yes—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—21no—
243C23—81no—
244C23—81no—
245C23—82no—
246C23—82no—
247C23—81no—
248C23C2×#12711yescharacteristic
249Q8—82no—
250Q8—82no—
251D4—82no—
252D4—84no—
253D4—84no—
254D4—82no—
255D4—164no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4C2×#12311yescharacteristic
280C2×C4#53611yescharacteristic
281C2×C4#54712yes—
282C2×C4#54712yes—
283C2×C4C2×#13512yes—
284C2×C4C2×AGL(1,ℤ/8)12yes—
285C2×C4C2×#13714yes—
286C2×C4#57614yes—
287C8—28no—
288C8—42no—
289C8—44no—
290C8#48218yes—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—42no—
295C22—42no—
296C22—161no—
297C22—161no—
298C22—161no—
299C22—162no—
300C22—162no—
301C22#250111yescharacteristic
302C22#314711yescharacteristic
303C22#315111yescharacteristic
304C22#315212yes—
305C22#315512yes—
306C4—41no—
307C4—42no—
308C4—161no—
309C4—161no—
310C4—162no—
311C4—161no—
312C4—162no—
313C4C2×#48212yes—
314C4#322812yes—
315C4#334614yes—
316C4#335714yes—
317C2—81no—
318C2—321no—
319C2#1964612yes—
320C2#1972711yescharacteristic
321C2#1974611yescharacteristic
322C2#2790711yescharacteristic
323C2#2794512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
2. #27949C2semidirect
4. #20824C2semidirect
5. #10679C2semidirect
6. #10672C2semidirect
7. #9842C2semidirect
8. #9820C2semidirect
9. #8819C2semidirect
10. #8818C2semidirect
11. #8146C2semidirect

Maximal subgroups

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

subgroupindex
#331742
#279492
#278912
#208242
#106792
#106722
#98422
#98202
#88192
#88182
#81462