Finite Group Databasebeta

← #315785

Subgroups — #315785

Order 512.

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

normal subgroups only

This group has 778 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#27907C212yesmaximal
3#27878C212yesmaximal, complemented
4#20824C211yesmaximal, characteristic, complemented
5#10680C212yesmaximal, complemented
6#10671C212yesmaximal, complemented
7#9842C211yesmaximal, characteristic, complemented
8#9820C211yesmaximal, characteristic, complemented
9#8819C211yesmaximal, characteristic, complemented
10#8818C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#482—24no—
14C2×#468—22no—
15C2×C4×M16C2211yescharacteristic
16#2857C2211yescharacteristic
17#2856C2211yescharacteristic
18#2285C2212yes—
19#2092—24no—
20#2076—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1973—22no—
24#1957C2212yes—
25#1957C2212yes—
26#1957C2211yescharacteristic
27#1956C2211yescharacteristic
28#1921—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—22no—
49C2×#121—21no—
50C2×#117—22no—
51C22×C42C2311yescharacteristic
52#492—22no—
53#492—22no—
54#482—416no—
55#468—48no—
56C2×#107—24no—
57C2×#107—22no—
58C2×#107—24no—
59C2×#107—22no—
60C2×#106—24no—
61C2×#106—22no—
62C2×#103—22no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—22no—
66C2×C22⋊C8D414yes—
67C4×M16—22no—
68C4×M16—22no—
69C2×#97D412yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#391—21no—
75#390—22no—
76#390—21no—
77#389—22no—
78#384C2311yescharacteristic
79#384C2311yescharacteristic
80#384C2312yes—
81#384C2312yes—
82#382—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—22no—
94C82—22no—
95C22×Q8—21no—
96C22×D4—21no—
97C22×D4—22no—
98C22×D4—41no—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C2×SD16—44no—
102C2×SD16—44no—
103C2×SD16—44no—
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C22×C8C2×D412yes—
108C22×C8C2×D414yes—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—22no—
114#127—21no—
115#127—21no—
116#127—22no—
117C22⋊Q8—44no—
118#121—44no—
119#121—48no—
120#117—48no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—41no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—42no—
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—48no—
145#107—44no—
146#107—44no—
147#107—48no—
148#106—48no—
149#106—44no—
150#103—44no—
151#102—44no—
152#102—48no—
153#102—48no—
154C22⋊C8—28no—
155#97—24no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8#4112yes—
163C4×C8#4112yes—
164#95—42no—
165#95—42no—
166#95—42no—
167C24—41no—
168C24#12711yescharacteristic
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—82no—
182C2×D4—84no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4#12712yes—
197C22×C4#12711yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D412yes—
200SD16—88no—
201M16—48no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—24no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—24no—
210C2×C8#12314yes—
211C2×C8#13514yes—
212C2×C8AGL(1,ℤ/8)18yes—
213C2×C8AGL(1,ℤ/8)14yes—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—82no—
227C22⋊C4—84no—
228C22⋊C4—82no—
229C22⋊C4—84no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—22no—
234C42—22no—
235C42—84no—
236C42C2×#4111yescharacteristic
237C42C2×#4111yescharacteristic
238C422−1+412yes—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—81no—
245C23—81no—
246C23—82no—
247C23—82no—
248C23—81no—
249C23C2×#12711yescharacteristic
250Q8—82no—
251Q8—82no—
252D4—82no—
253D4—84no—
254D4—84no—
255D4—82no—
256D4—164no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4C2×#12311yescharacteristic
281C2×C4#53611yescharacteristic
282C2×C4#54712yes—
283C2×C4#54712yes—
284C2×C4C2×#13512yes—
285C2×C4C2×AGL(1,ℤ/8)12yes—
286C2×C4C2×AGL(1,ℤ/8)14yes—
287C2×C4#57814yes—
288C8—28no—
289C8—42no—
290C8—44no—
291C8#48218yescomplemented
292C22—41no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—161no—
298C22—161no—
299C22—161no—
300C22—162no—
301C22—162no—
302C22#250111yescharacteristic
303C22#314711yescharacteristic
304C22#315111yescharacteristic
305C22#315112yes—
306C22#315612yes—
307C4—41no—
308C4—42no—
309C4—161no—
310C4—162no—
311C4—161no—
312C4—161no—
313C4—162no—
314C4C2×#48212yes—
315C4#322812yes—
316C4#334314yes—
317C4#335314yes—
318C2—81no—
319C2—321no—
320C2#1964512yes—
321C2#1973111yescharacteristic
322C2#1974511yescharacteristic
323C2#2790711yescharacteristic
324C2#2794412yes—
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
3. #27878C2semidirect
4. #20824C2semidirect
5. #10680C2semidirect
6. #10671C2semidirect
7. #9842C2semidirect
8. #9820C2semidirect
9. #8819C2semidirect
10. #8818C2semidirect
11. #8146C2semidirect
291. C8#482semidirect

Maximal subgroups

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

subgroupindex
#331742
#279072
#278782
#208242
#106802
#106712
#98422
#98202
#88192
#88182
#81462