Finite Group Databasebeta

← #315788

Subgroups — #315788

Order 512.

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

normal subgroups only

This group has 722 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#33174C211yesmaximal, characteristic
2#27974C212yesmaximal
3#27875C212yesmaximal, complemented
4#20824C211yesmaximal, characteristic, complemented
5#10682C212yesmaximal, complemented
6#10677C212yesmaximal, complemented
7#9842C211yesmaximal, characteristic, complemented
8#9820C211yesmaximal, characteristic, complemented
9#8819C211yesmaximal, characteristic, complemented
10#8816C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#468—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2211yescharacteristic
16#2856C2211yescharacteristic
17#2285C2212yes—
18#2077—24no—
19#2073—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1973—22no—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1956C2211yescharacteristic
27#1920—24no—
28#1875—24no—
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1844C2211yescharacteristic
32#1844C2212yes—
33#1834C2212yes—
34#1834C2212yes—
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#468—48no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#106—22no—
58C2×#106—24no—
59C2×#103—22no—
60C2×#102—22no—
61C2×#102—24no—
62C2×#102—24no—
63C2×C22⋊C8D414yes—
64C4×M16—22no—
65C4×M16—22no—
66C2×#97D412yes—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70C2×C4×C8C2311yescharacteristic
71#391—21no—
72#390—22no—
73#390—21no—
74#389—22no—
75#384C2311yescharacteristic
76#384C2311yescharacteristic
77#384C2312yes—
78#384C2312yes—
79#382—22no—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85#340—24no—
86#340—24no—
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—48no—
118#121—44no—
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—42no—
135C2×C22⋊C4—41no—
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#106—48no—
146#106—44no—
147#103—44no—
148#102—44no—
149#102—84no—
150#102—84no—
151C22⋊C8—44no—
152#97—24no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8#4112yes—
160C4×C8#4112yes—
161#95—42no—
162#95—42no—
163#95—42no—
164C24—41no—
165C24#12711yescharacteristic
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—82no—
179C2×D4—84no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4#12712yes—
194C22×C4#12711yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D412yes—
197SD16—88no—
198M16—48no—
199C2×C8—22no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8#12314yes—
209C2×C8#13514yes—
210C2×C8AGL(1,ℤ/8)14yes—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—84no—
226C22⋊C4—84no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—22no—
231C42—22no—
232C42—84no—
233C42C2×#4111yescharacteristic
234C42C2×#4111yescharacteristic
235C422−1+412yes—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—81no—
242C23—81no—
243C23—82no—
244C23—82no—
245C23—81no—
246C23C2×#12711yescharacteristic
247Q8—82no—
248Q8—82no—
249D4—82no—
250D4—84no—
251D4—84no—
252D4—82no—
253D4—164no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4C2×#12311yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4#54712yes—
280C2×C4#54712yes—
281C2×C4C2×#13512yes—
282C2×C4C2×AGL(1,ℤ/8)12yes—
283C2×C4#57514yes—
284C2×C4#57714yes—
285C8—42no—
286C8—44no—
287C8—44no—
288C8#48218yes—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—42no—
293C22—42no—
294C22—161no—
295C22—161no—
296C22—161no—
297C22—162no—
298C22—162no—
299C22#250111yescharacteristic
300C22#314711yescharacteristic
301C22#315111yescharacteristic
302C22#315312yes—
303C22#315412yes—
304C4—41no—
305C4—42no—
306C4—161no—
307C4—162no—
308C4—161no—
309C4—161no—
310C4—162no—
311C4C2×#48212yes—
312C4#322812yes—
313C4#335014yes—
314C4#335114yes—
315C2—81no—
316C2—321no—
317C2#1964912yes—
318C2#1972111yescharacteristic
319C2#1974911yescharacteristic
320C2#2790711yescharacteristic
321C2#2794712yes—
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. #27875C2semidirect
4. #20824C2semidirect
5. #10682C2semidirect
6. #10677C2semidirect
7. #9842C2semidirect
8. #9820C2semidirect
9. #8819C2semidirect
10. #8816C2semidirect
11. #8146C2semidirect

Maximal subgroups

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

subgroupindex
#331742
#279742
#278752
#208242
#106822
#106772
#98422
#98202
#88192
#88162
#81462