Finite Group Databasebeta

← #331269

Subgroups — #331269

Order 512.

774 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 162 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#33174C211yesmaximal, characteristic
3#29877C212yesmaximal, complemented
4#29807C212yesmaximal
5#10679C212yesmaximal, complemented
6#10676C211yesmaximal, characteristic, complemented
7#10671C211yesmaximal, characteristic, complemented
8#10316C211yesmaximal, characteristic, complemented
9#10315C211yesmaximal, characteristic, complemented
10#10315C212yesmaximal, complemented
11#7850C211yesmaximal, characteristic, complemented
12C2×#498C2212yes—
13C2×#498C2211yescharacteristic
14C2×#492C2212yes—
15C2×#492C2211yescharacteristic
16C2×#468—22no—
17C2×#431C2212yes—
18C2×C4×M16C2211yescharacteristic
19#2857C2212yes—
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2088—24no—
23#2077—24no—
24#2055—24no—
25#2000—24no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1973—22no—
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1943—22no—
35#1845C2211yescharacteristic
36#1845C2211yescharacteristic
37#1834C2212yes—
38#1834C2212yes—
39#1809—22no—
40#1273C2211yescharacteristic
41#1273C2212yes—
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C22×SD16—24no—
45C22×M16D412yes—
46C2×C4⋊Q8C2312yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×#121—21no—
52C2×#121—22no—
53C22×C42C2311yescharacteristic
54#498—42no—
55#498—42no—
56#498—42no—
57#492—42no—
58#492—42no—
59#492—42no—
60#468—48no—
61#431—42no—
62#431—42no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#106—22no—
66C2×#106—24no—
67C2×C4⋊C8D414yes—
68C2×#103—22no—
69C2×#103—24no—
70C2×#102—22no—
71C2×#102—24no—
72C2×C22⋊C8D414yes—
73C4×M16—42no—
74C2×#97D412yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391—22no—
80#391—21no—
81#390—22no—
82#390—21no—
83#389—22no—
84#389—22no—
85#389—22no—
86#384C2311yescharacteristic
87#384C2312yes—
88#384C2311yescharacteristic
89#380—22no—
90#380—22no—
91#380—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×D4—21no—
100C22×D4—22no—
101C22×D4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C2×SD16—44no—
105C2×SD16—44no—
106C2×SD16—48no—
107C2×SD16—44no—
108C2×SD16—84no—
109C2×SD16—84no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—42no—
113C22×C8C2×D414yes—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—22no—
123C4⋊Q8—21no—
124#127—22no—
125#127—21no—
126#127—21no—
127C22⋊Q8—44no—
128C22⋊Q8—84no—
129#121—44no—
130#121—84no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D412yes—
152C2×C42C2411yescharacteristic
153#107—44no—
154#107—44no—
155#106—44no—
156#106—48no—
157C4⋊C8—44no—
158#103—44no—
159#103—84no—
160#102—44no—
161#102—84no—
162C22⋊C8—44no—
163#97—42no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170#95—42no—
171#95—42no—
172#95—42no—
173C24—41no—
174C24#12711yescharacteristic
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—82no—
192C2×D4—84no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4#12711yescharacteristic
207C22×C4#12712yes—
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D412yes—
210SD16—88no—
211SD16—816no—
212M16—48no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8AGL(1,ℤ/8)14yes—
224C2×C8#13714yes—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—82no—
239C4⋊C4—84no—
240C22⋊C4—82no—
241C22⋊C4—84no—
242C22⋊C4—82no—
243C22⋊C4—84no—
244C42—21no—
245C42—22no—
246C42—21no—
247C42—22no—
248C42H(5,𝔽2)12yes—
249C42H(5,𝔽2)11yescharacteristic
250C422−1+411yescharacteristic
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—81no—
257C23—81no—
258C23—82no—
259C23—82no—
260C23—81no—
261C23C2×#12711yescharacteristic
262Q8—82no—
263Q8—84no—
264Q8—82no—
265Q8—84no—
266D4—82no—
267D4—84no—
268D4—82no—
269D4—84no—
270D4—164no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4#53411yescharacteristic
295C2×C4#53412yes—
296C2×C4#53711yescharacteristic
297C2×C4#53712yes—
298C2×C4C2×AGL(1,ℤ/8)12yes—
299C2×C4C2×#13712yes—
300C2×C4#57714yes—
301C2×C4#57714yes—
302C8—24no—
303C8—24no—
304C8—44no—
305C8—44no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—161no—
312C22—162no—
313C22—161no—
314C22—161no—
315C22—162no—
316C22#265611yescharacteristic
317C22#315111yescharacteristic
318C22#315211yescharacteristic
319C22#315412yes—
320C22#315412yes—
321C4—41no—
322C4—42no—
323C4—161no—
324C4—161no—
325C4—162no—
326C4—161no—
327C4—162no—
328C4#319312yes—
329C4#320014yes—
330C4#325412yes—
331C4#326014yes—
332C2—81no—
333C2—321no—
334C2#1971311yescharacteristic
335C2#1971412yes—
336C2#1972111yescharacteristic
337C2#3024811yescharacteristic
338C2#3025412yes—
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
1. #33210C2semidirect
3. #29877C2semidirect
5. #10679C2semidirect
6. #10676C2semidirect
7. #10671C2semidirect
8. #10316C2semidirect
9. #10315C2semidirect
10. #10315C2semidirect
11. #7850C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#331742
#298772
#298072
#106792
#106762
#106712
#103162
#103152
#78502