Finite Group Databasebeta

← #319600

Subgroups — #319600

Order 512.

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

normal subgroups only

This group has 830 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#33210C211yesmaximal, characteristic
2#27892C212yesmaximal
3#25811C212yesmaximal
4#21202C211yesmaximal, characteristic
5#10676C211yesmaximal, characteristic, complemented
6#10674C211yesmaximal, characteristic, complemented
7#10316C211yesmaximal, characteristic, complemented
8#10316C211yesmaximal, characteristic, complemented
9#9841C212yesmaximal, complemented
10#9828C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#498C2212yes—
14C2×#498C2212yes—
15C2×#498C2211yescharacteristic
16C2×#419—22no—
17C2×C4×M16C2211yescharacteristic
18#2857C2212yes—
19#2285C2211yescharacteristic
20#2058—24no—
21#2036—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1960—22no—
26#1956C2212yes—
27#1956C2212yes—
28#1956C2211yescharacteristic
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1943—22no—
33#1928—24no—
34#1853—24no—
35#1845C2212yes—
36#1845C2211yescharacteristic
37#1834C2212yes—
38#1809—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C23×Q8—21no—
44C22×M16D412yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—42no—
50C2×C4⋊Q8C2312yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C22⋊Q8—22no—
54C2×#117—21no—
55C22×C4⋊C4—22no—
56C22×C42C2311yescharacteristic
57#498—24no—
58#498—24no—
59#498—42no—
60#498—42no—
61#498—42no—
62C2×#107—22no—
63C2×#107—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—22no—
67#419—48no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#103—24no—
74C2×C22⋊C8D414yes—
75C4×M16—42no—
76C2×#97D412yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81#391—22no—
82#389—22no—
83#389—22no—
84#384C2312yes—
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#382—21no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#336D414yes—
97#322—42no—
98C22×Q8—24no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—22no—
102C22×Q8—42no—
103C22×Q8—44no—
104C22×Q8—41no—
105C22×Q8—44no—
106C23×C4—41no—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—42no—
112C22×C8C2×D412yes—
113C22×C8C2×D414yes—
114C22×C8C2×D412yes—
115C22×C8C2×D414yes—
116C4⋊Q8—24no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—42no—
128C4⋊Q8#4112yes—
129C4⋊Q8#4112yes—
130C22⋊Q8—84no—
131#117—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—42no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2411yescharacteristic
156#107—44no—
157#107—84no—
158#106—48no—
159#106—48no—
160#106—44no—
161#103—44no—
162#103—44no—
163#103—44no—
164#103—44no—
165#103—48no—
166#103—84no—
167C22⋊C8—28no—
168#97—42no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175#95—42no—
176#95—42no—
177#95—42no—
178C24#12711yescharacteristic
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—44no—
182C2×Q8—416no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—82no—
194C2×Q8—88no—
195C2×Q8—82no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—44no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4#12712yes—
213C22×C4#12711yescharacteristic
214C22×C4#13714yes—
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D412yes—
217M16—48no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8AGL(1,ℤ/8)14yes—
229C2×C8#13718yes—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—48no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—84no—
247C22⋊C4—82no—
248C22⋊C4—84no—
249C42—21no—
250C42—21no—
251C42—22no—
252C42—22no—
253C42—82no—
254C42C2×#4111yescharacteristic
255C42C2×#4111yescharacteristic
256C42H(5,𝔽2)12yes—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—21no—
261C23#49712yes—
262C23C2×#12711yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—88no—
266Q8—82no—
267Q8—82no—
268Q8—84no—
269Q8—84no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—88no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4#49712yes—
297C2×C4#52511yescharacteristic
298C2×C4#53411yescharacteristic
299C2×C4#54612yes—
300C2×C4#54612yes—
301C2×C4C2×AGL(1,ℤ/8)12yes—
302C2×C4C2×#13714yes—
303C2×C4C2×#13712yes—
304C2×C4#57714yes—
305C8—24no—
306C8—28no—
307C8—44no—
308C8—42no—
309C22—22no—
310C22—24no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22#253911yescharacteristic
315C22C2×#49711yescharacteristic
316C22#315111yescharacteristic
317C22#315212yes—
318C22#315412yes—
319C4—22no—
320C4—42no—
321C4—82no—
322C4—82no—
323C4—161no—
324C4—162no—
325C4—161no—
326C4—162no—
327C4#311012yes—
328C4#319312yes—
329C4#332014yes—
330C4#332914yes—
331C2—42no—
332C2#1968012yes—
333C2#1972111yescharacteristic
334C2#1974611yescharacteristic
335C2#2831111yescharacteristic
336C2#2836712yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
5. #10676C2semidirect
6. #10674C2semidirect
7. #10316C2semidirect
8. #10316C2semidirect
9. #9841C2semidirect
10. #9828C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#278922
#258112
#212022
#106762
#106742
#103162
#98412
#98282
#81472