Finite Group Databasebeta

← #325727

Subgroups — #325727

Order 512.

585 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 154 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#30277C211yesmaximal, characteristic
2#29226C211yesmaximal, characteristic, complemented
3#28752C211yesmaximal, characteristic
4#22303C211yesmaximal, characteristic, complemented
5#10544C212yesmaximal, complemented
6#10539C212yesmaximal
7#9747C212yesmaximal
8#9741C212yesmaximal, complemented
9#9704C211yesmaximal, characteristic
10#9691C211yesmaximal, characteristic, complemented
11#8278C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#498C2211yescharacteristic
14C2×#488—21no—
15C2×#486C2211yescharacteristic
16C2×#486C2211yescharacteristic
17#2868C2211yescharacteristic
18#2858C2211yescharacteristic
19C4×#125C2211yescharacteristic
20#2099—22no—
21#2094—22no—
22#2094—22no—
23#2042—22no—
24#1992—21no—
25#1987C2212yes—
26#1987C2212yes—
27#1982C2212yes—
28#1981C2212yes—
29#1958—21no—
30#1940—22no—
31#1939C2212yes—
32#1917—22no—
33#1842C2211yescharacteristic
34#1836—21no—
35#1770C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#127C2311yescharacteristic
50C2×#125D411yescharacteristic
51C2×#124—21no—
52C2×#124—21no—
53#498—24no—
54#498—42no—
55#488—44no—
56#486—24no—
57#486—42no—
58C2×#106—21no—
59C2×#106—21no—
60C2×#106—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#103—22no—
67C2×#103—22no—
68C2×#102—21no—
69C2×#102—21no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#97—21no—
74C2×#97—22no—
75C2×C4×C8—21no—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#390—22no—
79#390—22no—
80#389D412yes—
81#387—22no—
82#384—21no—
83#384—22no—
84#384—21no—
85#382D411yescharacteristic
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#336—21no—
97#336D412yes—
98#336D412yes—
99#335—24no—
100#333—42no—
101#333—42no—
102#332—42no—
103#331—42no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—84no—
123#127—22no—
124#127—22no—
125#125—22no—
126#125—22no—
127#124—42no—
128#124—42no—
129#124—82no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C42—22no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42#4112yes—
149C2×C42C2411yescharacteristic
150#106—42no—
151#106—42no—
152#106—44no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—82no—
160C4⋊C8—82no—
161#103—44no—
162#103—82no—
163#102—44no—
164#102—42no—
165#102—42no—
166#102—44no—
167#102—82no—
168#97—42no—
169#97—82no—
170C4×C8—22no—
171C4×C8—42no—
172C4×C8#4114yes—
173#95—22no—
174#95—22no—
175#95—22no—
176C24—41no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—82no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4#12112yes—
201C22×C4#12112yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4111yescharacteristic
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8#12414yes—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—44no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—42no—
244C42—42no—
245C42—42no—
246C42AGL(1,ℤ/8)12yes—
247C42#13712yes—
248C42C2×#4112yes—
249C42H(5,𝔽2)12yes—
250C23—82no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—82no—
255C23C2×#12111yescharacteristic
256Q8—82no—
257Q8—82no—
258Q8—162no—
259Q8—162no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4—81no—
264D4—81no—
265D4—162no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#46814yes—
286C2×C4C2×#12411yescharacteristic
287C2×C4#53511yescharacteristic
288C2×C4#55012yes—
289C2×C4#55912yes—
290C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
291C2×C4C2×#13711yescharacteristic
292C2×C4#57712yes—
293C2×C4#57712yes—
294C2×C4#57812yes—
295C8—24no—
296C8—41no—
297C8—41no—
298C8—82no—
299C8—82no—
300C8—82no—
301C22—161no—
302C22—162no—
303C22—161no—
304C22—161no—
305C22—162no—
306C22#254811yescharacteristic
307C22#304612yes—
308C22C2×#46811yescharacteristic
309C22#305711yescharacteristic
310C22#306812yes—
311C4—22no—
312C4—24no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—161no—
317C4—161no—
318C4—161no—
319C4#313712yes—
320C4#322111yescharacteristic
321C4#322311yescharacteristic
322C4#339712yes—
323C4#340012yes—
324C2—321no—
325C2#1945012yes—
326C2#1954412yes—
327C2#2874511yescharacteristic
328C2#2876011yescharacteristic
329C2#2876811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #29226C2semidirect
4. #22303C2semidirect
5. #10544C2semidirect
8. #9741C2semidirect
10. #9691C2semidirect
11. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#302772
#292262
#287522
#223032
#105442
#105392
#97472
#97412
#97042
#96912
#82782