Finite Group Databasebeta

← #303588

Subgroups — #303588

Order 512.

940 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 170 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#36559C211yesmaximal, characteristic
2#33069C211yesmaximal, characteristic, complemented
3#30220C211yesmaximal, characteristic, complemented
4#29149C211yesmaximal, characteristic
5#10413C214yesmaximal, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9642C211yesmaximal, characteristic
8#8450C214yesmaximal, complemented
9#8175C211yesmaximal, characteristic, complemented
10C2×#554C2212yes—
11#2868C2211yescharacteristic
12#2859—21no—
13#2859C2211yescharacteristic
14#2566—22no—
15#2563C2212yes—
16#2538—22no—
17#2452—24no—
18D4×C4⋊C4—22no—
19C4×#127—21no—
20C4×#127C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#2043—24no—
23#2025—22no—
24#1961—21no—
25#1956C2214yes—
26#1947C2212yes—
27#1921—22no—
28#1863—22no—
29#1836—21no—
30#1831C2211yescharacteristic
31#1803C2212yes—
32#1802C2214yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#333C2214yes—
36C2×Q8⋊C8C2212yes—
37C2×D4⋊C8C2214yes—
38#1279C2212yes—
39#554—28no—
40#554—44no—
41C2×C4⋊Q8D414yes—
42C2×C4⋊Q8C2312yes—
43C2×#127D411yescharacteristic
44C2×#125—22no—
45C2×#125—21no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—22no—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—22no—
55C2×#107—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—22no—
59C2×#103—28no—
60C2×#102—24no—
61C2×#97—21no—
62C2×C4×C8C2312yes—
63#396—28no—
64#393—24no—
65#390—24no—
66#390—22no—
67#390—22no—
68#390D412yes—
69#389—24no—
70#389—24no—
71#389—22no—
72#389—22no—
73#385—28no—
74#384—21no—
75#384—21no—
76#384—22no—
77#384D411yescharacteristic
78#384C2312yes—
79#382—22no—
80#382—21no—
81#380—24no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4C2311yescharacteristic
86C43—21no—
87C43C2311yescharacteristic
88#340—24no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—21no—
94#333—44no—
95Q8⋊C8—24no—
96D4⋊C8—44no—
97C22×Q8C2×D411yescharacteristic
98C22×Q8C2×D414yes—
99C22×D4—22no—
100C22×D4C2×D412yes—
101C23×C4—24no—
102C23×C4—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—84no—
109#127—22no—
110#127Q1614yes—
111#125—48no—
112#125—82no—
113C22⋊Q8—416no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C4×Q8—22no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8#4114yes—
120C4×D4—24no—
121C4×D4—24no—
122C4×D4—48no—
123C4×D4—44no—
124C4×D4—44no—
125C4×D4—88no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—28no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42C2×D412yes—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D412yes—
155C2×C42C2×D411yescharacteristic
156C2×C42C2411yescharacteristic
157#107—44no—
158C4⋊C8—22no—
159C4⋊C8—24no—
160C4⋊C8—22no—
161#103—44no—
162#103—88no—
163#102—84no—
164#97—42no—
165C4×C8—22no—
166C4×C8—22no—
167#95—28no—
168#95—22no—
169#95—21no—
170#95—42no—
171C24—22no—
172C2×Q8—21no—
173C2×Q8—28no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—48no—
177C2×Q8#12412yes—
178C2×D4—24no—
179C2×D4—22no—
180C2×D4—24no—
181C2×D4—28no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4C2×Q1614yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—48no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—88no—
197C22×C4#12012yes—
198C22×C4#12012yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D411yescharacteristic
202C2×C8—44no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C4⋊C4—21no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—48no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4#12412yes—
230C22⋊C4—48no—
231C22⋊C4—48no—
232C22⋊C4—44no—
233C22⋊C4—88no—
234C42—24no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—42no—
239C42—44no—
240C42—44no—
241C42—42no—
242C42C2×Q1612yes—
243C42#13712yes—
244C42C2×#4112yes—
245C42H(5,𝔽2)12yes—
246C23—24no—
247C23—42no—
248C23—44no—
249C23—42no—
250C23—44no—
251C23C2×#12011yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—42no—
255Q8—88no—
256D4—28no—
257D4—416no—
258D4—44no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—48no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—168no—
283C2×C4#45114yes—
284C2×C4C2×#12411yescharacteristic
285C2×C4#53511yescharacteristic
286C2×C4#54612yes—
287C2×C4#54612yes—
288C2×C4C22×Q1611yescharacteristic
289C2×C4C2×#13711yescharacteristic
290C2×C4#57512yes—
291C2×C4#57512yes—
292C2×C4#57712yes—
293C8—82no—
294C8—82no—
295C8—82no—
296C22—44no—
297C22—48no—
298C22—82no—
299C22—82no—
300C22—84no—
301C22#243511yescharacteristic
302C22C2×#45111yescharacteristic
303C22#300512yes—
304C22#301011yescharacteristic
305C22#301912yes—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—82no—
312C4—84no—
313C4—82no—
314C4—162no—
315C4#313511yescharacteristic
316C4#314211yescharacteristic
317C4#321912yes—
318C4#329912yes—
319C4#332912yes—
320C2—84no—
321C2#1874912yes—
322C2#1882012yes—
323C2#2586211yescharacteristic
324C2#2587811yescharacteristic
325C2#2588811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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. #33069C2semidirect
3. #30220C2semidirect
5. #10413C2semidirect
6. #9702C2semidirect
8. #8450C2semidirect
9. #8175C2semidirect

Maximal subgroups

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

subgroupindex
#365592
#330692
#302202
#291492
#104132
#97022
#96422
#84502
#81752