Finite Group Databasebeta

← #303597

Subgroups — #303597

Order 512.

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

normal subgroups only

This group has 824 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#36560C211yesmaximal, characteristic
2#33069C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic
4#29150C211yesmaximal, characteristic
5#10422C214yesmaximal
6#9703C211yesmaximal, characteristic
7#9641C211yesmaximal, characteristic
8#8448C214yesmaximal
9#8175C211yesmaximal, characteristic
10C2×#557C2212yes—
11#2868C2211yescharacteristic
12#2859—21no—
13#2859C2211yescharacteristic
14#2568C2212yes—
15#2565—22no—
16#2546—22no—
17#2506—24no—
18#2365—22no—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#2100—24no—
23#2026—22no—
24#1990C2214yes—
25#1960—21no—
26#1947C2212yes—
27#1921—22no—
28#1863—22no—
29#1836—21no—
30#1831C2211yescharacteristic
31#1803C2212yes—
32#1803C2214yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#332C2214yes—
36C2×Q8⋊C8C2212yes—
37C2×Q8⋊C8C2214yes—
38#1279C2212yes—
39#557—28no—
40#557—44no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8C2312yes—
45C2×#125—22no—
46C2×#125—24no—
47C2×#125—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8D412yes—
51C2×C4×Q8C2312yes—
52C2×C4×Q8C2311yescharacteristic
53C2×#107—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—24no—
58C2×#97—21no—
59C2×C4×C8C2312yes—
60#400—24no—
61#398—28no—
62#391—22no—
63#391—24no—
64#391D412yes—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—22no—
69#389—22no—
70#389—22no—
71#387—28no—
72#384—21no—
73#384—21no—
74#384D411yescharacteristic
75#384D414yes—
76#384C2312yes—
77#382—22no—
78#382—21no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4D412yes—
84C4×C4⋊C4C2311yescharacteristic
85C43—21no—
86C43C2311yescharacteristic
87#340—24no—
88#340—22no—
89#340—28no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—21no—
94#332—44no—
95Q8⋊C8—24no—
96Q8⋊C8—44no—
97C22×Q8—22no—
98C22×Q8C2×D411yescharacteristic
99C22×Q8C2×D412yes—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—44no—
107C4⋊Q8—44no—
108C4⋊Q8—44no—
109C4⋊Q8—44no—
110C4⋊Q8Q1614yes—
111#125—48no—
112#125—416no—
113#125—82no—
114C4×Q8—22no—
115C4×Q8—24no—
116C4×Q8—24no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8—48no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×Q8#4114yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
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—42no—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C42—21no—
143C2×C42—24no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D412yes—
151C2×C42C2×D411yescharacteristic
152C2×C42C2411yescharacteristic
153#107—44no—
154C4⋊C8—22no—
155C4⋊C8—24no—
156C4⋊C8—22no—
157#103—44no—
158#103—84no—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162#95—28no—
163#95—28no—
164#95—22no—
165#95—24no—
166#95—21no—
167#95—42no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—24no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—24no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8#12412yes—
179C2×Q8C2×Q1614yes—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—42no—
187C22×C4#12012yes—
188C22×C4#12012yes—
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C2×C8—44no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C4⋊C4—21no—
201C4⋊C4—24no—
202C4⋊C4—28no—
203C4⋊C4—24no—
204C4⋊C4—28no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C4⋊C4—48no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—84no—
221C4⋊C4#12412yes—
222C42—24no—
223C42—22no—
224C42—22no—
225C42—22no—
226C42—42no—
227C42—44no—
228C42—44no—
229C42—48no—
230C42—42no—
231C42—42no—
232C42—42no—
233C42C2×Q1612yes—
234C42#13712yes—
235C42C2×#4112yes—
236C42H(5,𝔽2)12yes—
237C23C2×#12011yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—28no—
241Q8—44no—
242Q8—42no—
243Q8—82no—
244Q8—82no—
245Q8—84no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—48no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4#45114yes—
270C2×C4C2×#12411yescharacteristic
271C2×C4#53511yescharacteristic
272C2×C4#54612yes—
273C2×C4#54612yes—
274C2×C4C22×Q1611yescharacteristic
275C2×C4C2×#13711yescharacteristic
276C2×C4#57512yes—
277C2×C4#57512yes—
278C2×C4#57712yes—
279C8—82no—
280C8—82no—
281C8—82no—
282C22#243511yescharacteristic
283C22C2×#45111yescharacteristic
284C22#300512yes—
285C22#301011yescharacteristic
286C22#302012yes—
287C4—22no—
288C4—24no—
289C4—42no—
290C4—42no—
291C4—44no—
292C4—42no—
293C4—84no—
294C4—82no—
295C4—82no—
296C4—162no—
297C4#313511yescharacteristic
298C4#314211yescharacteristic
299C4#321912yes—
300C4#329912yes—
301C4#332912yes—
302C2#1877912yes—
303C2#1883012yes—
304C2#2586211yescharacteristic
305C2#2587911yescharacteristic
306C2#2588811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#365602
#330692
#302302
#291502
#104222
#97032
#96412
#84482
#81752