Finite Group Databasebeta

← #327872

Subgroups — #327872

Order 512.

882 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 160 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#35167C211yesmaximal, characteristic
2#27614C212yesmaximal
3#15111C211yesmaximal, characteristic
4#10569C214yesmaximal
5#10343C214yesmaximal
6#8717C212yesmaximal
7#8701C211yesmaximal, characteristic
8C2×#544—21no—
9#2857—22no—
10#2659C2214yes—
11#2552—22no—
12#2551—22no—
13#2550—24no—
14#2316—21no—
15C2×#389C2211yescharacteristic
16#2285C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18#2100—22no—
19#2083—22no—
20#1921—24no—
21#1852C2214yes—
22#1824C2214yes—
23#1817C2212yes—
24#1445—21no—
25#1445C2212yes—
26#1442—21no—
27#1442C2212yes—
28#1436—22no—
29#1434C2211yescharacteristic
30#1294C2218yes—
31#1280C2214yes—
32#544—48no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—22no—
35C2×#125—24no—
36C2×C22⋊Q8—24no—
37C2×#117—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4C2311yescharacteristic
42C22×C4⋊C4C2311yescharacteristic
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45C2×#107—24no—
46C2×C22⋊C8C2314yes—
47#398—28no—
48#395—28no—
49#391—24no—
50#389—24no—
51#389—24no—
52#389—24no—
53#389—24no—
54#389—22no—
55#389—22no—
56#384—22no—
57#384D412yes—
58#384D412yes—
59#382—22no—
60#380—24no—
61#380—22no—
62#380C2314yes—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4D414yes—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95C2312yes—
75C2×#95C2312yes—
76#340—24no—
77#340—24no—
78#340—24no—
79#336—22no—
80#336—22no—
81C22×Q8—24no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2411yescharacteristic
90C22×C8—24no—
91C4⋊Q8—416no—
92C4⋊Q8—84no—
93#125—416no—
94C22⋊Q8—48no—
95C22⋊Q8—48no—
96#117—48no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4Q1614yes—
117C2×C4⋊C4C2×D414yes—
118C2×C4⋊C4C2×D414yes—
119C2×C22⋊C4—24no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—82no—
125C2×C42C2×D412yes—
126C2×C42C2×D412yes—
127#107—48no—
128C22⋊C8—28no—
129#95—28no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—42no—
135#95—44no—
136#95—42no—
137#95—44no—
138#95—42no—
139#95—82no—
140#95—82no—
141C24C22×D411yescharacteristic
142C2×Q8—48no—
143C2×Q8—84no—
144C2×Q8—84no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4#12011yescharacteristic
180C22×C4#12012yes—
181C22×C4#12011yescharacteristic
182C22×C4C2×Q1612yes—
183C22×C4C2×Q1614yes—
184C22×C4AGL(1,ℤ/8)14yes—
185C22×C4#13712yes—
186C22×C4C22×D412yes—
187C22×C42−1+414yes—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C4⋊C4—28no—
192C4⋊C4—28no—
193C4⋊C4—24no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—48no—
200C4⋊C4—416no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C4⋊C4—82no—
208C22⋊C4—48no—
209C42—24no—
210C42—24no—
211C42—44no—
212C42—48no—
213C42—82no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23C2×#12011yescharacteristic
219C23#53612yes—
220C23C22×Q1611yescharacteristic
221C23C2×AGL(1,ℤ/8)12yes—
222C23C2×#13711yescharacteristic
223Q8—168no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#45114yes—
271C2×C4#45412yes—
272C2×C4#53614yes—
273C8—84no—
274C22—22no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22#261411yescharacteristic
283C22C2×#45111yescharacteristic
284C22#300311yescharacteristic
285C22C2×#45411yescharacteristic
286C22#301911yescharacteristic
287C22#302511yescharacteristic
288C22#302811yescharacteristic
289C22#323012yes—
290C22#323912yes—
291C4—24no—
292C4—22no—
293C4—24no—
294C4—44no—
295C4—44no—
296C4—41no—
297C4—84no—
298C4—82no—
299C4—84no—
300C4—162no—
301C4—161no—
302C2—22no—
303C2—22no—
304C2#1818611yescharacteristic
305C2#1821311yescharacteristic
306C2#1836311yescharacteristic
307C2#1841111yescharacteristic
308C2#2964211yescharacteristic
309C2#2965111yescharacteristic
310C2#2965911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351672
#276142
#151112
#105692
#103432
#87172
#87012