Finite Group Databasebeta

← #232372

Subgroups — #232372

Order 512.

1332 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1332 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C2×#3229C212yesmaximal
2#34642C211yesmaximal, characteristic
3C2×#1518C212yesmaximal
4#14274C211yesmaximal, characteristic
5#9973C218yesmaximal
6#7906C211yesmaximal, characteristic
7C2×#533C2212yes—
8C22×C4⋊Q8C2211yescharacteristic
9#3229—28no—
10#3229—28no—
11C2×#498—22no—
12C2×#487—22no—
13C2×#483C2218yes—
14C2×#475—24no—
15C2×#431C2212yes—
16#2660—22no—
17#2660—22no—
18#2659—22no—
19#2659C2214yes—
20#2399—24no—
21C2×#391—22no—
22#2285—22no—
23C2×#382C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#1518—28no—
26C2×#328C2218yes—
27#1445—22no—
28#1445C2214yes—
29#1280C2214yes—
30C23×Q8—21no—
31#533—24no—
32#533—24no—
33#533—24no—
34#533—24no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8D412yes—
41C2×C4⋊Q8D414yes—
42C2×#125—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8C2314yes—
45C2×C4×Q8—22no—
46C2×C4×Q8D414yes—
47C2×#117—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C42D412yes—
52C22×C42C2311yescharacteristic
53#498—216no—
54#487—216no—
55#483—216no—
56#483—216no—
57#475—232no—
58#431—24no—
59#431D418yes—
60C2×#106—28no—
61C2×C4⋊C8—22no—
62C2×#103—28no—
63C2×C22⋊C8C2314yes—
64C2×C4×C8—22no—
65#398—28no—
66#398—28no—
67#395—28no—
68#395—28no—
69#391—48no—
70#384—22no—
71#384—22no—
72#382—24no—
73#382—24no—
74#380—28no—
75C4×C4⋊C4—28no—
76#376—22no—
77#376—22no—
78C2×#95—21no—
79C2×#95—28no—
80C2×#95D412yes—
81C2×#95C2312yes—
82C2×#95C2312yes—
83#328—216no—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—42no—
89C22×Q8—44no—
90C22×Q8—44no—
91C22×Q8C2×D414yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4C2×D414yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2411yescharacteristic
98C22×C8—24no—
99C4⋊Q8—28no—
100C4⋊Q8—28no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107#125—44no—
108#125—44no—
109C22⋊Q8—24no—
110C22⋊Q8—24no—
111C22⋊Q8—24no—
112C22⋊Q8—24no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—44no—
118C4×Q8—44no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C22⋊C4—24no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—24no—
142C2×C42—24no—
143C2×C42SD1618yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D414yes—
148C2×C42#4112yes—
149#106—232no—
150C4⋊C8—28no—
151#103—232no—
152C22⋊C8—24no—
153C22⋊C8C2×D418yes—
154C4×C8—28no—
155#95—216no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—42no—
160#95—416no—
161C24C22×D411yescharacteristic
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—48no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—416no—
169C2×Q8—84no—
170C2×Q8—84no—
171C2×Q8—84no—
172C2×Q8—84no—
173C2×Q8—84no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4#12012yes—
192C22×C4#12011yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4#12712yes—
197C22×C4#12712yes—
198C22×C4C2×SD1618yes—
199C22×C4C2×SD1614yes—
200C22×C4#13714yes—
201C22×C4C22×D412yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D412yes—
204C22×C4C2×#4111yescharacteristic
205C2×C8—28no—
206C2×C8—48no—
207C4⋊C4—28no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C4⋊C4—44no—
218C4⋊C4—88no—
219C4⋊C4—88no—
220C4⋊C4—84no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C42—28no—
224C42—28no—
225C42—48no—
226C42#12014yes—
227C42#12014yes—
228C42#12114yes—
229C42#12114yes—
230C23—22no—
231C23—22no—
232C23C2×#12011yescharacteristic
233C23C2×#12111yescharacteristic
234C23C2×#12711yescharacteristic
235C23C22×SD1612yes—
236C23C2×#13712yes—
237Q8—48no—
238Q8—84no—
239Q8—88no—
240Q8—88no—
241Q8—88no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—28no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—84no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4—88no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4#39214yes—
268C2×C4#39212yes—
269C2×C4#39212yes—
270C2×C4#44814yes—
271C2×C4#44814yes—
272C2×C4#46118yes—
273C2×C4#49214yes—
274C2×C4#49714yes—
275C2×C4C2×#12012yes—
276C2×C4C2×#12112yes—
277C8—48no—
278C22—21no—
279C22—21no—
280C22—24no—
281C22—24no—
282C22—21no—
283C22—21no—
284C22C2×#39211yescharacteristic
285C22C2×#44812yes—
286C22#302212yes—
287C22#302812yes—
288C22C2×#46112yes—
289C22C2×#49211yescharacteristic
290C22C2×#49711yescharacteristic
291C4—24no—
292C4—22no—
293C4—22no—
294C4—48no—
295C4—84no—
296C4—84no—
297C4—84no—
298C4—162no—
299C4#201314yes—
300C4#201314yes—
301C2—21no—
302C2—21no—
303C2—22no—
304C2#1831014yes—
305C2#2340811yescharacteristic
306C2C2×#201311yescharacteristic
307C2#2344611yescharacteristic
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
C2×#32292
#346422
C2×#15182
#142742
#99732
#79062