Finite Group Databasebeta

← #232413

Subgroups — #232413

Order 512.

1384 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 266 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#35173C211yesmaximal, characteristic
3C2×#1503C212yesmaximal
4#14155C211yesmaximal, characteristic
5#9973C218yesmaximal
6#7908C211yesmaximal, 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#2857—22no—
17#2659C2214yes—
18#2584—24no—
19#2567—22no—
20#2546—22no—
21#2316—22no—
22C2×#391—22no—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1503—28no—
26C2×#328C2218yes—
27#1445C2214yes—
28C4⋊#95—22no—
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—22no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8D412yes—
41C2×C4⋊Q8D412yes—
42C2×C4⋊Q8D414yes—
43C2×#125—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2314yes—
46C2×C4×Q8—22no—
47C2×C4×Q8D414yes—
48C2×#117—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2312yes—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#498—216no—
57#487—216no—
58#483—216no—
59#483—216no—
60#475—232no—
61#431—24no—
62#431D418yes—
63C2×#106—28no—
64C2×C4⋊C8—22no—
65C2×#103—28no—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—22no—
68#398—28no—
69#395—28no—
70#391—24no—
71#391—48no—
72#389—28no—
73#389—28no—
74#389—28no—
75#384—22no—
76#380—28no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—24no—
80C2×#95—28no—
81C2×#95C2312yes—
82C2×#95C2312yes—
83#328—216no—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—44no—
89C22×Q8—44no—
90C22×Q8C2×D414yes—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D414yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2411yescharacteristic
97C22×C8—24no—
98C4⋊Q8—28no—
99C4⋊Q8—28no—
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106C4⋊Q8—84no—
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—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4SD1618yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×D414yes—
143C2×C22⋊C4—24no—
144C2×C42—24no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—44no—
148C2×C42—42no—
149C2×C42—44no—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D414yes—
154C2×C42#4112yes—
155#106—232no—
156C4⋊C8—28no—
157#103—232no—
158C22⋊C8—24no—
159C22⋊C8C2×D418yes—
160C4×C8—28no—
161#95—216no—
162#95—24no—
163#95—24no—
164#95—416no—
165C24C22×D411yescharacteristic
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—48no—
169C2×Q8—44no—
170C2×Q8—48no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—416no—
174C2×Q8—88no—
175C2×Q8—84no—
176C2×Q8—84no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4#12011yescharacteristic
197C22×C4#12012yes—
198C22×C4#12011yescharacteristic
199C22×C4#12112yes—
200C22×C4#12112yes—
201C22×C4#12712yes—
202C22×C4#12712yes—
203C22×C4C2×SD1618yes—
204C22×C4C2×SD1614yes—
205C22×C4#13714yes—
206C22×C4C22×D412yes—
207C22×C4C22×D411yescharacteristic
208C22×C4C22×D412yes—
209C22×C4C2×#4111yescharacteristic
210C2×C8—28no—
211C2×C8—48no—
212C4⋊C4—216no—
213C4⋊C4—28no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—48no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—44no—
226C4⋊C4—88no—
227C4⋊C4—88no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C42—48no—
231C42—44no—
232C42—82no—
233C42—84no—
234C42#12014yes—
235C42#12014yes—
236C42#12114yes—
237C42#12114yes—
238C23—21no—
239C23—21no—
240C23—22no—
241C23C2×#12011yescharacteristic
242C23C2×#12111yescharacteristic
243C23C2×#12711yescharacteristic
244C23C22×SD1612yes—
245C23C2×#13712yes—
246Q8—48no—
247Q8—84no—
248Q8—88no—
249Q8—816no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—28no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4#39212yes—
279C2×C4#39212yes—
280C2×C4#39214yes—
281C2×C4#44818yes—
282C2×C4#45512yes—
283C2×C4#45612yes—
284C2×C4#46118yes—
285C2×C4#49214yes—
286C2×C4#49714yes—
287C2×C4C2×#12012yes—
288C2×C4C2×#12112yes—
289C8—48no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22C2×#39211yescharacteristic
299C22C2×#44812yes—
300C22C2×#45511yescharacteristic
301C22C2×#45611yescharacteristic
302C22#302812yes—
303C22C2×#46112yes—
304C22C2×#49211yescharacteristic
305C22C2×#49711yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—48no—
310C4—84no—
311C4—84no—
312C4—84no—
313C4—84no—
314C4#200714yes—
315C4#201318yes—
316C2—21no—
317C2—21no—
318C2—22no—
319C2#1839612yes—
320C2#1841012yes—
321C2C2×#200711yescharacteristic
322C2C2×#201311yescharacteristic
323C2#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
#351732
C2×#15032
#141552
#99732
#79082