Finite Group Databasebeta

← #237820

Subgroups — #237820

Order 512.

1148 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 186 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#35020C211yesmaximal, characteristic, complemented
2#21764C212yesmaximal
3#14290C211yesmaximal, characteristic
4#10347C214yesmaximal, complemented
5#10344C214yesmaximal, complemented
6#8015C212yesmaximal, complemented
7#7911C211yesmaximal, characteristic, complemented
8C22×#121C2211yescharacteristic
9C2×#407C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2569—24no—
13#2442—24no—
14#2315—22no—
15C2×#381C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1895—24no—
19#1874—24no—
20#1824C2214yes—
21#1786C2214yes—
22#1445C2212yes—
23#1438—22no—
24#1438C2212yes—
25#1437—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×D4—21no—
29C23×D4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C22×M16—22no—
32C2×#125—22no—
33C2×#121—22no—
34C2×#121—22no—
35C2×#121—22no—
36C2×#121—24no—
37C2×#121Q814yes—
38C22×C4⋊C4D414yes—
39C22×C4⋊C4C2311yescharacteristic
40C22×C22⋊C4—22no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4D412yes—
45C22×C42D412yes—
46C2×#102—28no—
47#407—24no—
48#407—24no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8C2314yes—
51#400—28no—
52#398—28no—
53#389—24no—
54#387—28no—
55#385—28no—
56#385—28no—
57#382—22no—
58#381—24no—
59#380—28no—
60#380—22no—
61#380—22no—
62#380D414yes—
63#380C2314yes—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—24no—
69C2×#95C2312yes—
70C2×#95C2312yes—
71#340—28no—
72C25—21no—
73C25C2×D411yescharacteristic
74C22×D4—24no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—21no—
78C22×D4—42no—
79C22×D4—42no—
80C22×D4—82no—
81C22×D4—84no—
82C22×D4C2×Q812yes—
83C23×C4—21no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—82no—
87C23×C4C2×D414yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×Q812yes—
92C23×C4#4114yes—
93C23×C4#4112yes—
94C23×C4C2411yescharacteristic
95C2×M16—44no—
96C2×M16—84no—
97C22×C8—24no—
98#125—84no—
99#121—28no—
100#121—28no—
101#121—44no—
102#121—44no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4C2×D418yes—
111C2×C4⋊C4C2×Q812yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—48no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—48no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—84no—
125C2×C22⋊C4—84no—
126C2×C22⋊C4—82no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—82no—
130#102—88no—
131C22⋊C8—24no—
132C22⋊C8—24no—
133C22⋊C8—44no—
134#95—28no—
135#95—28no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—42no—
141#95—84no—
142C24—24no—
143C24—22no—
144C24—42no—
145C24—42no—
146C24—44no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24—44no—
151C24#12012yes—
152C24C22⋊Q812yes—
153C24#12412yes—
154C24C22×D411yescharacteristic
155C2×D4—24no—
156C2×D4—48no—
157C2×D4—48no—
158C2×D4—42no—
159C2×D4—82no—
160C2×D4—84no—
161C2×D4—82no—
162C2×D4—88no—
163C2×D4—88no—
164C2×D4—88no—
165C22×C4—24no—
166C22×C4—28no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—48no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—84no—
188C22×C4—84no—
189C22×C4—84no—
190C22×C4#12011yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4C22⋊Q814yes—
193C22×C4C22⋊Q814yes—
194C22×C4C22⋊Q812yes—
195C22×C4#12414yes—
196C22×C4#12414yes—
197C22×C4#12412yes—
198C22×C4C22×D412yes—
199C22×C4C22×Q811yescharacteristic
200C22×C4C2×#4112yes—
201C22×C4C2×#4111yescharacteristic
202M16—88no—
203C2×C8—48no—
204C2×C8—44no—
205C4⋊C4—24no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C4⋊C4—42no—
209C4⋊C4—88no—
210C4⋊C4—88no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—48no—
214C22⋊C4—48no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—816no—
218C22⋊C4—84no—
219C42—44no—
220C42—82no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—42no—
237C23—84no—
238C23—84no—
239C23—84no—
240C23—82no—
241C23—82no—
242C23—84no—
243C23—88no—
244C23—82no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23#39314yes—
249C23C2×#12011yescharacteristic
250C23C2×C22⋊Q811yescharacteristic
251C23C2×#12411yescharacteristic
252C23#57514yes—
253D4—48no—
254D4—84no—
255D4—816no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—44no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—84no—
280C2×C4—82no—
281C2×C4—84no—
282C2×C4—82no—
283C2×C4—84no—
284C2×C4—82no—
285C2×C4—168no—
286C2×C4#39312yes—
287C2×C4#39312yes—
288C2×C4C2×C22⋊Q812yes—
289C2×C4C2×#12412yes—
290C8—84no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—24no—
300C22—44no—
301C22—44no—
302C22—84no—
303C22—84no—
304C22—88no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—164no—
309C22—164no—
310C22C2×#39311yescharacteristic
311C22#300512yes—
312C22#302112yes—
313C22#302612yes—
314C22#308012yes—
315C22#314212yes—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—82no—
321C4—84no—
322C4—88no—
323C4—162no—
324C2—22no—
325C2—22no—
326C2—44no—
327C2—162no—
328C2#1835414yes—
329C2#2345311yescharacteristic
330C2#2346311yescharacteristic
331C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #35020C2semidirect
4. #10347C2semidirect
5. #10344C2semidirect
6. #8015C2semidirect
7. #7911C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#217642
#142902
#103472
#103442
#80152
#79112