Finite Group Databasebeta

← #237798

Subgroups — #237798

Order 512.

1166 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 194 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#35181C211yesmaximal, characteristic, complemented
2#20738C212yesmaximal
3#13847C211yesmaximal, characteristic
4#10347C214yesmaximal, complemented
5#10344C214yesmaximal
6#8022C212yesmaximal, complemented
7#7912C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#432C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317—22no—
16#2285C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4—22no—
19C2×#376C2211yescharacteristic
20#1824C2214yes—
21#1802—22no—
22#1802—22no—
23#1786C2214yes—
24#1762—24no—
25#1445C2212yes—
26#1439—22no—
27#1438C2212yes—
28#1294C2218yes—
29#1280C2214yes—
30C23×D4C2311yescharacteristic
31C24×C4—21no—
32C2×#125—22no—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—24no—
37C2×C4×D4Q814yes—
38C2×#117—22no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—24no—
43C22×C42—21no—
44C22×C42D412yes—
45C22×C42C2311yescharacteristic
46#432—24no—
47#432—24no—
48C2×C4⋊C8—22no—
49C2×#102—24no—
50C2×#102—24no—
51C2×C22⋊C8C2314yes—
52C2×#97—22no—
53#400—28no—
54#398—28no—
55#389—24no—
56#389—24no—
57#389—24no—
58#387—28no—
59#385—28no—
60#384—24no—
61#384D412yes—
62#384D412yes—
63#382—24no—
64#382—22no—
65#382—22no—
66#380—22no—
67#380C2314yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C4×C22⋊C4—48no—
73#376—24no—
74C2×#95—21no—
75C2×#95—24no—
76C2×#95C2312yes—
77C2×#95C2312yes—
78#340—24no—
79#340—24no—
80C25—21no—
81C22×D4—22no—
82C22×D4—22no—
83C22×D4—21no—
84C22×D4—44no—
85C22×D4C2×Q812yes—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—24no—
89C23×C4—44no—
90C23×C4—44no—
91C23×C4—41no—
92C23×C4—41no—
93C23×C4—42no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D414yes—
97C23×C4C2411yescharacteristic
98C22×C8—24no—
99#125—84no—
100C4×D4—28no—
101C4×D4—28no—
102C4×D4—44no—
103C4×D4—44no—
104#117—48no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4C2×D414yes—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×Q812yes—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—48no—
123C2×C22⋊C4—48no—
124C2×C22⋊C4—48no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—44no—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C2×C42C2×Q812yes—
137C2×C42#4114yes—
138C2×C42#4112yes—
139C4⋊C8—44no—
140#102—48no—
141#102—48no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144#97—44no—
145#95—28no—
146#95—28no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—48no—
152#95—42no—
153C24—24no—
154C24—22no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—41no—
159C24—41no—
160C24—44no—
161C24C22×D411yescharacteristic
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—48no—
169C2×D4—44no—
170C2×D4—42no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—416no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—88no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4#12011yescharacteristic
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4#12412yes—
206C22×C4#12412yes—
207C22×C4C22×D412yes—
208C22×C4C22×Q811yescharacteristic
209C22×C4C2×#4112yes—
210C22×C4C2×#4111yescharacteristic
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—44no—
214C4⋊C4—24no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—88no—
225C4⋊C4—88no—
226C22⋊C4—432no—
227C22⋊C4—48no—
228C22⋊C4—48no—
229C42—48no—
230C42—48no—
231C42—44no—
232C42—82no—
233C42C22⋊Q814yes—
234C42C22⋊Q814yes—
235C42#12414yes—
236C42#12414yes—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—24no—
241C23—48no—
242C23—44no—
243C23—44no—
244C23—42no—
245C23—48no—
246C23—44no—
247C23—42no—
248C23—44no—
249C23—82no—
250C23—84no—
251C23—84no—
252C23—82no—
253C23—82no—
254C23—81no—
255C23—81no—
256C23C2×#12011yescharacteristic
257C23C2×C22⋊Q811yescharacteristic
258C23C2×#12411yescharacteristic
259C23#57514yes—
260D4—28no—
261D4—44no—
262D4—44no—
263D4—44no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—88no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—88no—
299C2×C4—84no—
300C2×C4#39314yes—
301C2×C4#39312yes—
302C2×C4#39312yes—
303C2×C4#45312yes—
304C2×C4#45412yes—
305C2×C4C2×C22⋊Q812yes—
306C2×C4C2×#12412yes—
307C8—84no—
308C22—24no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—48no—
315C22—44no—
316C22—88no—
317C22—84no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22C2×#39311yescharacteristic
323C22#300512yes—
324C22C2×#45311yescharacteristic
325C22C2×#45411yescharacteristic
326C22#302512yes—
327C22#308012yes—
328C22#314212yes—
329C4—24no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—84no—
334C4—84no—
335C4—84no—
336C4—84no—
337C4—82no—
338C4#202714yes—
339C2—22no—
340C2—22no—
341C2—84no—
342C2#1818212yes—
343C2#1819012yes—
344C2#2345311yescharacteristic
345C2C2×#202711yescharacteristic
346C2#2346711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
4. #10347C2semidirect
6. #8022C2semidirect
7. #7912C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#207382
#138472
#103472
#103442
#80222
#79122