Finite Group Databasebeta

← #237793

Subgroups — #237793

Order 512.

1170 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 202 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#35002C211yesmaximal, characteristic
2#20702C212yesmaximal
3#13853C211yesmaximal, characteristic
4#10343C214yesmaximal, complemented
5#10342C214yesmaximal, complemented
6#8019C212yesmaximal
7#7911C211yesmaximal, characteristic
8C2×#543—21no—
9C22×#117C2211yescharacteristic
10C2×#407C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2549—24no—
14#2440—24no—
15#2315—21no—
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1825—24no—
20#1824C2214yes—
21#1823C2214yes—
22#1736—24no—
23#1445C2212yes—
24#1445C2212yes—
25#1440—21no—
26#1440—21no—
27C4×#95—22no—
28#1294C2218yes—
29#1280C2214yes—
30C24×C4C2311yescharacteristic
31C22×M16—22no—
32#543—48no—
33C2×C4⋊Q8—22no—
34C2×C22⋊Q8—28no—
35C2×C22⋊Q8—24no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—24no—
40C2×#117Q814yescomplemented
41C22×C4⋊C4D414yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4—22no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42D412yes—
52#407—24no—
53#407—24no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8C2314yes—
56#398—28no—
57#395—28no—
58#391—24no—
59#389—24no—
60#389—24no—
61#385—28no—
62#384—22no—
63#380—24no—
64#380—22no—
65#380—22no—
66#380D414yes—
67#380C2314yes—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—24no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95C2312yes—
77C2×#95C2312yes—
78#340—28no—
79#340—28no—
80C25C2×D411yescharacteristic
81C22×Q8—24no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—44no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×Q812yes—
93C23×C4#4114yes—
94C23×C4#4112yes—
95C23×C4C2411yescharacteristic
96C2×M16—48no—
97C2×M16—44no—
98C22×C8—24no—
99C4⋊Q8—84no—
100C22⋊Q8—432no—
101C22⋊Q8—48no—
102C22⋊Q8—48no—
103#117—28no—
104#117—28no—
105#117—44no—
106#117—44no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4C2×D418yes—
120C2×C4⋊C4C2×Q812yes—
121C2×C4⋊C4C2×Q812yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—48no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—22no—
135C2×C42—24no—
136C2×C42—22no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—42no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—42no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—44no—
146#95—28no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—44no—
152#95—44no—
153#95—44no—
154#95—44no—
155C24—22no—
156C24—22no—
157C24—42no—
158C24—42no—
159C24#12012yes—
160C24C22⋊Q812yes—
161C24#12412yes—
162C24C22×D411yescharacteristic
163C2×Q8—48no—
164C2×Q8—84no—
165C2×Q8—84no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—44no—
177C22×C4—48no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—48no—
200C22×C4—84no—
201C22×C4—84no—
202C22×C4#12011yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4C22⋊Q814yes—
205C22×C4C22⋊Q812yes—
206C22×C4C22⋊Q814yes—
207C22×C4#12414yes—
208C22×C4#12412yes—
209C22×C4#12414yes—
210C22×C4AGL(1,ℤ/8)14yes—
211C22×C4#13714yes—
212C22×C4C22×D412yes—
213C22×C4C22×Q811yescharacteristic
214C22×C4C2×#4112yes—
215C22×C4C2×#4111yescharacteristic
216M16—88no—
217C2×C8—48no—
218C2×C8—44no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—416no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—48no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C22⋊C4—28no—
230C22⋊C4—28no—
231C22⋊C4—48no—
232C22⋊C4—48no—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C42—44no—
236C42—44no—
237C42—44no—
238C42—82no—
239C42—84no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23#39314yes—
257C23C2×#12011yescharacteristic
258C23C2×C22⋊Q811yescharacteristic
259C23C2×#12411yescharacteristic
260C23C2×AGL(1,ℤ/8)12yes—
261C23C2×#13712yes—
262Q8—168no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—168no—
305C2×C4#39312yes—
306C2×C4#39312yes—
307C2×C4#45814yes—
308C2×C4C2×C22⋊Q812yes—
309C2×C4C2×#12412yes—
310C8—84no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—24no—
320C22—24no—
321C22—24no—
322C22—44no—
323C22—44no—
324C22—84no—
325C22—84no—
326C22C2×#39311yescharacteristic
327C22#300311yescharacteristic
328C22#300411yescharacteristic
329C22#301812yes—
330C22C2×#45812yes—
331C22#307912yes—
332C22#314112yes—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—44no—
337C4—44no—
338C4—84no—
339C4—82no—
340C4—82no—
341C4—88no—
342C4#203014yes—
343C2—22no—
344C2—22no—
345C2—44no—
346C2#1817312yes—
347C2#1817412yes—
348C2#2345211yescharacteristic
349C2#2346111yescharacteristic
350C2C2×#203011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
4. #10343C2semidirect
5. #10342C2semidirect
40. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#207022
#138532
#103432
#103422
#80192
#79112