Finite Group Databasebeta

← #237821

Subgroups — #237821

Order 512.

1138 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1138 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#35002C211yesmaximal, characteristic
2#21765C212yesmaximal
3#14292C211yesmaximal, characteristic
4#10342C214yesmaximal, complemented
5#10341C214yesmaximal, complemented
6#8015C212yesmaximal
7#7911C211yesmaximal, characteristic
8C2×#543—21no—
9C22×C22⋊Q8C2211yescharacteristic
10C2×#407C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2549—24no—
14#2440—24no—
15#2315—21no—
16C2×#381C2211yescharacteristic
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1894—24no—
20#1872—24no—
21#1823C2214yes—
22#1787C2214yes—
23#1446C2212yes—
24#1445C2212yes—
25#1440—21no—
26#1440—21no—
27#1436—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×Q8C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C22×M16—22no—
33#543—48no—
34C2×C4⋊Q8—22no—
35C2×C22⋊Q8—28no—
36C2×C22⋊Q8—24no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8Q814yescomplemented
42C22×C4⋊C4—21no—
43C22×C4⋊C4D414yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4D411yescharacteristic
49C22×C22⋊C4D411yescharacteristic
50C22×C42D412yes—
51C2×#103—28no—
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#381—24no—
64#380—24no—
65#380—22no—
66#380—22no—
67#380D414yes—
68#380C2314yes—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—22no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95C2312yes—
78C2×#95C2312yes—
79#340—28no—
80C25C2×D411yescharacteristic
81C22×Q8—24no—
82C22×Q8—21no—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—82no—
86C22×Q8C2×Q812yes—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—82no—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×Q812yes—
98C23×C4#4114yes—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C2×M16—44no—
102C2×M16—84no—
103C22×C8—24no—
104C4⋊Q8—84no—
105C22⋊Q8—28no—
106C22⋊Q8—28no—
107C22⋊Q8—432no—
108C22⋊Q8—48no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C22⋊Q8—48no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—84no—
125C2×C4⋊C4C2×D418yes—
126C2×C4⋊C4C2×Q812yes—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—48no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—82no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—82no—
142#103—88no—
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—42no—
152#95—44no—
153#95—44no—
154#95—82no—
155#95—82no—
156C24—22no—
157C24—22no—
158C24—42no—
159C24—42no—
160C24#12012yes—
161C24C22⋊Q812yes—
162C24#12412yes—
163C24C22×D411yescharacteristic
164C2×Q8—24no—
165C2×Q8—48no—
166C2×Q8—48no—
167C2×Q8—42no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—84no—
171C2×Q8—84no—
172C2×Q8—84no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—28no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—44no—
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—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—48no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4#12011yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4C22⋊Q814yes—
212C22×C4C22⋊Q814yes—
213C22×C4C22⋊Q812yes—
214C22×C4#12414yes—
215C22×C4#12414yes—
216C22×C4#12412yes—
217C22×C4AGL(1,ℤ/8)14yes—
218C22×C4#13714yes—
219C22×C4C22×D412yes—
220C22×C4C22×Q811yescharacteristic
221C22×C4C2×#4112yes—
222C22×C4C2×#4111yescharacteristic
223M16—88no—
224C2×C8—48no—
225C2×C8—44no—
226C4⋊C4—24no—
227C4⋊C4—48no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—416no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C22⋊C4—28no—
237C22⋊C4—28no—
238C22⋊C4—48no—
239C22⋊C4—48no—
240C22⋊C4—48no—
241C22⋊C4—84no—
242C42—44no—
243C42—82no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23#39314yes—
261C23C2×#12011yescharacteristic
262C23C2×C22⋊Q811yescharacteristic
263C23C2×#12411yescharacteristic
264C23C2×AGL(1,ℤ/8)12yes—
265C23C2×#13712yes—
266Q8—48no—
267Q8—84no—
268Q8—168no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—168no—
312C2×C4#39312yes—
313C2×C4#39312yes—
314C2×C4C2×C22⋊Q812yes—
315C2×C4C2×#12412yes—
316C8—84no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—44no—
329C22—44no—
330C22—84no—
331C22—84no—
332C22C2×#39311yescharacteristic
333C22#300311yescharacteristic
334C22#300411yescharacteristic
335C22#302212yes—
336C22#302612yes—
337C22#307912yes—
338C22#314112yes—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—44no—
344C4—44no—
345C4—82no—
346C4—82no—
347C4—88no—
348C4—162no—
349C2—22no—
350C2—22no—
351C2—44no—
352C2#1835112yes—
353C2#1835312yes—
354C2#2345211yescharacteristic
355C2#2346411yescharacteristic
356C2#2346811yescharacteristic
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. #10342C2semidirect
5. #10341C2semidirect
41. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#217652
#142922
#103422
#103412
#80152
#79112