Finite Group Databasebeta

← #237776

Subgroups — #237776

Order 512.

1132 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1132 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
1C2×#2646C211yesmaximal, characteristic
2#21765C212yesmaximal
3#14159C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10345C214yesmaximal, complemented
6#8019C212yesmaximal
7#7909C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2212yes—
10#2646—24no—
11#2646—22no—
12#2646C2214yes—
13C2×#400—22no—
14C2×#398—22no—
15C2×#385—22no—
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1894—24no—
20#1872—22no—
21#1872—22no—
22#1823C2214yes—
23#1787C2214yes—
24#1446C2212yes—
25#1445C2212yes—
26#1436—22no—
27C22⋊#95—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×Q8C2311yescharacteristic
31C24×C4D412yes—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8Q814yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C42—21no—
49C2×#103—24no—
50C2×#103—24no—
51#407—24no—
52#407—24no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8C2314yes—
55#400—28no—
56#400—28no—
57#398—28no—
58#398—28no—
59#385—28no—
60#385—28no—
61#380—22no—
62#380—22no—
63#380D414yes—
64#380C2314yes—
65C4×C22⋊C4—24no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—24no—
72C2×#95—24no—
73C2×#95D412yes—
74C2×#95C2312yes—
75C2×#95C2312yes—
76#340—24no—
77#340—24no—
78C25C2×D411yescharacteristic
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—82no—
83C22×Q8C2×Q812yes—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4—82no—
94C23×C4—82no—
95C23×C4C2×D414yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2×Q812yes—
100C23×C4#4114yes—
101C23×C4#4112yes—
102C23×C4C2411yescharacteristic
103C2×M16—44no—
104C2×M16—82no—
105C2×M16—82no—
106C22×C8—24no—
107C22⋊Q8—28no—
108C22⋊Q8—28no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4—84no—
124C2×C4⋊C4—84no—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×Q812yes—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—48no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C42—42no—
139C2×C42—82no—
140#103—84no—
141#103—84no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144C22⋊C8—44no—
145#95—28no—
146#95—28no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—48no—
153#95—84no—
154C24—24no—
155C24—22no—
156C24—22no—
157C24—42no—
158C24#12011yescharacteristic
159C24#12011yescharacteristic
160C24C22⋊Q812yes—
161C24#12412yes—
162C24C22×D411yescharacteristic
163C2×Q8—24no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—42no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—84no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—82no—
207C22×C4—84no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—816no—
212C22×C4#12012yes—
213C22×C4C22⋊Q814yes—
214C22×C4C22⋊Q814yes—
215C22×C4C22⋊Q812yes—
216C22×C4#12414yes—
217C22×C4#12414yes—
218C22×C4#12412yes—
219C22×C4C22×D412yes—
220C22×C4C22×Q811yescharacteristic
221C22×C4C2×#4112yes—
222C22×C4C2×#4111yescharacteristic
223M16—88no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C4⋊C4—24no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—48no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—88no—
235C4⋊C4—88no—
236C22⋊C4—48no—
237C22⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C42—84no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—24no—
248C23—24no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—24no—
254C23—24no—
255C23—42no—
256C23—42no—
257C23—48no—
258C23—48no—
259C23#39312yes—
260C23#39312yes—
261C23#45512yes—
262C23#45612yes—
263C23C2×#12011yescharacteristic
264C23C2×C22⋊Q811yescharacteristic
265C23C2×#12411yescharacteristic
266C23#57514yes—
267Q8—48no—
268Q8—84no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—88no—
301C2×C4—88no—
302C2×C4—88no—
303C2×C4—82no—
304C2×C4—88no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4#39314yes—
311C2×C4C2×C22⋊Q812yes—
312C2×C4C2×#12412yes—
313C8—84no—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—24no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—48no—
328C22—48no—
329C22—42no—
330C22—42no—
331C22#202814yes—
332C22C2×#39311yescharacteristic
333C22#300512yes—
334C22C2×#45511yescharacteristic
335C22C2×#45611yescharacteristic
336C22#302612yes—
337C22#308012yes—
338C22#314212yes—
339C4—22no—
340C4—22no—
341C4—24no—
342C4—84no—
343C4—84no—
344C4—84no—
345C4—84no—
346C4—82no—
347C4—82no—
348C2—22no—
349C2—22no—
350C2—24no—
351C2—42no—
352C2#1832112yes—
353C2#1832912yes—
354C2#2345311yescharacteristic
355C2C2×#202811yescharacteristic
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. #10346C2semidirect
5. #10345C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
#217652
#141592
#103462
#103452
#80192
#79092