Finite Group Databasebeta

← #237749

Subgroups — #237749

Order 512.

1124 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 198 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#34578C211yesmaximal, characteristic
2#20700C212yesmaximal
3#13904C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10342C214yesmaximal, complemented
6#8016C212yesmaximal
7#7911C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2212yes—
10#2667—22no—
11#2665—22no—
12#2659C2212yes—
13#2646C2212yes—
14#2626—22no—
15#2396—24no—
16C4×C22⋊Q8—22no—
17#2280C2212yes—
18C2×#381—22no—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21#1824—22no—
22#1824—22no—
23#1823C2218yes—
24#1736—24no—
25#1445C2212yes—
26#1445C2212yes—
27C4⋊#95—22no—
28#1294C2218yes—
29#1280C2212yes—
30#1280C2212yes—
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C2×C22⋊Q8—22no—
35C2×C4×Q8—22no—
36C2×#117D414yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4Q814yescomplemented
44C22×C4⋊C4C2312yes—
45C22×C22⋊C4—21no—
46C22×C22⋊C4D412yes—
47C22×C42D412yes—
48#407—24no—
49#407—24no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8C2314yes—
52#400—24no—
53#399—24no—
54#398—24no—
55#398—24no—
56#396—24no—
57#396—24no—
58#395—24no—
59#393—24no—
60#391—22no—
61#389—22no—
62#387—24no—
63#385—24no—
64#381—22no—
65#381—48no—
66#380—22no—
67#380—24no—
68#380D412yes—
69#380D412yes—
70#380C2314yes—
71#379—22no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—22no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—22no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82C2×#95C2311yescharacteristic
83#340—28no—
84#340—28no—
85C25C2×D411yescharacteristic
86C22×Q8—22no—
87C23×C4—24no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—44no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×Q812yes—
101C23×C4#4114yes—
102C23×C4#4112yes—
103C23×C4C2411yescharacteristic
104C2×M16—48no—
105C2×M16—44no—
106C22×C8—24no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C4×Q8—84no—
110#117—48no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×Q814yes—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—42no—
142C2×C42—24no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—44no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—44no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—24no—
157#95—24no—
158#95—48no—
159#95—48no—
160C24—22no—
161C24—22no—
162C24—42no—
163C24—41no—
164C24—41no—
165C24#12012yes—
166C24C22⋊Q812yes—
167C24#12412yes—
168C24C22×D411yescharacteristic
169C2×Q8—44no—
170C2×Q8—82no—
171C2×Q8—82no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—44no—
183C22×C4—48no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—416no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—88no—
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4C22⋊Q814yes—
208C22×C4C22⋊Q812yes—
209C22×C4C22⋊Q814yes—
210C22×C4#12414yes—
211C22×C4#12412yes—
212C22×C4#12414yes—
213C22×C4AGL(1,ℤ/8)14yes—
214C22×C4C22×D412yes—
215C22×C4C22×Q811yescharacteristic
216C22×C4C2×#4112yes—
217C22×C4C2×#4111yescharacteristic
218M16—88no—
219C2×C8—48no—
220C2×C8—44no—
221C4⋊C4—28no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—48no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—84no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—48no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—84no—
240C22⋊C4—84no—
241C42—44no—
242C42—48no—
243C42—82no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—22no—
252C23—22no—
253C23—24no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—44no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23#39314yes—
263C23C2×#12011yescharacteristic
264C23C2×C22⋊Q811yescharacteristic
265C23C2×#12411yescharacteristic
266C23C2×AGL(1,ℤ/8)12yes—
267C23#57512yes—
268Q8—88no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—816no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—82no—
307C2×C4#39312yes—
308C2×C4#39312yes—
309C2×C4#45814yes—
310C2×C4C2×C22⋊Q812yes—
311C2×C4C2×#12412yes—
312C8—84no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—44no—
325C22—48no—
326C22—44no—
327C22—44no—
328C22—44no—
329C22C2×#39311yescharacteristic
330C22#300612yes—
331C22#302211yescharacteristic
332C22#302211yescharacteristic
333C22C2×#45812yes—
334C22#307712yes—
335C22#313811yescharacteristic
336C22#314011yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—44no—
341C4—84no—
342C4—84no—
343C4—84no—
344C4—82no—
345C4—162no—
346C4#203014yes—
347C2—21no—
348C2—22no—
349C2—21no—
350C2—44no—
351C2#1814012yes—
352C2#1815012yes—
353C2#2345411yescharacteristic
354C2#2346411yescharacteristic
355C2C2×#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. #10346C2semidirect
5. #10342C2semidirect
43. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#345782
#207002
#139042
#103462
#103422
#80162
#79112