Finite Group Databasebeta

← #237799

Subgroups — #237799

Order 512.

1090 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1090 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#35167C211yesmaximal, characteristic
2#20740C212yesmaximal
3#13850C211yesmaximal, characteristic
4#10342C214yesmaximal
5#10341C214yesmaximal
6#8022C212yesmaximal
7#7912C211yesmaximal, characteristic
8C2×#544—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2552—22no—
14#2551—22no—
15#2550—24no—
16#2316—21no—
17#2285C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#376C2211yescharacteristic
20#1823C2214yes—
21#1803—22no—
22#1803—22no—
23#1787C2214yes—
24#1763—24no—
25#1446C2212yes—
26#1445C2212yes—
27#1441—21no—
28#1441—21no—
29C4×#95—22no—
30#1294C2218yes—
31#1280C2214yes—
32C23×Q8C2311yescharacteristic
33#544—48no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—22no—
36C2×#125—24no—
37C2×C22⋊Q8—24no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8Q814yescomplemented
43C2×#117—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C42—21no—
49C22×C42—21no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#432—24no—
53#432—24no—
54C2×C4⋊C8—22no—
55C2×#103—24no—
56C2×#103—24no—
57C2×C22⋊C8C2314yes—
58C2×#97—22no—
59#398—28no—
60#395—28no—
61#391—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#384—22no—
67#384D412yes—
68#384D412yes—
69#382—22no—
70#380—24no—
71#380—22no—
72#380C2314yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4D414yes—
77#376—24no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95C2312yes—
84C2×#95C2312yes—
85#340—24no—
86#340—24no—
87C22×Q8—24no—
88C22×Q8—21no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—42no—
92C22×Q8—42no—
93C22×Q8C2×Q812yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2411yescharacteristic
102C22×C8—24no—
103C4⋊Q8—416no—
104C4⋊Q8—84no—
105#125—416no—
106C22⋊Q8—48no—
107C22⋊Q8—48no—
108C4×Q8—28no—
109C4×Q8—28no—
110C4×Q8—44no—
111C4×Q8—44no—
112#117—48no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×Q812yes—
134C2×C22⋊C4—24no—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42—22no—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—42no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—44no—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×Q812yes—
149C2×C42#4114yes—
150C2×C42#4112yes—
151C4⋊C8—44no—
152#103—48no—
153#103—48no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156#97—44no—
157#95—28no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—44no—
163#95—44no—
164#95—42no—
165#95—44no—
166#95—44no—
167C24C22×D411yescharacteristic
168C2×Q8—24no—
169C2×Q8—24no—
170C2×Q8—24no—
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—84no—
179C2×Q8—84no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4#12011yescharacteristic
216C22×C4#12012yes—
217C22×C4#12011yescharacteristic
218C22×C4C22⋊Q812yes—
219C22×C4C22⋊Q812yes—
220C22×C4#12412yes—
221C22×C4#12412yes—
222C22×C4AGL(1,ℤ/8)14yes—
223C22×C4#13714yes—
224C22×C4C22×D412yes—
225C22×C4C22×Q811yescharacteristic
226C22×C4C2×#4112yes—
227C22×C4C2×#4111yescharacteristic
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—44no—
231C4⋊C4—28no—
232C4⋊C4—28no—
233C4⋊C4—24no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—416no—
240C4⋊C4—42no—
241C4⋊C4—48no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C22⋊C4—48no—
247C42—48no—
248C42—48no—
249C42—44no—
250C42—44no—
251C42—44no—
252C42—82no—
253C42C22⋊Q814yes—
254C42C22⋊Q814yes—
255C42#12414yes—
256C42#12414yes—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23C2×#12011yescharacteristic
262C23C2×C22⋊Q811yescharacteristic
263C23C2×#12411yescharacteristic
264C23C2×AGL(1,ℤ/8)12yes—
265C23C2×#13712yes—
266Q8—28no—
267Q8—44no—
268Q8—44no—
269Q8—44no—
270Q8—168no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4#39312yes—
320C2×C4#39314yes—
321C2×C4#39312yes—
322C2×C4#45512yes—
323C2×C4#45612yes—
324C2×C4C2×C22⋊Q812yes—
325C2×C4C2×#12412yes—
326C8—84no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22C2×#39311yescharacteristic
336C22#300311yescharacteristic
337C22#300411yescharacteristic
338C22C2×#45511yescharacteristic
339C22C2×#45611yescharacteristic
340C22#302511yescharacteristic
341C22#302511yescharacteristic
342C22#307912yes—
343C22#314112yes—
344C4—24no—
345C4—22no—
346C4—22no—
347C4—22no—
348C4—44no—
349C4—44no—
350C4—44no—
351C4—84no—
352C4—84no—
353C4—84no—
354C4—82no—
355C4#202814yes—
356C2—22no—
357C2—22no—
358C2#1817911yescharacteristic
359C2#1818111yescharacteristic
360C2#1818711yescharacteristic
361C2#1818911yescharacteristic
362C2#2345211yescharacteristic
363C2C2×#202811yescharacteristic
364C2#2346711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
42. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#207402
#138502
#103422
#103412
#80222
#79122