Finite Group Databasebeta

← #237791

Subgroups — #237791

Order 512.

1296 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 206 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#35052C211yesmaximal, characteristic
2#20700C212yesmaximal
3C2×#1445C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10342C214yesmaximal, complemented
6#8020C212yesmaximal
7#7909C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2212yes—
10#2848—22no—
11#2659C2212yes—
12#2646C2212yes—
13#2569—22no—
14#2549—22no—
15#2438—24no—
16#2329—22no—
17#2280C2212yes—
18C22×#95—22no—
19C22×#95C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1824—22no—
22#1824—22no—
23#1823C2218yes—
24#1736—24no—
25#1445—22no—
26#1445C2212yes—
27#1445C2212yes—
28#1294C2218yes—
29#1280C2212yes—
30#1280C2212yes—
31C24×C4—21no—
32C24×C4—21no—
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35C22×M16—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×#117D414yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4Q814yescomplemented
46C22×C4⋊C4C2312yes—
47#407—24no—
48#407—24no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8C2314yes—
51#400—24no—
52#398—24no—
53#398—24no—
54#395—24no—
55#391—24no—
56#389—24no—
57#387—24no—
58#387—24no—
59#385—24no—
60#385—24no—
61#385—24no—
62#381—22no—
63#380—22no—
64#380—22no—
65#380—22no—
66#380D412yes—
67#380D412yes—
68#380C2314yes—
69#379—22no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—24no—
75C2×#95—21no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95—48no—
79C2×#95D412yes—
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82C2×#95C2311yescharacteristic
83#340—28no—
84#340—28no—
85C25C2×D411yescharacteristic
86C22×Q8—22no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—44no—
95C23×C4—42no—
96C23×C4—44no—
97C23×C4—44no—
98C23×C4—42no—
99C23×C4—44no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—44no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×Q812yes—
113C23×C4#4114yes—
114C23×C4#4112yes—
115C23×C4C2411yescharacteristic
116C2×M16—48no—
117C2×M16—44no—
118C22×C8—24no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C22⋊Q8—84no—
122C22⋊Q8—84no—
123#117—48no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×Q814yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C42—24no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—44no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—24no—
160#95—24no—
161#95—48no—
162#95—42no—
163#95—48no—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—24no—
169C24#12011yescharacteristic
170C24#12011yescharacteristic
171C24C22⋊Q812yes—
172C24#12412yes—
173C24C22×D411yescharacteristic
174C2×Q8—44no—
175C2×Q8—82no—
176C2×Q8—82no—
177C22×C4—28no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—48no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—416no—
200C22×C4—416no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—416no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—88no—
218C22×C4—88no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4#12012yes—
226C22×C4C22⋊Q814yes—
227C22×C4C22⋊Q814yes—
228C22×C4C22⋊Q812yes—
229C22×C4#12414yes—
230C22×C4#12414yes—
231C22×C4#12412yes—
232C22×C4AGL(1,ℤ/8)14yes—
233C22×C4C22×D412yes—
234C22×C4C22×Q811yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4C2×#4111yescharacteristic
237M16—88no—
238C2×C8—48no—
239C2×C8—44no—
240C4⋊C4—28no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—48no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—48no—
256C42—84no—
257C23—28no—
258C23—28no—
259C23—28no—
260C23—28no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23#39312yes—
277C23#39312yes—
278C23#45512yes—
279C23#45612yes—
280C23#45814yes—
281C23C2×#12011yescharacteristic
282C23C2×C22⋊Q811yescharacteristic
283C23C2×#12411yescharacteristic
284C23C2×AGL(1,ℤ/8)12yes—
285C23#57512yes—
286Q8—164no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4—88no—
318C2×C4—88no—
319C2×C4—816no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—88no—
323C2×C4—88no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4#39314yes—
330C2×C4C2×C22⋊Q812yes—
331C2×C4C2×#12412yes—
332C8—84no—
333C22—28no—
334C22—216no—
335C22—28no—
336C22—22no—
337C22—22no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—24no—
347C22—24no—
348C22—22no—
349C22—22no—
350C22—24no—
351C22—24no—
352C22#202814yes—
353C22#203014yes—
354C22C2×#39311yescharacteristic
355C22#300612yes—
356C22C2×#45511yescharacteristic
357C22C2×#45611yescharacteristic
358C22C2×#45812yes—
359C22#307712yes—
360C22#313811yescharacteristic
361C22#314011yescharacteristic
362C4—22no—
363C4—22no—
364C4—24no—
365C4—44no—
366C4—84no—
367C4—84no—
368C4—82no—
369C4—84no—
370C4—84no—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—24no—
375C2—24no—
376C2#1816912yes—
377C2#1817112yes—
378C2#2345411yescharacteristic
379C2C2×#202811yescharacteristic
380C2C2×#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
45. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350522
#207002
C2×#14452
#103462
#103422
#80202
#79092