Finite Group Databasebeta

← #7487625

Subgroups — #7487625

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46153C211yesmaximal, characteristic, complemented
2#39513C211yesmaximal, characteristic
3#32105C214yesmaximal
4#32083C214yesmaximal
5#31572C214yesmaximal, complemented
6#30220C212yesmaximal, complemented
7#29574C212yesmaximal
8#29038C214yesmaximal, complemented
9#27765C214yesmaximal
10#26775C214yesmaximal, complemented
11#23266C211yesmaximal, characteristic, complemented
12#2868C2211yescharacteristic
13#2859C2212yes—
14#2845—21no—
15#2793—24no—
16#2726—22no—
17#2726—24no—
18#2629—21no—
19#2552C2214yes—
20#2547C2212yes—
21#2545C2214yes—
22#2507C2214yes—
23#2507C2214yes—
24#2497—22no—
25#2446C2214yes—
26#2382—21no—
27#2365C2212yes—
28#2314—21no—
29C4×#125C2211yescharacteristic
30#2293C2211yescharacteristic
31#2100C2212yes—
32#2096C2214yes—
33#2094C2212yes—
34#2084C2212yes—
35#2072C2214yes—
36#2071C2212yes—
37#2066C2214yes—
38#2066C2214yes—
39#2043C2214yes—
40#2043C2214yes—
41#2043C2214yes—
42#2043C2214yes—
43#2042C2214yes—
44#2042C2214yes—
45#1983C2214yes—
46#1959C2214yes—
47#1956C2214yes—
48#1956C2214yes—
49#1941C2214yes—
50#1935C2214yes—
51#1935C2214yes—
52#1934C2214yes—
53#1927C2214yes—
54#1927C2214yes—
55#1920C2214yes—
56#1917C2214yes—
57#1916C2214yes—
58#1865C2214yes—
59#1865C2214yes—
60#1864C2214yes—
61#1803C2214yes—
62#1771C2214yes—
63#1763C2214yes—
64#1744C2212yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2314yes—
67C2×#125—24no—
68C2×#125—22no—
69C2×#125D411yescharacteristic
70C2×#124C2314yes—
71C2×C4×Q8C2314yes—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#106C2314yes—
75C2×#106C2314yes—
76C2×C4⋊C8C2314yes—
77C2×#103C2314yes—
78C2×#103C2314yes—
79C2×#103C2314yes—
80C2×#103C2314yes—
81C2×#103C2314yes—
82C2×#103C2314yes—
83C2×#103C2314yes—
84C2×#103C2314yes—
85C2×#102C2314yes—
86C2×#102C2314yes—
87C2×#97C2314yes—
88C2×C4×C8C2314yes—
89#400—24no—
90#398—24no—
91#398—24no—
92#398—24no—
93#398—24no—
94#398—24no—
95#398—24no—
96#398—24no—
97#397—24no—
98#396—24no—
99#396—24no—
100#391C2314yes—
101#391C2314yes—
102#390C2312yes—
103#389—22no—
104#389—24no—
105#389—22no—
106#389—22no—
107#389—24no—
108#389—24no—
109#389D412yes—
110#389C2314yes—
111#389C2314yes—
112#389C2312yes—
113#389C2314yes—
114#389C2314yes—
115#388—22no—
116#388—24no—
117#388—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#387—24no—
122#387—24no—
123#387D414yes—
124#384—21no—
125#384—21no—
126#384—22no—
127#384C2314yes—
128#384C2312yes—
129#384C2312yes—
130#382—22no—
131#382D411yescharacteristic
132#380—22no—
133#380—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2312yes—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—21no—
140C43C2311yescharacteristic
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#336C2312yes—
146#336C2314yes—
147#336C2312yes—
148#336C2314yes—
149#336C2314yes—
150C22×Q8C2414yes—
151C22×D4C2411yescharacteristic
152C23×C4—21no—
153C22×C8C2414yes—
154C22×C8C2414yes—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159#125—24no—
160#125—84no—
161#125—88no—
162#124—44no—
163#124—44no—
164C4×Q8—44no—
165C4×Q8—44no—
166C4×D4—42no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2411yescharacteristic
189C2×C4⋊C4C2412yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204C2×C42C2414yes—
205C2×C42C2412yes—
206#106—44no—
207#106—44no—
208C4⋊C8—44no—
209#103—44no—
210#103—44no—
211#103—44no—
212#103—44no—
213#103—44no—
214#103—44no—
215#103—44no—
216#103—44no—
217#102—44no—
218#102—44no—
219#97—44no—
220C4×C8—44no—
221#95—22no—
222#95—24no—
223#95—21no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95C2×D414yes—
233#95C2×D414yes—
234#95C2×D414yes—
235C24—21no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×D4—22no—
243C2×D4—24no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4C22×D414yes—
253C22×C4C22×D414yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C22×D412yes—
258C22×C4C2511yescharacteristic
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C4⋊C4—24no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—48no—
280C4⋊C4—82no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—41no—
302C42—42no—
303C42—84no—
304C42—84no—
305C42#13714yes—
306C23—41no—
307C23—41no—
308C23—42no—
309C23—41no—
310C23—42no—
311C23C23×D411yescharacteristic
312Q8—44no—
313Q8—44no—
314Q8—44no—
315Q8—44no—
316D4—44no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—44no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4C2×#13712yes—
348C2×C4C2×#13714yes—
349C2×C4#57714yes—
350C2×C4#58512yes—
351C2×C4#58514yes—
352C8—44no—
353C8—44no—
354C22—81no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22#348711yescharacteristic
360C22#350012yes—
361C22C22×#13711yescharacteristic
362C22C2×#57712yes—
363C22#358611yescharacteristic
364C4—24no—
365C4—24no—
366C4—22no—
367C4—44no—
368C4—44no—
369C4—41no—
370C4—84no—
371C4—82no—
372C4—84no—
373C4—81no—
374C4—84no—
375C4#356714yes—
376C2—161no—
377C2#5197911yescharacteristic
378C2#6188911yescharacteristic
379C2#6193511yescharacteristic
380C2#6254212yes—
381C2#6259712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #46153C2semidirect
5. #31572C2semidirect
6. #30220C2semidirect
8. #29038C2semidirect
10. #26775C2semidirect
11. #23266C2semidirect

Maximal subgroups

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

subgroupindex
#461532
#395132
#321052
#320832
#315722
#302202
#295742
#290382
#277652
#267752
#232662