Finite Group Databasebeta

← #7487623

Subgroups — #7487623

Order 512.

1337 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46153C211yesmaximal, characteristic, complemented
2#39770C211yesmaximal, characteristic
3#32109C214yesmaximal, complemented
4#32079C214yesmaximal, complemented
5#31570C214yesmaximal, complemented
6#30218C212yesmaximal
7#29573C212yesmaximal, complemented
8#29032C214yesmaximal, complemented
9#27763C214yesmaximal, complemented
10#26771C214yesmaximal, complemented
11#23266C211yesmaximal, characteristic, complemented
12#2867C2211yescharacteristic
13#2859C2212yes—
14#2845—21no—
15#2726—22no—
16#2726—24no—
17#2709—24no—
18#2641—21no—
19#2552C2214yes—
20#2548C2212yes—
21#2544C2214yes—
22#2507C2214yes—
23#2507C2214yes—
24#2445—22no—
25#2445C2214yes—
26#2382—21no—
27#2366C2212yes—
28#2328—21no—
29C4×#125C2211yescharacteristic
30#2295C2211yescharacteristic
31#2100C2212yes—
32#2095C2212yes—
33#2093C2214yes—
34#2083C2212yes—
35#2072C2212yes—
36#2071C2214yes—
37#2067C2214yes—
38#2067C2214yes—
39#2042C2214yes—
40#2042C2214yes—
41#2041C2214yes—
42#2041C2214yes—
43#2041C2214yes—
44#2041C2214yes—
45#1983C2214yes—
46#1959C2214yes—
47#1957C2214yes—
48#1957C2214yes—
49#1942C2214yes—
50#1936C2214yes—
51#1936C2214yes—
52#1933C2214yes—
53#1925C2214yes—
54#1925C2214yes—
55#1921C2214yes—
56#1917C2214yes—
57#1916C2214yes—
58#1863C2214yes—
59#1862C2214yes—
60#1862C2214yes—
61#1802C2214yes—
62#1771C2214yes—
63#1762C2214yes—
64#1744C2212yes—
65C2×#127—21no—
66C2×#127—21no—
67C2×#127—22no—
68C2×#127C2312yes—
69C2×#127C2314yes—
70C2×#125—24no—
71C2×#125—22no—
72C2×#125D411yescharacteristic
73C2×#124C2314yes—
74C2×#123—24no—
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—22no—
77C2×C4×D4C2314yes—
78C2×#107C2314yes—
79C2×#107C2314yes—
80C2×C4⋊C8C2314yes—
81C2×#103C2314yes—
82C2×#103C2314yes—
83C2×#102C2314yes—
84C2×#102C2314yes—
85C2×#102C2314yes—
86C2×#102C2314yes—
87C2×#102C2314yes—
88C2×#102C2314yes—
89C2×#102C2314yes—
90C2×#102C2314yes—
91C2×#97C2314yes—
92C2×C4×C8C2314yes—
93#398—24no—
94#398—24no—
95#398—24no—
96#398—24no—
97#398—24no—
98#398—24no—
99#394—24no—
100#394—24no—
101#394—24no—
102#394—24no—
103#394—24no—
104#391C2312yes—
105#390—22no—
106#390—22no—
107#390—22no—
108#390C2314yes—
109#390C2314yes—
110#389—24no—
111#389—22no—
112#389—22no—
113#389—24no—
114#389—24no—
115#389D412yes—
116#389C2314yes—
117#389C2314yes—
118#389C2312yes—
119#389C2314yes—
120#389C2314yes—
121#387—24no—
122#387—24no—
123#387—22no—
124#387D414yes—
125#386—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#384C2314yes—
130#384C2312yes—
131#384C2312yes—
132#382—22no—
133#382D411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2312yes—
138C4×C4⋊C4C2314yes—
139#376—21no—
140C43C2311yescharacteristic
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#336C2314yes—
146#336C2312yes—
147#336C2312yes—
148#336C2314yes—
149#336C2314yes—
150C22×Q8C2411yescharacteristic
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4C2414yes—
156C23×C4—24no—
157C22×C8C2414yes—
158C22×C8C2414yes—
159#127—42no—
160#127—42no—
161#127—44no—
162#127—44no—
163#127—44no—
164#127—84no—
165#127—82no—
166#125—24no—
167#125—84no—
168#125—88no—
169#124—44no—
170#124—44no—
171#123—88no—
172C4×Q8—42no—
173C4×D4—44no—
174C4×D4—44no—
175C4×D4—84no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2412yes—
193C2×C4⋊C4C2411yescharacteristic
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—24no—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2414yes—
210C2×C42C2411yescharacteristic
211C2×C42C2412yes—
212#107—44no—
213#107—44no—
214C4⋊C8—44no—
215#103—44no—
216#103—44no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—44no—
221#102—44no—
222#102—44no—
223#102—44no—
224#102—44no—
225#97—44no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95C2×D414yes—
237#95C2×D414yes—
238#95C2×D414yes—
239C24—24no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—48no—
253C2×D4—48no—
254C2×D4—84no—
255C2×D4—84no—
256C2×D4—84no—
257C2×D4—84no—
258C2×D4—82no—
259C2×D4—82no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C22×D414yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D414yes—
271C22×C4C22×D412yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D412yes—
274C22×C4C2511yescharacteristic
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C4⋊C4—24no—
282C4⋊C4—28no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—82no—
296C4⋊C4—84no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C22⋊C4—84no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C22⋊C4—84no—
309C22⋊C4—84no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—22no—
315C42—41no—
316C42—42no—
317C42—42no—
318C42—84no—
319C42#13714yes—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23C23×D411yescharacteristic
328Q8—44no—
329D4—44no—
330D4—44no—
331D4—44no—
332D4—44no—
333D4—88no—
334D4—168no—
335D4—168no—
336D4—164no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—168no—
365C2×C4C2×#13714yes—
366C2×C4C2×#13712yes—
367C2×C4#57614yes—
368C2×C4#58512yes—
369C2×C4#58514yes—
370C8—44no—
371C8—44no—
372C22—84no—
373C22—84no—
374C22—84no—
375C22—84no—
376C22—84no—
377C22—84no—
378C22—84no—
379C22#348711yescharacteristic
380C22#350012yes—
381C22C22×#13711yescharacteristic
382C22C2×#57612yes—
383C22#358511yescharacteristic
384C4—24no—
385C4—24no—
386C4—22no—
387C4—44no—
388C4—44no—
389C4—41no—
390C4—81no—
391C4—84no—
392C4—82no—
393C4—81no—
394C4—84no—
395C4#356714yes—
396C2—164no—
397C2#5197911yescharacteristic
398C2#6189211yescharacteristic
399C2#6193211yescharacteristic
400C2#6253712yes—
401C2#6259812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
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
3. #32109C2semidirect
4. #32079C2semidirect
5. #31570C2semidirect
7. #29573C2semidirect
8. #29032C2semidirect
9. #27763C2semidirect
10. #26771C2semidirect
11. #23266C2semidirect

Maximal subgroups

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

subgroupindex
#461532
#397702
#321092
#320792
#315702
#302182
#295732
#290322
#277632
#267712
#232662