Finite Group Databasebeta

← #6510724

Subgroups — #6510724

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37481C211yesmaximal, characteristic, complemented
2#35157C211yesmaximal, characteristic
3#29119C214yesmaximal, complemented
4#27970C214yesmaximal, complemented
5#27718C214yesmaximal, complemented
6#27716C214yesmaximal, complemented
7#27702C214yesmaximal, complemented
8#27700C214yesmaximal, complemented
9#21760C212yesmaximal
10#20699C212yesmaximal, complemented
11#20528C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2853—21no—
16#2853—21no—
17#2842—22no—
18#2667—24no—
19#2629C2214yes—
20#2622—22no—
21#2608—21no—
22#2608—21no—
23#2500C2214yes—
24#2499C2214yes—
25#2485—24no—
26#2441C2214yes—
27#2441C2214yes—
28#2317C2212yes—
29#2311C2212yes—
30#2280C2212yes—
31#2280C2211yescharacteristic
32#2101C2214yescomplemented
33#2101C2214yescomplemented
34#2100C2214yescomplemented
35#2099C2214yescomplemented
36#2072C2214yescomplemented
37#2071C2214yescomplemented
38#2026C2214yescomplemented
39#2025C2214yescomplemented
40#1992C2214yescomplemented
41#1991C2214yescomplemented
42#1936C2214yescomplemented
43#1935C2214yescomplemented
44#1934C2214yescomplemented
45#1933C2214yescomplemented
46#1893C2214yes—
47#1892C2214yes—
48#1875C2212yes—
49#1875C2212yes—
50#1873C2212yes—
51#1873C2212yes—
52#1836C2214yescomplemented
53#1834C2214yescomplemented
54#1825C2214yes—
55#1825C2214yes—
56#1825C2214yes—
57#1824C2214yes—
58#1823C2214yes—
59#1788C2214yes—
60#1752C2214yes—
61C8×C22⋊C4C2214yes—
62#1750C2214yes—
63#1735C2214yes—
64#1735C2214yes—
65C22×#41C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C23×C8C2314yes—
68C2×#125C2314yes—
69#525—28no—
70C2×#123—21no—
71C2×C22⋊Q8C2312yes—
72C2×#121C2312yes—
73C2×#120—21no—
74C2×C4×D4—22no—
75C2×#117—22no—
76C2×#117—28no—
77C2×#117C2311yescharacteristic
78C2×#117C2314yes—
79C2×#117C2312yes—
80C2×#117C2312yes—
81C2×#117C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C2×#103C2314yes—
86C2×#103C2314yes—
87C2×#102C2314yes—
88C2×#102C2314yes—
89C2×C22⋊C8—28no—
90C2×C22⋊C8C2314yes—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×#97C2314yes—
94C2×C4×C8C2314yes—
95#399—24no—
96#399—24no—
97#398—24no—
98#398—24no—
99#397—24no—
100#397—24no—
101#396—24no—
102#396—24no—
103#396—24no—
104#396—24no—
105#394—24no—
106#394—24no—
107#391C2314yes—
108#390C2314yes—
109#389—24no—
110#389—24no—
111#389—24no—
112#389C2314yes—
113#389C2314yes—
114#388—24no—
115#388—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#386—22no—
121#386—22no—
122#385—22no—
123#385—22no—
124#384C2314yes—
125#384C2314yes—
126#382C2314yes—
127#381—22no—
128#381D411yescharacteristic
129#381D411yescharacteristic
130#380—24no—
131#380D414yes—
132#380C2314yes—
133#380C2314yes—
134#379—22no—
135#379D412yes—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4C2314yes—
139#340C2314yes—
140#340C2314yes—
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#340C2314yes—
146#340C2314yes—
147#340C2314yes—
148#340C2314yes—
149#340C2314yes—
150#340C2314yes—
151#336C2314yes—
152#336C2314yes—
153C25C2×D411yescharacteristic
154C2×#41—24no—
155C2×#41—44no—
156C22×Q8C2411yescharacteristic
157C22×D4—21no—
158C22×D4—22no—
159C22×D4C2411yescharacteristic
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—44no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4#4118yes—
166C23×C4C2411yescharacteristic
167C23×C4C2412yes—
168C22×C8—28no—
169C22×C8—44no—
170C22×C8—44no—
171C22×C8C2414yes—
172C22×C8C2414yes—
173#125—44no—
174#125—44no—
175#123—44no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178#121—24no—
179#121—24no—
180#120—44no—
181C4×D4—48no—
182#117—28no—
183#117—28no—
184#117—24no—
185#117—44no—
186#117—48no—
187#117—44no—
188#117—44no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4C2×D412yes—
217C2×C22⋊C4C2×D412yes—
218C2×C22⋊C4C2×D414yes—
219C2×C22⋊C4C2×D414yes—
220C2×C42—24no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42C2414yes—
226#103—44no—
227#103—44no—
228#102—44no—
229#102—44no—
230C22⋊C8—28no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233#97—44no—
234C4×C8—44no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95C2×D414yes—
244#95C2×D414yes—
245C24—21no—
246C24—42no—
247C24—44no—
248C24#12414yes—
249C24C22×D412yes—
250C24C22×D411yescharacteristic
251#41—88no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—42no—
259C2×D4—84no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—416no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—84no—
276C22×C4#12414yes—
277C22×C4#12418yes—
278C22×C4C22×D414yes—
279C22×C4C22×D414yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C2×#4114yes—
283C22×C4C2×#4114yes—
284C22×C4C2×#4114yes—
285C22×C4C2511yescharacteristic
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—88no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—48no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—24no—
326C42—24no—
327C42—48no—
328C42—42no—
329C42—82no—
330C42—82no—
331C23—24no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—44no—
338C23—48no—
339C23—44no—
340C23—41no—
341C23—42no—
342C23—88no—
343C23C2×#12414yes—
344C23C2×#12412yes—
345C23C23×D411yescharacteristic
346Q8—44no—
347D4—44no—
348D4—84no—
349D4—88no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4C2×#12414yes—
380C2×C4C2×#12412yes—
381C2×C4C22×#4112yes—
382C2×C4#58514yes—
383C8—44no—
384C8—44no—
385C22—22no—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—88no—
394C22—84no—
395C22—81no—
396C22—82no—
397C22C22×#12411yescharacteristic
398C22#348512yes—
399C22#358414yes—
400C4—22no—
401C4—24no—
402C4—44no—
403C4—44no—
404C4—84no—
405C4—84no—
406C4—81no—
407C4—84no—
408C4—82no—
409C4#349114yes—
410C2—42no—
411C2—44no—
412C2—161no—
413C2#3545511yescharacteristic
414C2#6061212yes—
415C2#6176814yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37481C2semidirect
3. #29119C2semidirect
4. #27970C2semidirect
5. #27718C2semidirect
6. #27716C2semidirect
7. #27702C2semidirect
8. #27700C2semidirect
10. #20699C2semidirect
11. #20528C2semidirect
32. #2101C22semidirect
33. #2101C22semidirect
34. #2100C22semidirect
35. #2099C22semidirect
36. #2072C22semidirect
37. #2071C22semidirect
38. #2026C22semidirect
39. #2025C22semidirect
40. #1992C22semidirect
41. #1991C22semidirect
42. #1936C22semidirect
43. #1935C22semidirect
44. #1934C22semidirect
45. #1933C22semidirect
52. #1836C22semidirect
53. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#374812
#351572
#291192
#279702
#277182
#277162
#277022
#277002
#217602
#206992
#205282