Finite Group Databasebeta

← #6510723

Subgroups — #6510723

Order 512.

1491 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic, complemented
2#35157C211yesmaximal, characteristic
3#29120C214yesmaximal, complemented
4#27971C214yesmaximal, complemented
5#27714C214yesmaximal, complemented
6#27712C214yesmaximal, complemented
7#27698C214yesmaximal, complemented
8#27696C214yesmaximal, complemented
9#21760C212yesmaximal
10#20699C212yesmaximal, complemented
11#20528C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2853—22no—
16#2842—21no—
17#2840—21no—
18#2665—24no—
19#2628C2214yes—
20#2622—21no—
21#2622—21no—
22#2607—22no—
23#2500C2214yes—
24#2499C2214yes—
25#2485—24no—
26#2439C2214yes—
27#2439C2214yes—
28#2317C2212yes—
29#2311C2212yes—
30#2280C2212yes—
31#2280C2211yescharacteristic
32#2101C2214yescomplemented
33#2101C2214yescomplemented
34#2100C2214yescomplemented
35#2099C2214yescomplemented
36#2074C2214yescomplemented
37#2073C2214yescomplemented
38#2024C2214yescomplemented
39#2023C2214yescomplemented
40#1993C2214yescomplemented
41#1990C2214yescomplemented
42#1936C2214yescomplemented
43#1935C2214yescomplemented
44#1934C2214yescomplemented
45#1933C2214yescomplemented
46#1893C2214yes—
47#1892C2214yes—
48#1875C2212yes—
49#1875C2212yes—
50#1873C2212yes—
51#1873C2212yes—
52#1836C2214yescomplemented
53#1835C2214yescomplemented
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×C4⋊Q8C2314yes—
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#400—24no—
96#399—24no—
97#397—24no—
98#397—24no—
99#396—24no—
100#396—24no—
101#395—24no—
102#394—24no—
103#394—24no—
104#393—24no—
105#393—24no—
106#393—24no—
107#391—24no—
108#391—24no—
109#391C2314yes—
110#390C2314yes—
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#382C2314yes—
126#382C2314yes—
127#381—22no—
128#381D412yes—
129#380—24no—
130#380D414yes—
131#380C2314yes—
132#380C2314yes—
133#379—22no—
134#379D411yescharacteristic
135#379D411yescharacteristic
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×Q8—24no—
157C22×Q8C2411yescharacteristic
158C22×D4—21no—
159C22×D4—22no—
160C22×D4C2411yescharacteristic
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—44no—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4#4118yes—
167C23×C4C2411yescharacteristic
168C23×C4C2412yes—
169C22×C8—28no—
170C22×C8—44no—
171C22×C8—44no—
172C22×C8C2414yes—
173C22×C8C2414yes—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#123—44no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179#121—24no—
180#121—24no—
181#120—44no—
182C4×D4—48no—
183#117—28no—
184#117—28no—
185#117—24no—
186#117—44no—
187#117—48no—
188#117—44no—
189#117—44no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2412yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2×D412yes—
216C2×C22⋊C4C2×D414yes—
217C2×C22⋊C4C2×D414yes—
218C2×C42—24no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—44no—
223C2×C42C2414yes—
224#103—44no—
225#103—44no—
226#102—44no—
227#102—44no—
228C22⋊C8—28no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231#97—44no—
232C4×C8—44no—
233#95—24no—
234#95—24no—
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×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—84no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—416no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—84no—
279C22×C4#12414yes—
280C22×C4#12418yes—
281C22×C4C22×D414yes—
282C22×C4C22×D414yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C2×#4114yes—
286C22×C4C2×#4114yes—
287C22×C4C2×#4114yes—
288C22×C4C2511yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—88no—
296C4⋊C4—28no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
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—
347Q8—168no—
348D4—44no—
349D4—84no—
350D4—88no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—48no—
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—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—88no—
380C2×C4C2×#12414yes—
381C2×C4C2×#12412yes—
382C2×C4C22×#4112yes—
383C2×C4#58514yes—
384C8—44no—
385C8—44no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—24no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—88no—
395C22—84no—
396C22—81no—
397C22—82no—
398C22C22×#12411yescharacteristic
399C22#348512yes—
400C22#358414yes—
401C4—22no—
402C4—24no—
403C4—44no—
404C4—44no—
405C4—84no—
406C4—84no—
407C4—81no—
408C4—84no—
409C4—82no—
410C4#349114yes—
411C2—42no—
412C2—44no—
413C2—161no—
414C2#3545511yescharacteristic
415C2#6061212yes—
416C2#6176914yes—
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. #37480C2semidirect
3. #29120C2semidirect
4. #27971C2semidirect
5. #27714C2semidirect
6. #27712C2semidirect
7. #27698C2semidirect
8. #27696C2semidirect
10. #20699C2semidirect
11. #20528C2semidirect
32. #2101C22semidirect
33. #2101C22semidirect
34. #2100C22semidirect
35. #2099C22semidirect
36. #2074C22semidirect
37. #2073C22semidirect
38. #2024C22semidirect
39. #2023C22semidirect
40. #1993C22semidirect
41. #1990C22semidirect
42. #1936C22semidirect
43. #1935C22semidirect
44. #1934C22semidirect
45. #1933C22semidirect
52. #1836C22semidirect
53. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#351572
#291202
#279712
#277142
#277122
#276982
#276962
#217602
#206992
#205282