Finite Group Databasebeta

← #6473513

Subgroups — #6473513

Order 512.

1588 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C212yesmaximal
2#33043C212yesmaximal, complemented
3#32937C212yesmaximal
4#30293C211yesmaximal, characteristic, complemented
5#30242C211yesmaximal, characteristic, complemented
6#29733C212yesmaximal
7#28925C214yesmaximal, complemented
8#28871C214yesmaximal, complemented
9#28088C214yesmaximal, complemented
10#26988C214yesmaximal, complemented
11#22183C212yesmaximal, complemented
12#21527C212yesmaximal, complemented
13C2×#1753C211yesmaximal, characteristic
14C23×M16C2211yescharacteristic
15C2×#487C2212yescomplemented
16C2×#470C2212yes—
17C2×#470C2212yes—
18C2×#467C2212yes—
19C2×#467C2212yes—
20C2×#441C2212yescomplemented
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2848C2214yes—
27#2646—22no—
28#2626—22no—
29#2589—24no—
30#2587—24no—
31#2497C2214yes—
32#2492C2214yes—
33#2429C2214yes—
34#2423C2214yes—
35#2329—22no—
36#2312—22no—
37C4×C22⋊Q8C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#2095C2214yescomplemented
40#2084C2214yescomplemented
41#2075C2214yes—
42#2058C2214yescomplemented
43#2035C2214yes—
44#2035C2214yes—
45#2016C2214yescomplemented
46#2001C2214yes—
47#1990C2212yescomplemented
48#1971C2212yescomplemented
49#1956C2212yescomplemented
50#1943C2212yescomplemented
51#1935C2214yescomplemented
52#1933C2214yescomplemented
53#1933C2214yescomplemented
54#1927C2214yescomplemented
55#1918C2214yescomplemented
56#1894C2214yes—
57#1894C2214yes—
58#1892C2214yes—
59#1892C2214yes—
60#1878C2214yescomplemented
61#1865C2214yescomplemented
62#1834C2212yescomplemented
63#1809C2212yes—
64#1809C2212yes—
65#1807C2212yes—
66#1807C2212yes—
67#1776C2212yescomplemented
68#1763C2214yescomplemented
69#1753C2214yes—
70#1753C2214yes—
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C24×C4C2311yescharacteristic
74C22×Q16C2312yes—
75C22×Q16C2312yes—
76C22×M16—24no—
77C22×M16—42no—
78C22×M16—42no—
79C22×M16C2312yes—
80C22×M16C2312yes—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C2314yes—
89C2×C22⋊Q8C2314yes—
90C2×C22⋊Q8C2314yes—
91C2×C4×Q8C2314yes—
92C22×C22⋊C4D412yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#487—44no—
96#470—44no—
97#470—44no—
98#467—28no—
99#467—28no—
100#441—44no—
101C2×#107C2312yes—
102C2×#107C2312yes—
103C2×#106C2312yes—
104C2×#106C2312yes—
105C2×#106C2312yes—
106C2×#106C2312yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×C22⋊C8—24no—
114C2×C22⋊C8C2314yes—
115C2×#97—42no—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×C4×C8—42no—
119C2×C4×C8C2312yes—
120C2×C4×C8C2312yes—
121#400—24no—
122#400—24no—
123#399—24no—
124#398—24no—
125#398—24no—
126#396—24no—
127#396—24no—
128#393—24no—
129#391—22no—
130#391—24no—
131#391C2314yes—
132#391C2314yes—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—24no—
137#389C2314yes—
138#389C2314yes—
139#388—24no—
140#388—24no—
141#388—24no—
142#388—24no—
143#387—24no—
144#387—24no—
145#387—24no—
146#385—24no—
147#385—24no—
148#385—24no—
149#385—24no—
150#384C2312yes—
151#384C2312yes—
152#380—22no—
153#380—22no—
154#380—22no—
155#380C2314yes—
156#380C2314yes—
157#380C2314yes—
158#379—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4—24no—
162C4×C22⋊C4D412yes—
163C4×C22⋊C4D412yes—
164C4×C22⋊C4C2314yes—
165C2×#95D412yes—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#336C2314yes—
173#336C2314yes—
174C25C2×D411yescharacteristic
175C22×Q8—22no—
176C22×Q8C2414yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—41no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4—41no—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C2×Q16—24no—
190C2×Q16—24no—
191C2×Q16—42no—
192C2×Q16—42no—
193C2×Q16—42no—
194C2×Q16—42no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—42no—
200C2×M16—42no—
201C2×M16—416no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—41no—
207C22×C8—41no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8—41no—
211C22×C8—41no—
212C22×C8#4114yes—
213C22×C8#4114yes—
214C22×C8C2412yes—
215C22×C8C2412yes—
216C22×C8C2412yes—
217C22×C8C2411yescharacteristic
218C22×C8C2411yescharacteristic
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C22⋊Q8—28no—
224C22⋊Q8—28no—
225C22⋊Q8—48no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C4×Q8—44no—
231C4×Q8—44no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2414yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C2×D414yes—
256C2×C22⋊C4C2×D414yes—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2412yes—
267C2×C42C2412yes—
268#107—24no—
269#107—24no—
270#106—42no—
271#106—42no—
272#106—42no—
273#106—42no—
274#103—28no—
275#103—28no—
276#103—28no—
277#103—28no—
278#103—44no—
279#103—44no—
280C22⋊C8—28no—
281#97—24no—
282#97—42no—
283C4×C8—24no—
284C4×C8—42no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95C2×D414yes—
296#95C2×D414yes—
297C24—22no—
298C24—24no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24C22×D412yes—
303C24C22×D411yescharacteristic
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—82no—
312C2×Q8—82no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4#13718yes—
343C22×C4C22×D412yes—
344C22×C4C22×D414yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4C2511yescharacteristic
351Q16—48no—
352Q16—84no—
353M16—48no—
354M16—48no—
355C2×C8—28no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8C2×#4114yes—
371C2×C8C2×#4114yes—
372C2×C8H(5,𝔽2)12yes—
373C2×C8H(5,𝔽2)12yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—84no—
409C42—84no—
410C23—22no—
411C23—21no—
412C23—21no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—84no—
422C23—84no—
423C23#53212yes—
424C23C2×#13714yes—
425C23C23×D411yescharacteristic
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—164no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—84no—
467C2×C4#53212yes—
468C2×C4C2×#13714yes—
469C2×C4C2×#13714yes—
470C2×C4C22×#4111yescharacteristic
471C2×C4C22×#4111yescharacteristic
472C2×C4C2×H(5,𝔽2)11yescharacteristic
473C2×C4C2×H(5,𝔽2)11yescharacteristic
474C2×C4#58511yescharacteristic
475C2×C4#58511yescharacteristic
476C8—24no—
477C8—22no—
478C8—22no—
479C8—24no—
480C8—42no—
481C22—22no—
482C22—24no—
483C22—24no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22—44no—
488C22—44no—
489C22—42no—
490C22—88no—
491C22#318514yes—
492C22C2×#53211yescharacteristic
493C22#349611yescharacteristic
494C22#349611yescharacteristic
495C22C22×#13712yes—
496C22#358412yes—
497C4—24no—
498C4—24no—
499C4—22no—
500C4—44no—
501C4—42no—
502C4—84no—
503C4—84no—
504C4—84no—
505C4—84no—
506C4#345911yescharacteristic
507C4#346011yescharacteristic
508C4#349011yescharacteristic
509C4#349311yescharacteristic
510C2—24no—
511C2—82no—
512C2#3500311yescharacteristic
513C2#6068511yescharacteristic
514C2C2×#318511yescharacteristic
515C2#6225412yes—
516C2#6226012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33043C2semidirect
4. #30293C2semidirect
5. #30242C2semidirect
7. #28925C2semidirect
8. #28871C2semidirect
9. #28088C2semidirect
10. #26988C2semidirect
11. #22183C2semidirect
12. #21527C2semidirect
15. C2×#487C22semidirect
20. C2×#441C22semidirect
39. #2095C22semidirect
40. #2084C22semidirect
42. #2058C22semidirect
45. #2016C22semidirect
47. #1990C22semidirect
48. #1971C22semidirect
49. #1956C22semidirect
50. #1943C22semidirect
51. #1935C22semidirect
52. #1933C22semidirect
53. #1933C22semidirect
54. #1927C22semidirect
55. #1918C22semidirect
60. #1878C22semidirect
61. #1865C22semidirect
62. #1834C22semidirect
67. #1776C22semidirect
68. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#350142
#330432
#329372
#302932
#302422
#297332
#289252
#288712
#280882
#269882
#221832
#215272
C2×#17532