Finite Group Databasebeta

← #6506555

Subgroups — #6506555

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37453C211yesmaximal, characteristic
2#35151C211yesmaximal, characteristic
3#29344C212yesmaximal, complemented
4#29341C212yesmaximal, complemented
5#27985C212yesmaximal, complemented
6#27723C214yesmaximal, complemented
7#27705C214yesmaximal, complemented
8#27283C214yesmaximal, complemented
9#27277C214yesmaximal, complemented
10#25914C212yesmaximal, complemented
11#21768C212yesmaximal
12#20911C212yesmaximal
13#20529C211yesmaximal, characteristic
14C2×#544C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×#117C2211yescharacteristic
17C2×#491C2212yescomplemented
18C2×#487C2212yescomplemented
19C2×#419C2214yes—
20C2×#407C2212yes—
21#2855—22no—
22#2817—22no—
23#2627—22no—
24#2627—22no—
25#2626—22no—
26#2621C2212yes—
27#2621C2212yes—
28#2503C2214yes—
29#2483—22no—
30#2441—22no—
31#2441C2214yes—
32#2439—22no—
33#2439C2214yes—
34#2434C2214yes—
35#2317C2211yescharacteristic
36#2317C2211yescharacteristic
37#2316C2211yescharacteristic
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2101C2214yescomplemented
41#2101C2214yescomplemented
42#2074C2214yescomplemented
43#2072C2214yescomplemented
44#2043C2214yescomplemented
45#2043C2214yescomplemented
46#2026C2214yescomplemented
47#2024C2214yescomplemented
48#1993C2212yescomplemented
49#1991C2212yescomplemented
50#1960C2212yescomplemented
51#1956C2212yescomplemented
52#1935C2214yescomplemented
53#1935C2214yescomplemented
54#1933C2214yescomplemented
55#1933C2214yescomplemented
56#1894C2214yes—
57#1894C2214yes—
58#1873C2214yes—
59#1873C2214yes—
60#1873C2214yes—
61#1873C2214yes—
62#1872C2214yes—
63#1872C2214yes—
64#1836C2212yescomplemented
65#1835C2212yescomplemented
66#1824C2212yes—
67#1824C2212yes—
68#1787C2212yes—
69#1787C2212yes—
70#1753C2214yes—
71#1753C2214yes—
72#1750C2212yes—
73#1750C2212yes—
74#1736C2214yes—
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C22×M16C2314yes—
78#544—44no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#125—21no—
84C2×#125C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8C2314yes—
91C2×C22⋊Q8C2314yes—
92C2×#117—22no—
93C2×#117—44no—
94C2×#117C2314yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C4⋊C4—22no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—22no—
105C22×C42—21no—
106#491—44no—
107#487—44no—
108#419—48no—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117#407—28no—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125#400—24no—
126#400—24no—
127#399—24no—
128#398—22no—
129#398—22no—
130#398—24no—
131#397—22no—
132#397—22no—
133#397—24no—
134#396—24no—
135#396—24no—
136#395—24no—
137#395—24no—
138#393—24no—
139#391—24no—
140#391C2314yes—
141#391C2314yes—
142#389—22no—
143#389—22no—
144#389—24no—
145#389C2314yes—
146#389C2314yes—
147#387—24no—
148#387—24no—
149#385—22no—
150#385—22no—
151#385—24no—
152#385—24no—
153#384—21no—
154#384C2312yes—
155#382—21no—
156#382C2312yes—
157#382C2312yes—
158#382C2312yes—
159#381—21no—
160#381—24no—
161#381—24no—
162#381D412yes—
163#380—22no—
164#380D412yes—
165#380D412yes—
166#380C2312yes—
167#380C2312yes—
168#379—21no—
169#379D412yes—
170C4×C22⋊C4—22no—
171C4×C22⋊C4C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2314yes—
182C25C2×D411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2414yes—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—21no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4#4118yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C2×M16—28no—
202C2×M16—44no—
203C2×M16—44no—
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—24no—
207C4⋊Q8—42no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210C4⋊Q8—42no—
211C4⋊Q8—42no—
212C4⋊Q8—44no—
213#125—24no—
214#125—42no—
215#125—44no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—28no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C22⋊Q8—48no—
223#117—28no—
224#117—42no—
225#117—42no—
226#117—44no—
227#117—42no—
228#117—42no—
229#117—48no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
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—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2411yescharacteristic
255C2×C4⋊C4C2411yescharacteristic
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4C2×D412yes—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D414yes—
270C2×C22⋊C4C2×D414yes—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—44no—
275C2×C42C2412yes—
276C2×C42C2412yes—
277#103—28no—
278#103—28no—
279#103—44no—
280#103—44no—
281#103—44no—
282#103—44no—
283#103—44no—
284#103—44no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—28no—
288#97—42no—
289#97—42no—
290C4×C8—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C2×D414yes—
305#95C2×D414yes—
306C24—41no—
307C24—41no—
308C24—42no—
309C24—42no—
310C24#12414yes—
311C24C22×D412yes—
312C24C22×D411yescharacteristic
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—44no—
320C2×Q8—416no—
321C2×Q8—44no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—84no—
325C2×Q8—88no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—416no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4#12418yes—
353C22×C4#12414yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4114yes—
360C22×C4C2×#4114yes—
361C22×C4C2×#4114yes—
362C22×C4C2511yescharacteristic
363M16—88no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4H(5,𝔽2)14yes—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—48no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—48no—
411C42—42no—
412C42—42no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—44no—
424C23—48no—
425C23—84no—
426C23—82no—
427C23—82no—
428C23C2×#12412yes—
429C23C2×#12414yes—
430C23C23×D411yescharacteristic
431Q8—44no—
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—88no—
436Q8—164no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—84no—
471C2×C4—88no—
472C2×C4—88no—
473C2×C4—84no—
474C2×C4C2×#12412yes—
475C2×C4C2×#12414yes—
476C2×C4C22×#4112yes—
477C2×C4C2×H(5,𝔽2)12yes—
478C2×C4#58512yes—
479C8—44no—
480C8—44no—
481C22—24no—
482C22—22no—
483C22—24no—
484C22—24no—
485C22—24no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—42no—
491C22—41no—
492C22—42no—
493C22—88no—
494C22C22×#12411yescharacteristic
495C22#348411yescharacteristic
496C22#348511yescharacteristic
497C22#358612yes—
498C22#358612yes—
499C4—22no—
500C4—24no—
501C4—44no—
502C4—42no—
503C4—44no—
504C4—42no—
505C4—84no—
506C4—82no—
507C4—84no—
508C4—81no—
509C4—81no—
510C4—82no—
511C4#350514yes—
512C2—44no—
513C2—42no—
514C2#3537711yescharacteristic
515C2#6062012yes—
516C2#6164911yescharacteristic
517C2#6165211yescharacteristic
518C2#6177912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #29344C2semidirect
4. #29341C2semidirect
5. #27985C2semidirect
6. #27723C2semidirect
7. #27705C2semidirect
8. #27283C2semidirect
9. #27277C2semidirect
10. #25914C2semidirect
17. C2×#491C22semidirect
18. C2×#487C22semidirect
40. #2101C22semidirect
41. #2101C22semidirect
42. #2074C22semidirect
43. #2072C22semidirect
44. #2043C22semidirect
45. #2043C22semidirect
46. #2026C22semidirect
47. #2024C22semidirect
48. #1993C22semidirect
49. #1991C22semidirect
50. #1960C22semidirect
51. #1956C22semidirect
52. #1935C22semidirect
53. #1935C22semidirect
54. #1933C22semidirect
55. #1933C22semidirect
64. #1836C22semidirect
65. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#374532
#351512
#293442
#293412
#279852
#277232
#277052
#272832
#272772
#259142
#217682
#209112
#205292