Finite Group Databasebeta

← #6433111

Subgroups — #6433111

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic, complemented
2#34044C211yesmaximal, characteristic
3#29965C214yesmaximal, complemented
4#29964C214yesmaximal, complemented
5#29963C214yesmaximal, complemented
6#29962C214yesmaximal, complemented
7#23734C212yesmaximal, complemented
8#23617C214yesmaximal, complemented
9#23453C212yesmaximal, complemented
10#20846C212yesmaximal, complemented
11#20627C211yesmaximal, characteristic, complemented
12#20601C212yesmaximal
13C23×#41C2211yescharacteristic
14C22×#125C2211yescharacteristic, complemented
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#428C2214yescomplemented
18C2×#423C2212yescomplemented
19C2×#417C2214yes—
20C2×#417C2214yes—
21C2×#407C2212yes—
22#2629—22no—
23#2629—22no—
24#2629C2214yes—
25#2629C2214yescomplemented
26C2×#398—22no—
27C2×#396—22no—
28#2311—22no—
29#2311C2212yes—
30#2311C2212yes—
31#2310C2212yes—
32#2309C2212yes—
33#2308—22no—
34#2297—22no—
35#2297C2212yes—
36#2297C2212yes—
37C2×#389C2212yescomplemented
38C2×#384C2211yescharacteristic, complemented
39#2280—22no—
40C2×#380C2212yes—
41#2096C2214yescomplemented
42#2095C2214yescomplemented
43#2094C2214yescomplemented
44#2093C2214yescomplemented
45#2068C2218yescomplemented
46#2067C2218yescomplemented
47#2066C2218yescomplemented
48#2065C2218yescomplemented
49#2026C2214yescomplemented
50#2026C2214yescomplemented
51#2025C2214yescomplemented
52#2025C2214yescomplemented
53#1854C2218yes—
54#1853C2214yescomplemented
55#1852C2214yescomplemented
56#1798C2214yes—
57#1798C2214yescomplemented
58#1788C2218yes—
59#1787C2214yes—
60#1786C2214yes—
61#1736C2214yescomplemented
62#1728C2212yescomplemented
63C22×#41—21no—
64C22×#41—21no—
65C22×#41—24no—
66C22×#41—44no—
67C22×#41—44no—
68C22×#41Q818yes—
69C22×#41C2312yes—
70C22×#41C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4—21no—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C24×C4C2311yescharacteristic
76C22×M16C2314yes—
77C2×#125—22no—
78C2×#125—44no—
79C2×#125C2314yes—
80C2×#117—21no—
81C2×#117—21no—
82C2×#117—44no—
83C2×#117—44no—
84C2×#117D412yes—
85C2×#117D412yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C2×#117C2312yes—
89C2×#117C2312yes—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4D412yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4D412yes—
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#428—48no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107#423—44no—
108C2×C4⋊C8C2318yes—
109#417—48no—
110#417—48no—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#102C2318yes—
114C2×#102C2318yes—
115#407—28no—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118#398—28no—
119#398—28no—
120#396—28no—
121#396—28no—
122#391—24no—
123#391C2312yes—
124#391C2312yes—
125#391C2312yes—
126#391C2312yes—
127#390—24no—
128#390C2312yes—
129#390C2312yes—
130#390C2312yes—
131#390C2312yes—
132#389—24no—
133#389C2318yes—
134#386—24no—
135#386—24no—
136#386—24no—
137#386—24no—
138#384—22no—
139#384C2314yes—
140#381—24no—
141#380—22no—
142#380—22no—
143#380—24no—
144#380—24no—
145#380C2314yes—
146#380C2314yes—
147#379—24no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—24no—
150C2×#95—22no—
151C2×#95—22no—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C25—21no—
157C25C2×D411yescharacteristic
158C2×#41—28no—
159C2×#41—28no—
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—28no—
163C2×#41—28no—
164C2×#41—44no—
165C2×#41—44no—
166C2×#41—432no—
167C2×#41—44no—
168C2×#41—44no—
169C2×#41—44no—
170C2×#41—44no—
171C2×#41—432no—
172C2×#41C2×Q818yes—
173C2×#41C2×Q818yes—
174C22×Q8—21no—
175C22×Q8—22no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8C2×Q814yes—
179C22×Q8C2412yes—
180C22×D4—28no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4C2×Q814yes—
190C22×D4C2412yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—21no—
194C23×C4—28no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×Q814yes—
209C23×C4#4114yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C2×M16—28no—
214C2×M16—28no—
215C2×M16—28no—
216C22×C8C2418yes—
217#125—48no—
218#125—48no—
219#117—28no—
220#117—44no—
221#117—44no—
222#117—44no—
223#117—44no—
224#117—44no—
225#117—44no—
226#117—44no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2414yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—84no—
259C2×C22⋊C4C2×D414yes—
260C2×C22⋊C4C2×D414yes—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—44no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2412yes—
269C2×C42C2412yes—
270#107—48no—
271#106—48no—
272C4⋊C8—48no—
273#103—48no—
274#103—48no—
275#102—48no—
276#102—48no—
277C22⋊C8—28no—
278C22⋊C8—28no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—82no—
295C24—82no—
296C24C22⋊Q814yes—
297C24C22×D411yescharacteristic
298C24C22×D411yescharacteristic
299C24C22×D411yescharacteristic
300#41—232no—
301#41—416no—
302#41—432no—
303#41—432no—
304#41—416no—
305#41—416no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—48no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—48no—
314C2×Q8C22×Q814yes—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—48no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—48no—
325C2×D4—432no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4C22×Q814yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—432no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4C22⋊Q814yes—
375C22×C4C22⋊Q814yes—
376C22×C4C22⋊Q814yes—
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q812yes—
383C22×C4C22×Q812yes—
384C22×C4C22×Q812yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4112yes—
388C22×C4H(5,𝔽2)14yes—
389C22×C4H(5,𝔽2)14yes—
390C22×C4C2511yescharacteristic
391M16—416no—
392C2×C8—28no—
393C2×C8—28no—
394C2×C8—28no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—416no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—88no—
407C4⋊C4—88no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—44no—
423C42—44no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—28no—
435C23—24no—
436C23—48no—
437C23—48no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—84no—
443C23—82no—
444C23—82no—
445C23—88no—
446C23—88no—
447C23—84no—
448C23—84no—
449C23C2×C22⋊Q812yes—
450C23C2×C22⋊Q812yes—
451C23C2×C22⋊Q812yes—
452C23#53412yes—
453C23#53512yes—
454C23C23×D411yescharacteristic
455Q8—28no—
456Q8—48no—
457Q8—44no—
458D4—28no—
459D4—48no—
460D4—416no—
461D4—416no—
462D4—44no—
463C2×C4—28no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—416no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—416no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—84no—
498C2×C4—88no—
499C2×C4—88no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4C2×C22⋊Q812yes—
503C2×C4C2×C22⋊Q812yes—
504C2×C4C2×C22⋊Q812yes—
505C2×C4#53412yes—
506C2×C4#53512yes—
507C2×C4#53818yes—
508C2×C4#54318yes—
509C2×C4C23×Q811yescharacteristic
510C2×C4C22×#4111yescharacteristic
511C2×C4C2×H(5,𝔽2)12yes—
512C2×C4C2×H(5,𝔽2)12yes—
513C8—48no—
514C22—24no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—48no—
529C22—44no—
530C22—88no—
531C22—816no—
532C22—84no—
533C22—82no—
534C22—82no—
535C22#262614yes—
536C22C22×C22⋊Q811yescharacteristic
537C22C2×#53411yescharacteristic
538C22C2×#53511yescharacteristic
539C22#358512yes—
540C22#358612yes—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—44no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4#262614yes—
551C4C2×#53812yes—
552C4C2×#54312yes—
553C2—24no—
554C2—42no—
555C2—42no—
556C2—84no—
557C2C2×#262611yescharacteristic
558C2#6049312yes—
559C2#6075512yes—
560C2#6080811yescharacteristic
561C2#6080911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2629C2semidirect
3. #29965C2semidirect
4. #29964C2semidirect
5. #29963C2semidirect
6. #29962C2semidirect
7. #23734C2semidirect
8. #23617C2semidirect
9. #23453C2semidirect
10. #20846C2semidirect
11. #20627C2semidirect
14. C22×#125C22semidirect
17. C2×#428C22semidirect
18. C2×#423C22semidirect
25. #2629C22semidirect
37. C2×#389C22semidirect
38. C2×#384C22semidirect
41. #2096C22semidirect
42. #2095C22semidirect
43. #2094C22semidirect
44. #2093C22semidirect
45. #2068C22semidirect
46. #2067C22semidirect
47. #2066C22semidirect
48. #2065C22semidirect
49. #2026C22semidirect
50. #2026C22semidirect
51. #2025C22semidirect
52. #2025C22semidirect
54. #1853C22semidirect
55. #1852C22semidirect
57. #1798C22semidirect
61. #1736C22semidirect
62. #1728C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
#340442
#299652
#299642
#299632
#299622
#237342
#236172
#234532
#208462
#206272
#206012