Finite Group Databasebeta

← #7623126

Subgroups — #7623126

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60707C211yesmaximal, characteristic, complemented
2C2×#2951C211yesmaximal, characteristic, complemented
3#33685C214yesmaximal, complemented
4#33679C214yesmaximal, complemented
5#33677C212yesmaximal, complemented
6#33669C212yesmaximal, complemented
7#33665C212yesmaximal, complemented
8#33663C212yesmaximal, complemented
9#33646C211yesmaximal, characteristic, complemented
10C2×#2233C211yesmaximal, characteristic, complemented
11#33634C212yesmaximal, complemented
12#33589C212yesmaximal, complemented
13#33520C211yesmaximal, characteristic, complemented
14#33516C211yesmaximal, characteristic, complemented
15#33460C212yesmaximal, complemented
16#33454C212yesmaximal, complemented
17#33422C211yesmaximal, characteristic, complemented
18C22×#135C2211yescharacteristic
19#3494—21no—
20C2×#520—21no—
21C2×#509C2212yes—
22C2×#508C2212yes—
23#3180—22no—
24#3179—22no—
25#3176C2212yescomplemented
26#3175C2212yescomplemented
27#3169C2212yes—
28#3168C2212yes—
29#3164C2211yescharacteristic, complemented
30#3155C2211yescharacteristic, complemented
31#3136—21no—
32#3055—21no—
33#3050C2211yescharacteristic
34C2×#466C2211yescharacteristic
35#2951—21no—
36#2951—21no—
37#2951—22no—
38#2951C2212yes—
39#2951C2212yes—
40#2951C2214yescomplemented
41#2951C2211yescharacteristic, complemented
42#2948C2211yescharacteristic, complemented
43#2946—21no—
44C2×#440—22no—
45C2×C4×Q16C2212yescomplemented
46C2×C4×D8C2212yescomplemented
47C22×#107C2211yescharacteristic
48C2×#417—21no—
49C2×#405C2211yescharacteristic
50#2236C2212yescomplemented
51#2235C2212yescomplemented
52#2233C2212yes—
53#2233C2212yes—
54#2233C2214yescomplemented
55#2233C2212yescomplemented
56#2232C2212yescomplemented
57#2229C2214yescomplemented
58#2228C2214yescomplemented
59#2226C2214yescomplemented
60#2225C2214yescomplemented
61#2211C2214yescomplemented
62#2210C2214yescomplemented
63#2207C2214yescomplemented
64#2206C2214yescomplemented
65#2189C2212yescomplemented
66#2188C2211yescharacteristic, complemented
67#2187C2211yescharacteristic, complemented
68#2186C2214yes—
69#2186C2212yescomplemented
70#2185C2212yes—
71#2185C2211yescharacteristic, complemented
72#2184C2212yes—
73#2184C2211yescharacteristic, complemented
74#2177C2214yescomplemented
75#2177C2214yescomplemented
76#2176C2214yescomplemented
77#2175C2214yescomplemented
78C4×SD32C2214yescomplemented
79C4×SD32C2214yescomplemented
80C4×SD32C2214yescomplemented
81C4×D16C2214yescomplemented
82#2164C2214yescomplemented
83#2163C2212yescomplemented
84#2162C2212yescomplemented
85#2159C2212yescomplemented
86C2×#367C2212yescomplemented
87#2151C2212yescomplemented
88#2139C2212yes—
89#2139C2212yescomplemented
90#2138C2212yes—
91C2×#357C2212yescomplemented
92#2112C2211yescharacteristic
93C2×C22⋊C16C2211yescharacteristic
94#2108C2212yescomplemented
95C22×#41—21no—
96C2×AGL(1,ℤ/8)—21no—
97C2×#135—28no—
98C2×#135—21no—
99C2×#135—21no—
100C2×#135C2311yescharacteristic
101C2×#135C2312yes—
102C2×#135C2312yes—
103C22×Q16C2311yescharacteristic
104C22×SD16—21no—
105C22×D8C2311yescharacteristic
106C22×D8C2311yescharacteristic
107C22×M16D411yescharacteristic
108C23×C8C2311yescharacteristic
109#555—41no—
110#553—41no—
111#553—41no—
112#551—42no—
113#547—41no—
114#546—42no—
115#544—21no—
116#542—22no—
117#540—22no—
118#535—21no—
119#532—22no—
120#529—22no—
121#525—21no—
122C2×#121—21no—
123#520—22no—
124#520—22no—
125#520—24no—
126#520—21no—
127#520—42no—
128#520—41no—
129#520—41no—
130#518—21no—
131C2×C4×Q8—22no—
132C2×C4×D4—22no—
133C2×C4×D4—42no—
134C2×#117—41no—
135C2×#117D411yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—41no—
138#509—28no—
139#508—28no—
140C2×Q32C2312yes—
141C2×SD32C2312yes—
142C2×SD32C2312yes—
143C2×SD32C2314yes—
144C2×D16C2312yes—
145C2×D16C2314yes—
146C2×M32C2312yes—
147C22×C16C2312yes—
148#501—22no—
149#500C2312yes—
150#499C2312yes—
151#496—22no—
152#495C2312yes—
153#493C2312yes—
154#490—22no—
155#489—22no—
156#488—22no—
157#486—22no—
158#469—22no—
159#468—22no—
160#467C2311yescharacteristic
161#467C2311yescharacteristic
162#466—22no—
163#466—21no—
164#466—21no—
165#466C2312yes—
166#466C2311yescharacteristic
167#466C2312yes—
168#466C2311yescharacteristic
169#465—22no—
170#465—21no—
171#465—21no—
172#442—22no—
173#442—22no—
174#440—22no—
175#440—22no—
176#440—24no—
177#440—24no—
178#440—24no—
179#440—24no—
180C4×Q16—22no—
181C4×Q16—22no—
182C4×Q16C2312yes—
183C4×Q16C2314yes—
184C4×Q16C2314yes—
185C4×SD16—22no—
186C4×D8—22no—
187C4×D8—22no—
188C4×D8—22no—
189C4×D8C2312yes—
190C4×D8C2312yes—
191C4×D8C2312yes—
192C4×D8C2314yes—
193C4×D8C2314yes—
194#428—21no—
195#427C2311yescharacteristic
196C2×#107—21no—
197C2×#107—21no—
198C2×#107—21no—
199C2×#107C2312yes—
200C2×#107C2311yescharacteristic
201C2×#107C2312yes—
202C2×#107C2312yes—
203#418—21no—
204#417—21no—
205#417—22no—
206#417—22no—
207#417—24no—
208#416—21no—
209C2×#103—22no—
210C2×#102—21no—
211C2×#102—22no—
212#405D411yescharacteristic
213#405D412yes—
214#405D411yescharacteristic
215#405C2311yescharacteristic
216#405C2311yescharacteristic
217#405C2312yes—
218C2×#97D412yes—
219C2×C4×C8C2312yes—
220#367C2314yes—
221#367C2314yes—
222#367C2314yes—
223#366C2314yes—
224C4⋊C16C2314yes—
225C4⋊C16C2314yes—
226#358C2312yes—
227#358C2312yes—
228#358C2314yes—
229#358C2314yes—
230#357C2314yes—
231#357C2314yes—
232#357C2312yes—
233#357C2312yes—
234#357C2314yes—
235#357C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2311yescharacteristic
238C22⋊C16C2312yes—
239C22⋊C16C2312yes—
240C22⋊C16C2312yes—
241#346C2314yes—
242C4×C16C2314yes—
243C2×#41—28no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—21no—
247C2×#41—44no—
248C2×#41—41no—
249C2×#41—41no—
250C2×#41—42no—
251C22×Q8—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—41no—
255C22×D4—41no—
256C23×C4—41no—
257C23×C4—41no—
258C23×C4—42no—
259C23×C4C2×D411yescharacteristic
260AGL(1,ℤ/8)—44no—
261#135—28no—
262#135—28no—
263#135—24no—
264#135—24no—
265#135—24no—
266#135—24no—
267#135#4118yescomplemented
268C2×Q16—24no—
269C2×Q16—21no—
270C2×Q16C2411yescharacteristic
271C2×Q16C2412yes—
272C2×SD16—28no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×SD16—21no—
276C2×SD16—22no—
277C2×D8—24no—
278C2×D8—21no—
279C2×D8—21no—
280C2×D8—22no—
281C2×D8—24no—
282C2×D8C2412yes—
283C2×D8C2411yescharacteristic
284C2×D8C2412yes—
285C2×M16—42no—
286C2×M16C2×D411yescharacteristic
287C2×M16C2×D411yescharacteristic
288C2×M16C2×D412yes—
289C22×C8—24no—
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D412yes—
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C4⋊Q8—22no—
298C4⋊Q8—41no—
299#127—22no—
300#126—42no—
301#126—42no—
302#125—22no—
303#125—41no—
304#124—22no—
305#124—42no—
306#124—42no—
307#123—41no—
308#123—42no—
309#123—42no—
310#123—42no—
311#123—42no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—42no—
315C22⋊Q8—42no—
316#121—22no—
317#121—22no—
318#121—21no—
319#121—21no—
320#121—41no—
321#121—41no—
322#121—42no—
323#121—41no—
324#121—41no—
325#121—42no—
326#121—42no—
327#120—41no—
328#120—42no—
329C4×Q8—24no—
330C4×Q8—24no—
331C4×Q8—22no—
332C4×Q8—42no—
333C4×Q8—42no—
334C4×D4—24no—
335C4×D4—24no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4—42no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—44no—
345C4×D4—44no—
346C4×D4—44no—
347C4×D4—44no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—42no—
352#117—21no—
353#117—42no—
354#117—42no—
355#117—44no—
356#117—41no—
357#117—42no—
358#117D812yes—
359#117D811yescharacteristic
360#117D811yescharacteristic
361#117C2×D411yescharacteristic
362#117C2×D411yescharacteristic
363#117C2×D412yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—44no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4D811yescharacteristic
377C2×C4⋊C4D811yescharacteristic
378C2×C22⋊C4—42no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4D812yes—
382C2×C42—42no—
383C2×C42C2×D412yes—
384Q32—24no—
385SD32—24no—
386SD32—28no—
387SD32—24no—
388D16—24no—
389D16—28no—
390M32—24no—
391C2×C16—24no—
392C2×C16C2412yes—
393C2×C16C2414yes—
394C2×C16C2412yes—
395#107—22no—
396#107—22no—
397#107—22no—
398#107—21no—
399#107—22no—
400#107C2414yes—
401#107C2411yescharacteristic
402#107C2412yes—
403#107C2411yescharacteristic
404#106—21no—
405#106—22no—
406#103—24no—
407#103—24no—
408#103—21no—
409#103—21no—
410#103—22no—
411#102—24no—
412#102—24no—
413#102—22no—
414#102—22no—
415#102—21no—
416#102—21no—
417#102—22no—
418#102—22no—
419#102—22no—
420#97C2×D412yes—
421#97C2×D412yes—
422#97C2×D412yes—
423#97C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2×D412yes—
426C4×C8C2412yes—
427C4×C8C2412yes—
428C24—41no—
429C24—41no—
430C24C2×D811yescharacteristic
431#41—28no—
432#41—28no—
433#41—48no—
434#41—48no—
435#41—44no—
436#41—44no—
437#41—42no—
438#41—42no—
439#41—164no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—42no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—42no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—162no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—48no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—164no—
483C22×C4—162no—
484C22×C4C2×D812yes—
485C22×C4C2×D811yescharacteristic
486C22×C4C2×D811yescharacteristic
487C22×C4C2×D811yescharacteristic
488C22×C4C2×D812yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492Q16—24no—
493Q16—22no—
494Q16—22no—
495Q16C2×#4114yes—
496SD16—28no—
497SD16—28no—
498SD16—44no—
499SD16—42no—
500D8—24no—
501D8—22no—
502D8—24no—
503D8—22no—
504D8—44no—
505D8—42no—
506D8C2×#4114yes—
507M16—22no—
508M16—22no—
509M16—42no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—42no—
516C2×C8C22×D411yescharacteristic
517C2×C8C22×D412yes—
518C2×C8C22×D412yes—
519C2×C8C22×D411yescharacteristic
520C2×C8C22×D411yescharacteristic
521C2×C8C2×#4114yes—
522C2×C8C2511yescharacteristic
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—24no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—41no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—41no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—41no—
539C4⋊C4C2×D811yescharacteristic
540C4⋊C4C2×D811yescharacteristic
541C4⋊C4C2×D812yes—
542C4⋊C4C2×D811yescharacteristic
543C4⋊C4C2×D811yescharacteristic
544C4⋊C4C2×D812yes—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—41no—
557C22⋊C4—42no—
558C22⋊C4—41no—
559C22⋊C4C2×D812yes—
560C22⋊C4C2×D812yes—
561C22⋊C4C2×D812yes—
562C22⋊C4C2×D812yes—
563C42—44no—
564C42—44no—
565C42—41no—
566C42—42no—
567C42C2×D812yes—
568C42C2×D812yes—
569C42C22×D412yes—
570C42C22×D412yes—
571C16—22no—
572C16—22no—
573C16—24no—
574C23—22no—
575C23—22no—
576C23—41no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—41no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—161no—
585C23—161no—
586C23C22×D811yescharacteristic
587C23C22×D811yescharacteristic
588C23C22×D811yescharacteristic
589Q8—24no—
590Q8—44no—
591Q8—42no—
592Q8—41no—
593Q8—41no—
594Q8—162no—
595D4—24no—
596D4—44no—
597D4—48no—
598D4—48no—
599D4—42no—
600D4—41no—
601D4—42no—
602D4—41no—
603D4—42no—
604D4—82no—
605D4—84no—
606D4—82no—
607D4—82no—
608D4—162no—
609D4—162no—
610D4—161no—
611D4—162no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—82no—
632C2×C4—162no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4—162no—
636C2×C4—162no—
637C2×C4—162no—
638C2×C4—162no—
639C2×C4—161no—
640C2×C4#52512yes—
641C2×C4C22×D812yes—
642C2×C4C22×D812yes—
643C2×C4C22×D812yes—
644C2×C4C22×D811yescharacteristic
645C2×C4C22×D812yes—
646C2×C4C22×D811yescharacteristic
647C2×C4C22×D811yescharacteristic
648C2×C4#57512yes—
649C2×C4C23×D411yescharacteristic
650C8—21no—
651C8—21no—
652C8—22no—
653C8—22no—
654C8#52512yes—
655C8C22×#4112yes—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—44no—
665C22—88no—
666C22—82no—
667C22—82no—
668C22—81no—
669C22—82no—
670C22—81no—
671C22—82no—
672C22—82no—
673C22—81no—
674C22—322no—
675C22#310012yes—
676C22#341612yes—
677C22C23×D811yescharacteristic
678C4—21no—
679C4—21no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—44no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—84no—
689C4—81no—
690C4#310012yes—
691C4C2×#52511yescharacteristic
692C4C2×#57511yescharacteristic
693C2—22no—
694C2—22no—
695C2—41no—
696C2—84no—
697C2—161no—
698C2—162no—
699C2C2×#310011yescharacteristic
700C2C2×#341611yescharacteristic
701C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60707C2semidirect
2. C2×#2951C2semidirect
3. #33685C2semidirect
4. #33679C2semidirect
5. #33677C2semidirect
6. #33669C2semidirect
7. #33665C2semidirect
8. #33663C2semidirect
9. #33646C2semidirect
10. C2×#2233C2semidirect
11. #33634C2semidirect
12. #33589C2semidirect
13. #33520C2semidirect
14. #33516C2semidirect
15. #33460C2semidirect
16. #33454C2semidirect
17. #33422C2semidirect
25. #3176C22semidirect
26. #3175C22semidirect
29. #3164C22semidirect
30. #3155C22semidirect
40. #2951C22semidirect
41. #2951C22semidirect
42. #2948C22semidirect
45. C2×C4×Q16C22semidirect
46. C2×C4×D8C22semidirect
50. #2236C22semidirect
51. #2235C22semidirect
54. #2233C22semidirect
55. #2233C22semidirect
56. #2232C22semidirect
57. #2229C22semidirect
58. #2228C22semidirect
59. #2226C22semidirect
60. #2225C22semidirect
61. #2211C22semidirect
62. #2210C22semidirect
63. #2207C22semidirect
64. #2206C22semidirect
65. #2189C22semidirect
66. #2188C22semidirect
67. #2187C22semidirect
69. #2186C22semidirect
71. #2185C22semidirect
73. #2184C22semidirect
74. #2177C22semidirect
75. #2177C22semidirect
76. #2176C22semidirect
77. #2175C22semidirect
78. C4×SD32C22semidirect
79. C4×SD32C22semidirect
80. C4×SD32C22semidirect
81. C4×D16C22semidirect
82. #2164C22semidirect
83. #2163C22semidirect
84. #2162C22semidirect
85. #2159C22semidirect
86. C2×#367C22semidirect
87. #2151C22semidirect
89. #2139C22semidirect
91. C2×#357C22semidirect
94. #2108C22semidirect
267. #135#41semidirect

Maximal subgroups

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

subgroupindex
#607072
C2×#29512
#336852
#336792
#336772
#336692
#336652
#336632
#336462
C2×#22332
#336342
#335892
#335202
#335162
#334602
#334542
#334222