Finite Group Databasebeta

← #6816575

Subgroups — #6816575

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3387C211yesmaximal, characteristic, complemented
2C2×#3342C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic
4#39172C211yesmaximal, characteristic
5#33314C211yesmaximal, characteristic, complemented
6#31109C214yesmaximal, complemented
7#31107C214yesmaximal, complemented
8#29867C211yesmaximal, characteristic, complemented
9#29866C211yesmaximal, characteristic, complemented
10#28706C211yesmaximal, characteristic
11#28683C211yesmaximal, characteristic, complemented
12#26070C214yesmaximal, complemented
13#25840C211yesmaximal, characteristic, complemented
14#25837C211yesmaximal, characteristic, complemented
15#25705C211yesmaximal, characteristic
16#25701C211yesmaximal, characteristic
17#25043C214yesmaximal, complemented
18#21633C211yesmaximal, characteristic
19#21278C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#556C2211yescharacteristic
24#3387—28no—
25#3342—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#485C2212yes—
32C2×#485C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#475C2211yescharacteristic
38C2×#475C2212yes—
39C2×#467C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2721—24no—
48#2716—24no—
49#2658—21no—
50#2640—21no—
51#2568—21no—
52#2563C2211yescharacteristic
53#2547C2211yescharacteristic
54#2546—21no—
55#2546C2211yescharacteristic
56#2546C2211yescharacteristic
57#2545—21no—
58#2545—21no—
59#2441C2214yes—
60#2439C2214yes—
61#2423C2214yes—
62#2421C2214yes—
63#2327—21no—
64#2327—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2295C2211yescharacteristic
68#2087C2214yes—
69#2085C2214yes—
70#2084C2214yes—
71#2077C2214yes—
72#2074C2214yes—
73#2072C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2036C2214yes—
77#2035C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1971C2212yes—
84#1954C2211yescharacteristic
85#1944C2214yes—
86#1943C2212yes—
87#1935C2212yes—
88#1935C2212yes—
89#1924C2214yes—
90#1865C2214yes—
91#1863C2214yes—
92#1854C2214yes—
93#1854C2214yes—
94#1847C2214yes—
95#1847C2214yes—
96#1831C2211yescharacteristic
97#1807C2214yes—
98C4×#106C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#103C2211yescharacteristic
101#1752C2214yes—
102C4×#97C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C23×C8C2312yes—
105Q82—24no—
106Q82—42no—
107#557—24no—
108#557—42no—
109#556—24no—
110#556—24no—
111#556—42no—
112#556—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#126—22no—
126C2×#126—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124D411yescharacteristic
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×D4C2311yescharacteristic
145C2×#117C2312yes—
146C2×#117C2312yes—
147#501—28no—
148#500—24no—
149#500—42no—
150#498—24no—
151#498—42no—
152#485—28no—
153#485—28no—
154#479—28no—
155#477—24no—
156#475—24no—
157#475—24no—
158#475—24no—
159#475—28no—
160#467—28no—
161#446—24no—
162#441—28no—
163C4×Q16D418yes—
164C8×D4—24no—
165#427—28no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2314yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2314yes—
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#103C2311yescharacteristic
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2314yes—
191C2×#97C2312yes—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#400—24no—
197#399—24no—
198#399—24no—
199#398—22no—
200#398—22no—
201#398—24no—
202#397—24no—
203#397—22no—
204#396—24no—
205#396—24no—
206#393—22no—
207#393—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391—24no—
215#391D411yescharacteristic
216#391C2312yes—
217#391C2312yes—
218#390—21no—
219#390D411yescharacteristic
220#389—24no—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#389C2312yes—
225#388—24no—
226#388—24no—
227#388D414yes—
228#387—22no—
229#387—22no—
230#387—24no—
231#387—24no—
232#387—24no—
233#387—24no—
234#384—22no—
235#384C2314yes—
236#382—21no—
237#382D411yescharacteristic
238#382C2314yes—
239#380C2314yes—
240#380C2314yes—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4C2314yes—
246#376—21no—
247#376—21no—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4C2×D411yescharacteristic
267C23×C4C2412yes—
268C2×Q16—24no—
269C2×Q16C2×D414yes—
270C22×C8—24no—
271C22×C8—28no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2411yescharacteristic
275C22×C8C2412yes—
276C22×C8C2411yescharacteristic
277C4⋊Q8—24no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—42no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—48no—
291C4⋊Q8—44no—
292C4⋊Q8—82no—
293C4⋊Q8—82no—
294#126—48no—
295#126—48no—
296#125—24no—
297#125—42no—
298#125—48no—
299#124—42no—
300#124—44no—
301#124—44no—
302#124—82no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—42no—
306C22⋊Q8—42no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—42no—
314C4×Q8—42no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8C2×D414yes—
320C4×Q8C2×D414yes—
321C4×D4—22no—
322C4×D4—22no—
323#117—24no—
324#117—24no—
325#117—42no—
326#117—42no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D414yes—
365C2×C22⋊C4C2×D414yes—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380#107—24no—
381#107—24no—
382#107—24no—
383#107—44no—
384#107C2×D414yes—
385#106—24no—
386#106—24no—
387#106—24no—
388#106—24no—
389#106—24no—
390#106—24no—
391#106—44no—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103C2×D414yes—
402#103C2×D414yes—
403C22⋊C8—24no—
404C22⋊C8—28no—
405#97—24no—
406#97—24no—
407#97—24no—
408C4×C8—22no—
409C4×C8C2×D414yes—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—21no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24C22×D412yes—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8C22×D412yes—
450C2×Q8C22×D412yes—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476Q16—28no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—48no—
485C2×C8C22×D412yes—
486C2×C8C22×D412yes—
487C2×C82−1+414yes—
488C2×C82−1+414yes—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—82no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22×D412yes—
526C4⋊C4C22×D412yes—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42C22×D412yes—
552C42H(5,𝔽2)12yes—
553C23—24no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23C23×D411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—44no—
564Q8—42no—
565Q8—42no—
566Q8—84no—
567Q8—84no—
568Q8—84no—
569Q8—162no—
570Q8—164no—
571D4—44no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4D4212yes—
615C2×C4D4212yes—
616C2×C4#57512yes—
617C2×C4#57712yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+412yes—
621C2×C4C2×2−1+412yes—
622C8—22no—
623C8—24no—
624C8—42no—
625C8—42no—
626C8#53614yes—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22C2×D4211yescharacteristic
633C22#349912yes—
634C22C2×#57511yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358612yes—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4#327114yes—
654C4C2×#53611yescharacteristic
655C4#349911yescharacteristic
656C4#355212yes—
657C2—82no—
658C2#3936011yescharacteristic
659C2C2×#327111yescharacteristic
660C2#6108311yescharacteristic
661C2#6250712yes—
662C2#6252212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3387C2semidirect
5. #33314C2semidirect
6. #31109C2semidirect
7. #31107C2semidirect
8. #29867C2semidirect
9. #29866C2semidirect
11. #28683C2semidirect
12. #26070C2semidirect
13. #25840C2semidirect
14. #25837C2semidirect
17. #25043C2semidirect
19. #21278C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33872
C2×#33422
#397442
#391722
#333142
#311092
#311072
#298672
#298662
#287062
#286832
#260702
#258402
#258372
#257052
#257012
#250432
#216332
#212782