Finite Group Databasebeta

← #6683857

Subgroups — #6683857

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic
2C2×#3369C211yesmaximal, characteristic
3#39490C211yesmaximal, characteristic
4Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
5#33027C211yesmaximal, characteristic, complemented
6#30965C214yesmaximal, complemented
7#30821C214yesmaximal
8#30279C211yesmaximal, characteristic, complemented
9#29868C211yesmaximal, characteristic
10#28682C211yesmaximal, characteristic, complemented
11#28647C211yesmaximal, characteristic, complemented
12#26395C214yesmaximal, complemented
13#26177C214yesmaximal
14#25936C211yesmaximal, characteristic, complemented
15#25841C211yesmaximal, characteristic, complemented
16#25839C211yesmaximal, characteristic
17#25646C211yesmaximal, characteristic
18#22242C211yesmaximal, characteristic, complemented
19#21115C211yesmaximal, characteristic
20C2×Q82C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3385—28no—
25#3369—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#489C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2212yes—
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#474C2212yes—
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×#446C2211yescharacteristic
42C2×#441C2212yes—
43C2×#440C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2868—21no—
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2689—24no—
49#2621—21no—
50#2568C2211yescharacteristic
51#2567—21no—
52#2567—21no—
53#2567C2211yescharacteristic
54#2567C2211yescharacteristic
55#2567C2211yescharacteristic
56#2551—24no—
57#2551C2214yes—
58#2503—21no—
59#2499—21no—
60#2453C2214yes—
61#2447C2214yes—
62Q8×C4⋊C4C2214yes—
63#2317—21no—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2086C2214yes—
71#2084C2214yes—
72#2071C2214yes—
73#2066C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2042C2214yes—
77#2024C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1944C2214yes—
85#1935C2214yes—
86#1928C2214yes—
87#1926C2212yes—
88#1919C2212yes—
89#1917C2214yes—
90#1917C2214yes—
91#1880C2212yes—
92#1878C2214yes—
93#1878C2214yes—
94#1865C2212yes—
95#1865C2214yes—
96#1863C2214yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×SD16C2311yescharacteristic
105Q82—24no—
106Q82—24no—
107Q82—42no—
108Q82—42no—
109#556—24no—
110#556—42no—
111#554—24no—
112#554—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—24no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2312yes—
126C2×#126—22no—
127C2×#125C2312yes—
128C2×#124C2312yes—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—22no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×#117—21no—
144C2×#117—21no—
145C22×C4⋊C4—21no—
146#501—28no—
147#500—24no—
148#500—42no—
149#489—28no—
150#487—24no—
151#487—42no—
152#479—28no—
153#477—24no—
154#476—24no—
155#476—24no—
156#475—28no—
157#474—28no—
158#464—28no—
159#462—24no—
160#446—24no—
161#446—24no—
162#441—28no—
163#440—28no—
164C4×SD16D418yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2314yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2314yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2311yescharacteristic
189C2×#97C2312yes—
190C2×#97C2312yes—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193#400—22no—
194#400—24no—
195#398—24no—
196#397—24no—
197#397—24no—
198#395—22no—
199#395—22no—
200#395—22no—
201#395—24no—
202#395—24no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—22no—
208#391D411yescharacteristic
209#391D411yescharacteristic
210#391C2314yes—
211#391C2314yes—
212#390C2312yes—
213#389—24no—
214#389—24no—
215#389—24no—
216#389—24no—
217#389D414yes—
218#389C2314yes—
219#389C2314yes—
220#389C2314yes—
221#389C2312yes—
222#388—22no—
223#387—24no—
224#385—24no—
225#384—21no—
226#384—21no—
227#384—21no—
228#384—24no—
229#384—21no—
230#384—21no—
231#384D411yescharacteristic
232#384C2314yes—
233#382—21no—
234#382—21no—
235#382—21no—
236#381—24no—
237C4×C4⋊C4—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2314yes—
244C4×C22⋊C4—21no—
245C43—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—22no—
270C2×Q16—24no—
271C2×Q16—28no—
272C2×SD16—24no—
273C2×SD16C2×D414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—42no—
291C4⋊Q8—42no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—48no—
295C4⋊Q8—48no—
296C4⋊Q8—84no—
297C4⋊Q8—88no—
298#126—48no—
299#125—24no—
300#125—42no—
301#124—24no—
302#124—42no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8C2×D414yes—
322C4×D4—24no—
323C4×D4—22no—
324C4×D4—42no—
325C4×D4C2×D414yes—
326#117—44no—
327#117—44no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—24no—
385#107—24no—
386#107—44no—
387#106—24no—
388#106—24no—
389#106—24no—
390#106C2×D414yes—
391C4⋊C8—24no—
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—28no—
401#103—44no—
402#103C2×D414yes—
403#102—24no—
404#102—24no—
405#102—24no—
406#102C2×D414yes—
407#97—24no—
408#97—24no—
409C4×C8—22no—
410C4×C8C2×D414yes—
411#95—22no—
412#95—21no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—21no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95C2×D414yes—
423#95C2×D414yes—
424C24—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—82no—
443C2×Q8—82no—
444C2×Q8—84no—
445C2×Q8—84no—
446C2×Q8—82no—
447C2×Q8C22×D412yes—
448C2×Q82−1+414yes—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4C22×D412yes—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—84no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476Q16—48no—
477SD16—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8C22×D412yes—
488C2×C8C22×D412yes—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—21no—
497C4⋊C4—21no—
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—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4C22×D412yes—
527C4⋊C4C22×D412yes—
528C4⋊C42−1+414yes—
529C22⋊C4—24no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C42—28no—
536C42—24no—
537C42—22no—
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—
551C42#13714yes—
552C42C22×D412yes—
553C42H(5,𝔽2)12yes—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23C23×D411yescharacteristic
560Q8—22no—
561Q8—24no—
562Q8—44no—
563Q8—44no—
564Q8—42no—
565Q8—42no—
566Q8—44no—
567Q8—162no—
568Q8—162no—
569Q8—162no—
570Q8—164no—
571Q8—164no—
572Q8#53614yes—
573D4—22no—
574D4—22no—
575D4—42no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—164no—
615C2×C4D4212yes—
616C2×C4D4212yes—
617C2×C4C2×#13714yes—
618C2×C4C2×#13712yes—
619C2×C4#57812yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+412yes—
623C2×C4C2×2−1+412yes—
624C8—22no—
625C8—22no—
626C8—42no—
627C8—42no—
628C8—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22C2×D4211yescharacteristic
635C22#348612yes—
636C22C22×#13711yescharacteristic
637C22C2×#57811yescharacteristic
638C22#358512yes—
639C4—24no—
640C4—24no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4#327614yes—
656C4C2×#53611yescharacteristic
657C4#349911yescharacteristic
658C4#355212yes—
659C2—82no—
660C2#3767511yescharacteristic
661C2C2×#327611yescharacteristic
662C2#6108211yescharacteristic
663C2#6183012yes—
664C2#6185912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. Q8×C4⋊Q8C2semidirect
5. #33027C2semidirect
6. #30965C2semidirect
8. #30279C2semidirect
10. #28682C2semidirect
11. #28647C2semidirect
12. #26395C2semidirect
14. #25936C2semidirect
15. #25841C2semidirect
18. #22242C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33692
#394902
Q8×C4⋊Q82
#330272
#309652
#308212
#302792
#298682
#286822
#286472
#263952
#261772
#259362
#258412
#258392
#256462
#222422
#211152