Finite Group Databasebeta

← #6804402

Subgroups — #6804402

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2#36574C211yesmaximal, characteristic, complemented
3#33286C211yesmaximal, characteristic
4#32626C214yesmaximal, complemented
5#31176C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic
7#30206C211yesmaximal, characteristic, complemented
8#29193C211yesmaximal, characteristic, complemented
9#29176C211yesmaximal, characteristic, complemented
10#28616C211yesmaximal, characteristic, complemented
11#28614C211yesmaximal, characteristic
12#28514C211yesmaximal, characteristic
13#28444C211yesmaximal, characteristic
14#26909C214yesmaximal, complemented
15#26189C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic
17#22221C211yesmaximal, characteristic, complemented
18#21691C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#478C2214yes—
24C2×#464C2214yes—
25C2×#441C2214yes—
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2592—21no—
33#2592C2212yes—
34#2584—21no—
35#2584—21no—
36#2581C2212yes—
37#2580—21no—
38#2513—21no—
39#2513—21no—
40#2509—21no—
41#2509—21no—
42#2508—21no—
43#2508—21no—
44#2453C2212yes—
45#2453C2212yes—
46#2448—21no—
47#2448C2212yes—
48#2447—21no—
49#2447C2212yes—
50#2370—21no—
51#2370—21no—
52#2367—21no—
53#2367C2212yes—
54#2357—21no—
55#2351C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C42×Q8C2211yescharacteristic
63#2092C2214yes—
64#2088C2214yes—
65#2074C2214yes—
66#2068C2214yes—
67#2056C2214yes—
68#2055C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2026C2214yes—
72#2015C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1985C2212yes—
77#1971C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1940C2212yes—
86#1936C2214yes—
87#1936C2214yes—
88#1928C2214yes—
89#1920C2214yes—
90#1917C2212yes—
91#1917C2212yes—
92#1880C2214yes—
93#1880C2214yes—
94#1878C2214yes—
95#1877C2214yes—
96#1863C2214yes—
97#1863C2214yes—
98#1845C2211yescharacteristic
99#1844C2211yescharacteristic
100#1834C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#124—21no—
117C2×#124—21no—
118C2×#124C2312yes—
119C2×#124C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131#494—24no—
132#492—24no—
133#487—24no—
134#478—48no—
135#464—48no—
136#441—48no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156#400—22no—
157#400—22no—
158#400—22no—
159#399—22no—
160#398—22no—
161#398—22no—
162#397—22no—
163#397—22no—
164#397—22no—
165#397—22no—
166#395—22no—
167#393—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#385—22no—
225#385—22no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382—22no—
235#382C2312yes—
236#381—22no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C4×C22⋊C4—22no—
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8C2×D411yescharacteristic
274C22×Q8C2412yes—
275C22×Q8C2412yes—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C2412yes—
279C23×C4—22no—
280C2×Q16—28no—
281C2×Q16—28no—
282C2×Q16—44no—
283C2×SD16—28no—
284C2×SD16—44no—
285C2×SD16—44no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8#4114yes—
296#127—22no—
297#127—22no—
298#125—44no—
299#125—44no—
300#125—44no—
301#124—44no—
302#124—44no—
303#124—44no—
304#124—82no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—42no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—82no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—82no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—44no—
390#106—28no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—42no—
394#103—28no—
395#103—44no—
396#103—44no—
397#103—44no—
398#103—44no—
399#102—28no—
400#102—44no—
401#102—44no—
402#97—24no—
403#97—42no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8—22no—
407C4×C8—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434C24—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8C2×#4114yes—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2511yescharacteristic
484Q16—416no—
485SD16—88no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8H(5,𝔽2)14yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—82no—
568C42—82no—
569C42—82no—
570C42—82no—
571C42C2×#4112yes—
572C422−1+412yes—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23C23×D411yescharacteristic
581Q8—28no—
582Q8—44no—
583Q8—44no—
584Q8—44no—
585Q8—44no—
586Q8—84no—
587D4—44no—
588D4—44no—
589D4—84no—
590D4—84no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—164no—
649C2×C4#54812yes—
650C2×C4#54812yes—
651C2×C4#57512yes—
652C2×C4#57512yes—
653C2×C4#57712yes—
654C2×C4#57812yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58512yes—
659C8—24no—
660C8—42no—
661C8—42no—
662C8—42no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22#344911yescharacteristic
671C22C2×#54811yescharacteristic
672C22#349811yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57711yescharacteristic
676C22C2×#57811yescharacteristic
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#337114yes—
697C4#337714yes—
698C4#351312yes—
699C4#355412yes—
700C2—162no—
701C2#3916111yescharacteristic
702C2#6097211yescharacteristic
703C2#6100811yescharacteristic
704C2C2×#337111yescharacteristic
705C2C2×#337711yescharacteristic
706C2#6242511yescharacteristic
707C2#6242711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36574C2semidirect
4. #32626C2semidirect
5. #31176C2semidirect
7. #30206C2semidirect
8. #29193C2semidirect
9. #29176C2semidirect
10. #28616C2semidirect
14. #26909C2semidirect
15. #26189C2semidirect
17. #22221C2semidirect
18. #21691C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365742
#332862
#326262
#311762
#302192
#302062
#291932
#291762
#286162
#286142
#285142
#284442
#269092
#261892
#222682
#222212
#216912
#213402