Finite Group Databasebeta

← #6804560

Subgroups — #6804560

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 694 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#32592C214yesmaximal, complemented
5#32489C214yesmaximal, complemented
6#30229C211yesmaximal, characteristic, complemented
7#30204C211yesmaximal, characteristic
8#29207C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic
10#29176C211yesmaximal, characteristic, complemented
11#28688C211yesmaximal, characteristic, complemented
12#28655C211yesmaximal, characteristic
13#28654C211yesmaximal, characteristic
14#27385C214yesmaximal, complemented
15#27157C214yesmaximal, complemented
16#22247C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic
18#21727C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23#2862C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2858C2211yescharacteristic
28#2858C2211yescharacteristic
29#2592—21no—
30#2592—21no—
31#2584—21no—
32#2584C2212yes—
33#2581—21no—
34#2580C2212yes—
35#2513C2212yes—
36#2513C2212yes—
37#2509—21no—
38#2509C2212yes—
39#2508—21no—
40#2508C2212yes—
41#2453—21no—
42#2453—21no—
43#2448—21no—
44#2448—21no—
45#2447—21no—
46#2447—21no—
47#2370—21no—
48#2370C2212yes—
49#2367—21no—
50#2367—21no—
51#2357C2212yes—
52#2351—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2101C2214yes—
62#2088C2214yes—
63#2085C2214yes—
64#2073C2214yes—
65#2067C2214yes—
66#2067C2214yes—
67#2065C2214yes—
68#2059C2214yes—
69#2025C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1985C2212yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1949C2214yes—
82#1949C2214yes—
83#1944C2214yes—
84#1940C2212yes—
85#1936C2214yes—
86#1936C2214yes—
87#1926C2214yes—
88#1926C2214yes—
89#1925C2214yes—
90#1920C2214yes—
91#1918C2214yes—
92#1918C2214yes—
93#1917C2212yes—
94#1917C2212yes—
95#1864C2214yes—
96#1862C2214yes—
97#1845C2211yescharacteristic
98#1844C2211yescharacteristic
99#1834C2211yescharacteristic
100#1776C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#124—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129#500—24no—
130#498—24no—
131#486—24no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#400—22no—
150#400—22no—
151#400—22no—
152#399—22no—
153#398—22no—
154#398—22no—
155#397—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#395—22no—
160#393—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#385—22no—
218#385—22no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#382—21no—
224#382—21no—
225#382—22no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#381—22no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4D411yescharacteristic
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C22⋊C4—22no—
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#336C2312yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269C22×Q8—21no—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8C2×D411yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2412yes—
276C23×C4—22no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8#4114yes—
287#127—22no—
288#127—22no—
289#125—44no—
290#125—44no—
291#125—82no—
292#124—44no—
293#124—44no—
294#124—82no—
295#124—82no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—44no—
300C4×Q8—82no—
301C4×Q8—82no—
302C4×Q8—82no—
303C4×Q8—82no—
304C4×D4—44no—
305C4×D4—82no—
306C4×D4—82no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—44no—
382#106—28no—
383#106—44no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—42no—
387#102—28no—
388#102—28no—
389#102—44no—
390#102—44no—
391#97—24no—
392#97—42no—
393C4×C8—22no—
394C4×C8—22no—
395C4×C8—22no—
396C4×C8—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—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#95C2×D411yescharacteristic
418#95C2×D411yescharacteristic
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423C24—22no—
424C2×Q8—24no—
425C2×Q8—41no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C82−1+414yes—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4C2×#4114yes—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—44no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42C2×#4112yes—
559C422−1+412yes—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C23×D411yescharacteristic
568Q8—44no—
569Q8—48no—
570Q8—84no—
571Q8—84no—
572D4—44no—
573D4—44no—
574D4—84no—
575D4—84no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
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—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—164no—
634C2×C4#54812yes—
635C2×C4#54812yes—
636C2×C4#57512yes—
637C2×C4#57512yes—
638C2×C4#57612yes—
639C2×C4#57712yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×2−1+412yes—
642C2×C4C2×2−1+411yescharacteristic
643C2×C4#58512yes—
644C8—24no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22#344911yescharacteristic
656C22C2×#54811yescharacteristic
657C22#349911yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57611yescharacteristic
661C22C2×#57711yescharacteristic
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#337014yes—
682C4#337114yes—
683C4#351612yes—
684C4#355512yes—
685C2—162no—
686C2#3916211yescharacteristic
687C2#6097211yescharacteristic
688C2#6100611yescharacteristic
689C2C2×#337011yescharacteristic
690C2C2×#337111yescharacteristic
691C2#6248111yescharacteristic
692C2#6248311yescharacteristic
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. #32592C2semidirect
5. #32489C2semidirect
6. #30229C2semidirect
8. #29207C2semidirect
10. #29176C2semidirect
11. #28688C2semidirect
14. #27385C2semidirect
15. #27157C2semidirect
16. #22247C2semidirect
18. #21727C2semidirect
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
#325922
#324892
#302292
#302042
#292072
#291922
#291762
#286882
#286552
#286542
#273852
#271572
#222472
#222192
#217272
#213402