Finite Group Databasebeta

← #6624309

Subgroups — #6624309

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2#36574C211yesmaximal, characteristic, complemented
3#33290C211yesmaximal, characteristic, complemented
4#33286C211yesmaximal, characteristic
5#33286C211yesmaximal, characteristic
6#29178C211yesmaximal, characteristic, complemented
7#29176C211yesmaximal, characteristic, complemented
8#28688C211yesmaximal, characteristic, complemented
9#28655C211yesmaximal, characteristic
10#28654C211yesmaximal, characteristic
11#28514C211yesmaximal, characteristic
12#28051C214yesmaximal, complemented
13#28049C214yesmaximal, complemented
14#25475C214yesmaximal, complemented
15#23071C214yesmaximal, complemented
16#21727C211yesmaximal, characteristic, complemented
17#21726C211yesmaximal, characteristic, complemented
18C4×#486C211yesmaximal, characteristic
19C4×#446C211yesmaximal, characteristic, complemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#478C2214yes—
24C2×#446C2212yes—
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2858C2211yescharacteristic
30#2858C2211yescharacteristic
31#2592—21no—
32#2592—21no—
33#2584—21no—
34#2584C2212yes—
35#2581—21no—
36#2580C2212yes—
37#2513C2212yes—
38#2513C2212yes—
39#2509—21no—
40#2509C2212yes—
41#2508—21no—
42#2508C2212yes—
43#2453—21no—
44#2453—21no—
45#2448—21no—
46#2448—21no—
47#2447—21no—
48#2447—21no—
49#2370—21no—
50#2370C2212yes—
51#2367—21no—
52#2367—21no—
53#2357C2212yes—
54#2351—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2092C2214yes—
63#2088C2214yes—
64#2086C2214yes—
65#2085C2214yes—
66#2067C2214yes—
67#2065C2214yes—
68#1991C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1949C2214yes—
80#1944C2214yes—
81#1943C2214yes—
82#1936C2214yes—
83#1936C2214yes—
84#1936C2214yes—
85#1936C2214yes—
86#1934C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1920C2214yes—
90#1918C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1835C2214yes—
94#1834C2211yescharacteristic
95#1776C2214yes—
96#1776C2214yes—
97#1771C2212yes—
98#1771C2212yes—
99#1770C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#102C2214yes—
104C4×#97C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
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—
132#478—216no—
133#446—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2312yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155#400—22no—
156#400—22no—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#397—22no—
162#397—22no—
163#397—22no—
164#397—22no—
165#395—22no—
166#393—22no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—21no—
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#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#385—22no—
224#385—22no—
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#382—21no—
230#382—21no—
231#382—22no—
232#382C2312yes—
233#382C2312yes—
234#382C2312yes—
235#381—22no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C4×C22⋊C4—22no—
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8C2×D411yescharacteristic
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8#4114yes—
289#127—22no—
290#127—22no—
291#125—28no—
292#125—44no—
293#125—82no—
294#124—44no—
295#124—44no—
296#124—82no—
297#124—82no—
298C4×Q8—24no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×Q8—82no—
303C4×Q8—82no—
304C4×Q8—82no—
305C4×Q8—82no—
306C4×D4—28no—
307C4×D4—82no—
308C4×D4—82no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
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—22no—
330C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
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×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—28no—
383#107—28no—
384#106—28no—
385#106—28no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388C4⋊C8#4118yes—
389#102—28no—
390#102—28no—
391#102—28no—
392#102—28no—
393#97—24no—
394#97—24no—
395#97—24no—
396#97—24no—
397C4×C8—24no—
398C4×C8—24no—
399C4×C8—22no—
400C4×C8—22no—
401C4×C8—22no—
402C4×C8—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#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429C24—22no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—82no—
461C22×C4—82no—
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×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2511yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8C2×#4114yes—
487C2×C8C2×#4114yes—
488C2×C82−1+414yes—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4C2×#4114yes—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
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—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—82no—
560C42—82no—
561C42—82no—
562C42—82no—
563C42C2×#4114yes—
564C42C2×#4112yes—
565C422−1+412yes—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23C23×D411yescharacteristic
574Q8—44no—
575Q8—48no—
576Q8—84no—
577Q8—84no—
578D4—44no—
579D4—44no—
580D4—84no—
581D4—84no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
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—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
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—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—164no—
638C2×C4#53212yes—
639C2×C4#53212yes—
640C2×C4#54812yes—
641C2×C4#54812yes—
642C2×C4#57512yes—
643C2×C4#57512yes—
644C2×C4#57612yes—
645C2×C4#57712yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4112yes—
648C2×C4C2×2−1+412yes—
649C2×C4C2×2−1+411yescharacteristic
650C8—24no—
651C8—24no—
652C8—24no—
653C8—42no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22C2×#53211yescharacteristic
662C22C2×#54811yescharacteristic
663C22#349911yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57611yescharacteristic
667C22C2×#57711yescharacteristic
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#337014yes—
688C4#337114yes—
689C4#348212yes—
690C4#352312yes—
691C2—162no—
692C2#3650811yescharacteristic
693C2#6069311yescharacteristic
694C2#6072211yescharacteristic
695C2C2×#337011yescharacteristic
696C2C2×#337111yescharacteristic
697C2#6248111yescharacteristic
698C2#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
3. #33290C2semidirect
6. #29178C2semidirect
7. #29176C2semidirect
8. #28688C2semidirect
12. #28051C2semidirect
13. #28049C2semidirect
14. #25475C2semidirect
15. #23071C2semidirect
16. #21727C2semidirect
17. #21726C2semidirect
19. C4×#446C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365742
#332902
#332862
#291782
#291762
#286882
#286552
#286542
#285142
#280512
#280492
#254752
#230712
#217272
#217262
C4×#4862
C4×#4462