Finite Group Databasebeta

← #6623862

Subgroups — #6623862

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2#36574C211yesmaximal, characteristic, complemented
3#33287C211yesmaximal, characteristic, complemented
4#33286C211yesmaximal, characteristic
5#33226C211yesmaximal, characteristic
6#29207C211yesmaximal, characteristic, complemented
7#29193C211yesmaximal, characteristic, complemented
8#29190C211yesmaximal, characteristic
9#28616C211yesmaximal, characteristic, complemented
10#28614C211yesmaximal, characteristic
11#28444C211yesmaximal, characteristic
12#26839C214yesmaximal, complemented
13#26833C214yesmaximal, complemented
14#25299C214yesmaximal, complemented
15#23006C214yesmaximal, complemented
16#21692C211yesmaximal, characteristic
17#21691C211yesmaximal, characteristic, complemented
18C4×#487C211yesmaximal, characteristic, complemented
19C4×#446C211yesmaximal, characteristic, complemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#488C2214yes—
23C2×#487C2211yescharacteristic
24C2×#464C2214yes—
25C2×#446C2212yes—
26C2×#441C2214yes—
27C2×#440C2214yes—
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2858C2211yescharacteristic
34#2592—21no—
35#2592C2212yes—
36#2584—21no—
37#2584—21no—
38#2581C2212yes—
39#2580—21no—
40#2513—21no—
41#2513—21no—
42#2509—21no—
43#2509—21no—
44#2508—21no—
45#2508—21no—
46#2453C2212yes—
47#2453C2212yes—
48#2448—21no—
49#2448C2212yes—
50#2447—21no—
51#2447C2212yes—
52#2370—21no—
53#2370—21no—
54#2367—21no—
55#2367C2212yes—
56#2357—21no—
57#2351C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65#2068C2214yes—
66#2065C2214yes—
67#2059C2214yes—
68#2056C2214yes—
69#2055C2214yes—
70#2015C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1973C2214yes—
75#1971C2214yes—
76#1958C2214yes—
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1949C2214yes—
85#1936C2214yes—
86#1936C2214yes—
87#1935C2214yes—
88#1935C2214yes—
89#1933C2214yes—
90#1880C2214yes—
91#1878C2214yes—
92#1864C2214yes—
93#1863C2214yes—
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1771C2212yes—
98#1771C2212yes—
99#1770C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#103C2214yes—
104C4×#97C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
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#488—48no—
134#487—24no—
135#464—216no—
136#446—28no—
137#441—48no—
138#440—48no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162#400—22no—
163#400—22no—
164#400—22no—
165#399—22no—
166#398—22no—
167#398—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#395—22no—
173#393—22no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
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#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#388—22no—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#388—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#385—22no—
231#385—22no—
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#382—21no—
237#382—21no—
238#382—21no—
239#382—21no—
240#382—22no—
241#382C2312yes—
242#381—22no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—22no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4C2312yes—
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C4×C4⋊C4C2311yescharacteristic
263C4×C22⋊C4—22no—
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—21no—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×Q8C2412yes—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C2×Q16—28no—
283C2×Q16—28no—
284C2×Q16—44no—
285C2×SD16—28no—
286C2×SD16—28no—
287C2×SD16—44no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8#4114yes—
298#127—22no—
299#127—22no—
300#125—44no—
301#125—44no—
302#125—44no—
303#124—28no—
304#124—44no—
305#124—44no—
306#124—82no—
307C4×Q8—24no—
308C4×Q8—28no—
309C4×Q8—42no—
310C4×Q8—42no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—82no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—82no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
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⋊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⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
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×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—44no—
392#106—44no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8#4118yes—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—44no—
401#102—28no—
402#102—28no—
403#102—44no—
404#97—24no—
405#97—24no—
406#97—24no—
407#97—24no—
408C4×C8—24no—
409C4×C8—24no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—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#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440C24—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8C2×#4114yes—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—82no—
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×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2511yescharacteristic
490Q16—416no—
491SD16—416no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8C2×#4114yes—
502C2×C8C2×#4114yes—
503C2×C8H(5,𝔽2)14yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
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—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—82no—
573C42—82no—
574C42—82no—
575C42—82no—
576C42C2×#4114yes—
577C42C2×#4112yes—
578C422−1+412yes—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—84no—
593D4—44no—
594D4—44no—
595D4—84no—
596D4—84no—
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—24no—
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—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
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—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—164no—
653C2×C4#53212yes—
654C2×C4#53212yes—
655C2×C4#54812yes—
656C2×C4#54812yes—
657C2×C4#57512yes—
658C2×C4#57512yes—
659C2×C4#57712yes—
660C2×C4#57812yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4112yes—
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×2−1+411yescharacteristic
665C8—24no—
666C8—24no—
667C8—24no—
668C8—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#53211yescharacteristic
677C22C2×#54811yescharacteristic
678C22#349811yescharacteristic
679C22C2×#57511yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57711yescharacteristic
682C22C2×#57811yescharacteristic
683C4—22no—
684C4—22no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—44no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#337114yes—
703C4#337714yes—
704C4#348112yes—
705C4#352212yes—
706C2—162no—
707C2#3649611yescharacteristic
708C2#6069311yescharacteristic
709C2#6072311yescharacteristic
710C2C2×#337111yescharacteristic
711C2C2×#337711yescharacteristic
712C2#6242511yescharacteristic
713C2#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
3. #33287C2semidirect
6. #29207C2semidirect
7. #29193C2semidirect
9. #28616C2semidirect
12. #26839C2semidirect
13. #26833C2semidirect
14. #25299C2semidirect
15. #23006C2semidirect
17. #21691C2semidirect
18. C4×#487C2semidirect
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
#332872
#332862
#332262
#292072
#291932
#291902
#286162
#286142
#284442
#268392
#268332
#252992
#230062
#216922
#216912
C4×#4872
C4×#4462