Finite Group Databasebeta

← #6630910

Subgroups — #6630910

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33283C214yesmaximal, complemented
6#33238C211yesmaximal, characteristic, complemented
7#33226C211yesmaximal, characteristic, complemented
8#33049C211yesmaximal, characteristic, complemented
9#33048C211yesmaximal, characteristic
10#33047C214yesmaximal, complemented
11#33043C211yesmaximal, characteristic, complemented
12#33042C211yesmaximal, characteristic
13#21647C211yesmaximal, characteristic, complemented
14#21646C211yesmaximal, characteristic, complemented
15#21643C214yesmaximal, complemented
16C4×#487C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18#21412C214yesmaximal, complemented
19C4×#405C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#488C2214yes—
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#405C2212yes—
27C2×#405C2212yes—
28C2×C4×M16C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2868C2211yescharacteristic
31#2868C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2856C2211yescharacteristic
38#2647C2212yes—
39#2640C2212yes—
40#2629C2212yes—
41#2620C2212yes—
42#2512—21no—
43#2512—21no—
44#2507—21no—
45#2507—21no—
46#2497—21no—
47#2497—21no—
48#2491—21no—
49#2491—21no—
50#2452—21no—
51#2452—21no—
52#2446—21no—
53#2446—21no—
54#2429—21no—
55#2429—21no—
56#2422—21no—
57#2422—21no—
58#2330C2212yes—
59#2326C2212yes—
60#2314C2212yes—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×#127C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#1993C2214yes—
67#1992C2214yes—
68#1990C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1959C2214yes—
74#1959C2214yes—
75#1958C2214yes—
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1936C2214yes—
81#1936C2214yes—
82#1935C2214yes—
83#1935C2214yes—
84#1934C2214yes—
85#1934C2214yes—
86#1933C2214yes—
87#1933C2214yes—
88#1895C2214yes—
89#1894C2214yes—
90#1893C2214yes—
91#1892C2214yes—
92#1834C2214yes—
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1763C2214yes—
98#1762C2214yes—
99C4×#103C2214yes—
100C4×#102C2214yes—
101#1726C2212yes—
102#1726C2212yes—
103C4×#97C2211yescharacteristic
104C4×#97C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108C23×C8C2312yes—
109#542—216no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#127C2311yescharacteristic
119C2×#126—22no—
120C2×#125C2312yes—
121C2×#124—22no—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×#123—22no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—22no—
128C2×C22⋊Q8C2312yes—
129C2×#121—21no—
130C2×#121—21no—
131C2×#121C2312yes—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#488—216no—
142#487—24no—
143#487—24no—
144#486—24no—
145#486—24no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154#405—28no—
155#405—28no—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×C4×C8—24no—
163C2×C4×C8C2312yes—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2312yes—
172#400—22no—
173#400—22no—
174#398—22no—
175#398—22no—
176#398—22no—
177#398—22no—
178#397—22no—
179#397—22no—
180#396—22no—
181#396—22no—
182#396—22no—
183#396—22no—
184#394—22no—
185#394—22no—
186#393—22no—
187#393—22no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391—21no—
192#391C2312yes—
193#391C2312yes—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390C2312yes—
203#390C2312yes—
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#389—21no—
216#389—22no—
217#389—22no—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
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#384—21no—
233#384—21no—
234#384—21no—
235#384—21no—
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#384C2312yes—
241#384C2312yes—
242#384C2312yes—
243#382C2312yes—
244#380—21no—
245#380—21no—
246#380—21no—
247#380—21no—
248#380C2312yes—
249#380C2312yes—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2312yes—
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264#376C2311yescharacteristic
265#376C2311yescharacteristic
266C43C2311yescharacteristic
267C43C2311yescharacteristic
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8C2412yes—
280C22×D4—22no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4C2412yes—
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C2×M16—24no—
289C2×M16—24no—
290C2×M16—24no—
291C22×C8—24no—
292C22×C8—24no—
293C22×C8—24no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—28no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—82no—
306C4⋊Q8—82no—
307C4⋊Q8—82no—
308C4⋊Q8—82no—
309#127—22no—
310#127—22no—
311#126—48no—
312#125—28no—
313#124—28no—
314#124—28no—
315#124—48no—
316#123—48no—
317C22⋊Q8—28no—
318C22⋊Q8—48no—
319C22⋊Q8—82no—
320C22⋊Q8—82no—
321#121—28no—
322#121—82no—
323#121—82no—
324C4×Q8—28no—
325C4×Q8—82no—
326C4×Q8—82no—
327C4×D4—28no—
328C4×D4—82no—
329C4×D4—82no—
330#117—24no—
331#117#4118yes—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#103—28no—
408#103—28no—
409#103—28no—
410#103—28no—
411#102—28no—
412#102—28no—
413#102—28no—
414#102—28no—
415#97—24no—
416#97—24no—
417#97—24no—
418#97—24no—
419#97—24no—
420C4×C8—24no—
421C4×C8—24no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—24no—
425C4×C8—24no—
426C4×C8#4114yes—
427C4×C8#4114yes—
428C4×C8#4118yes—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451C24—22no—
452C24C22×D411yescharacteristic
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—24no—
457C2×Q8—44no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—82no—
465C2×Q8—82no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—24no—
470C2×D4—44no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—82no—
478C2×D4—82no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—44no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4114yes—
512C22×C4C2511yescharacteristic
513M16—216no—
514C2×C8—216no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8C2×#4114yes—
524C2×C8C2×#4114yes—
525C2×C8C2×#4114yes—
526C2×C8C2×#4114yes—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4#12414yes—
565C4⋊C4#12414yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4#12414yes—
582C22⋊C4#12414yes—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—21no—
594C42—22no—
595C42—21no—
596C42—44no—
597C42—82no—
598C42—82no—
599C42C2×#4114yes—
600C42C2×#4114yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C42C2×#4114yes—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23C2×#12412yes—
618C23C23×D411yescharacteristic
619Q8—44no—
620Q8—44no—
621Q8—84no—
622Q8—84no—
623Q8—164no—
624D4—44no—
625D4—44no—
626D4—84no—
627D4—84no—
628D4—164no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—164no—
681C2×C4C2×#12412yes—
682C2×C4C2×#12412yes—
683C2×C4C2×#12412yes—
684C2×C4C2×#12412yes—
685C2×C4C2×#12412yes—
686C2×C4#53212yes—
687C2×C4#53212yes—
688C2×C4#53212yes—
689C2×C4#53212yes—
690C2×C4#57612yes—
691C2×C4#57712yes—
692C2×C4#57712yes—
693C2×C4#57812yes—
694C2×C4C22×#4112yes—
695C2×C4C22×#4111yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4112yes—
698C8—24no—
699C8—24no—
700C8—24no—
701C8—24no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22C22×#12411yescharacteristic
717C22C2×#53211yescharacteristic
718C22C2×#53211yescharacteristic
719C22C2×#57611yescharacteristic
720C22C2×#57711yescharacteristic
721C22C2×#57711yescharacteristic
722C22C2×#57811yescharacteristic
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4#317814yes—
742C4#317914yes—
743C4#344412yes—
744C4#344412yes—
745C2—42no—
746C2—162no—
747C2#3663911yescharacteristic
748C2#6062211yescharacteristic
749C2#6062311yescharacteristic
750C2C2×#317811yescharacteristic
751C2C2×#317911yescharacteristic
752C2#6071611yescharacteristic
753C2#6071811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #33297C2semidirect
4. #33289C2semidirect
5. #33283C2semidirect
6. #33238C2semidirect
7. #33226C2semidirect
8. #33049C2semidirect
10. #33047C2semidirect
11. #33043C2semidirect
13. #21647C2semidirect
14. #21646C2semidirect
15. #21643C2semidirect
16. C4×#487C2semidirect
17. C4×#486C2semidirect
18. #21412C2semidirect
19. C4×#405C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#352772
#332972
#332892
#332832
#332382
#332262
#330492
#330482
#330472
#330432
#330422
#216472
#216462
#216432
C4×#4872
C4×#4862
#214122
C4×#4052