Finite Group Databasebeta

← #6493824

Subgroups — #6493824

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic, complemented
3#33287C211yesmaximal, characteristic, complemented
4#33286C211yesmaximal, characteristic, complemented
5#33232C211yesmaximal, characteristic, complemented
6#33230C211yesmaximal, characteristic, complemented
7#30220C211yesmaximal, characteristic, complemented
8#30219C211yesmaximal, characteristic, complemented
9#30205C211yesmaximal, characteristic, complemented
10#30204C211yesmaximal, characteristic, complemented
11#27774C214yesmaximal, complemented
12#27754C214yesmaximal, complemented
13#27267C214yesmaximal, complemented
14#27265C214yesmaximal, complemented
15#21864C211yesmaximal, characteristic, complemented
16#21805C211yesmaximal, characteristic
17#21400C211yesmaximal, characteristic, complemented
18#21195C211yesmaximal, characteristic
19#20473C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#532C2211yescharacteristic
22C2×#491C2214yes—
23C2×#489C2214yes—
24C2×#431C2211yescharacteristic
25C2×#416C2214yes—
26C22×C4×C8C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2858C2211yescharacteristic
35#2629—21no—
36#2628—21no—
37#2620—21no—
38#2620—21no—
39#2570—21no—
40#2570C2212yes—
41#2566—21no—
42#2566—21no—
43#2550—21no—
44#2550C2212yes—
45#2545C2212yes—
46#2545C2212yes—
47#2500—21no—
48#2500—21no—
49#2441—21no—
50#2441C2212yes—
51#2439—21no—
52#2439C2212yes—
53#2432C2212yes—
54#2432C2212yes—
55#2331—21no—
56#2330—21no—
57#2326—21no—
58#2326—21no—
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#2073C2214yes—
63#2072C2214yes—
64#2043C2214yes—
65#2043C2214yes—
66#2042C2214yes—
67#2042C2214yes—
68#2025C2214yes—
69#2024C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1983C2214yes—
75#1960C2214yes—
76#1959C2214yes—
77#1958C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1956C2211yescharacteristic
89#1956C2211yescharacteristic
90#1936C2214yes—
91#1936C2214yes—
92#1935C2214yes—
93#1935C2214yes—
94#1892C2214yes—
95#1873C2214yes—
96#1805C2214yes—
97#1788C2214yes—
98#1770C2212yes—
99#1770C2212yes—
100C4×#103C2214yes—
101#1752C2212yes—
102#1752C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×#41C2312yes—
108C23×C8C2312yes—
109#533—24no—
110#533—24no—
111#532—24no—
112#532—24no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#127C2311yescharacteristic
122C2×#125—21no—
123C2×#125—21no—
124C2×#125—21no—
125C2×#125C2312yes—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124C2312yes—
129C2×#124C2312yes—
130C2×#124C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C2312yes—
133C2×#121—21no—
134C2×#121—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×#117—21no—
140C2×#117—21no—
141C2×#117D411yescharacteristic
142C2×#117D411yescharacteristic
143C2×#117C2312yes—
144C22×C42C2311yescharacteristic
145#491—48no—
146#489—48no—
147#431—24no—
148C2×C4⋊C8C2312yes—
149#416—48no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8—24no—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173#400—22no—
174#399—22no—
175#398—22no—
176#398—22no—
177#396—22no—
178#396—22no—
179#396—22no—
180#396—22no—
181#396—22no—
182#396—22no—
183#395—22no—
184#394—22no—
185#394—22no—
186#394—22no—
187#394—22no—
188#393—22no—
189#391—21no—
190#391—21no—
191#391—22no—
192#391—22no—
193#391C2312yes—
194#391C2312yes—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390C2312yes—
202#390C2312yes—
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—22no—
212#389—22no—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#388—22no—
218#388—22no—
219#388—22no—
220#388—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#385—22no—
226#385—22no—
227#385—22no—
228#385—22no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384—21no—
235#384C2312yes—
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#382—21no—
241#382—21no—
242#382—21no—
243#382—21no—
244#382—21no—
245#382C2312yes—
246#382C2312yes—
247#380—21no—
248#380—21no—
249#380—21no—
250#380C2312yes—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263#376D411yescharacteristic
264#376D411yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C2×#41—24no—
276C2×#41—24no—
277C2×#41—44no—
278C2×#41—44no—
279C22×Q8—21no—
280C22×Q8—22no—
281C22×Q8C2412yes—
282C22×Q8C2412yes—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C22×C8—24no—
292C22×C8—24no—
293C22×C8—42no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308C4⋊Q8—82no—
309#127—22no—
310#127—22no—
311#125—44no—
312#125—44no—
313#125—44no—
314#125—82no—
315#124—24no—
316#124—24no—
317#124—44no—
318#124—44no—
319#124—44no—
320#124—44no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—44no—
324#121—44no—
325#121—44no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—44no—
329C4×D4—44no—
330C4×D4—44no—
331#117—42no—
332#117—42no—
333#117—44no—
334#117—44no—
335#117—82no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
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⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C22⋊C4C2×D411yescharacteristic
389C2×C22⋊C4C2×D411yescharacteristic
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
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×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42#4114yes—
404C2×C42#4114yes—
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C4⋊C8—24no—
412#103—28no—
413#103—28no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103—44no—
418#103—44no—
419#103—44no—
420#102—28no—
421#102—44no—
422#102—44no—
423#102—44no—
424C22⋊C8—24no—
425#97—24no—
426#97—24no—
427#97—42no—
428#97—42no—
429C4×C8—24no—
430C4×C8—22no—
431C4×C8—22no—
432C4×C8—22no—
433C4×C8—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#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455#95C2×D411yescharacteristic
456C24—22no—
457C24C22×D411yescharacteristic
458#41—416no—
459#41—88no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—24no—
467C2×Q8—24no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—82no—
473C2×Q8—82no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—24no—
477C2×D4—24no—
478C2×D4—44no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—82no—
486C2×D4—82no—
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—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4#12414yes—
504C22×C4#12414yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4C2511yescharacteristic
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—48no—
533C2×C8H(5,𝔽2)14yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C22⋊C4—41no—
573C22⋊C4—41no—
574C22⋊C4—41no—
575C22⋊C4—41no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—44no—
602C42—82no—
603C42—82no—
604C42#12414yes—
605C42#12414yes—
606C42C2×#4112yes—
607C42C2×#4112yes—
608C42C2×#4112yes—
609C42C2×#4112yes—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23C2×#12412yes—
624C23C23×D411yescharacteristic
625Q8—44no—
626Q8—44no—
627Q8—44no—
628Q8—44no—
629Q8—84no—
630Q8—164no—
631D4—44no—
632D4—44no—
633D4—88no—
634D4—84no—
635D4—84no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
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—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
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—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—88no—
692C2×C4C2×#12412yes—
693C2×C4C2×#12412yes—
694C2×C4C2×#12412yes—
695C2×C4C2×#12412yes—
696C2×C4C2×#12412yes—
697C2×C4#57512yes—
698C2×C4#57512yes—
699C2×C4#57712yes—
700C2×C4#57812yes—
701C2×C4C22×#4111yescharacteristic
702C2×C4C22×#4111yescharacteristic
703C2×C4C2×H(5,𝔽2)12yes—
704C2×C4#58512yes—
705C8—24no—
706C8—42no—
707C8—42no—
708C8—42no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22C22×#12411yescharacteristic
724C22#349811yescharacteristic
725C22#350011yescharacteristic
726C22C2×#57511yescharacteristic
727C22C2×#57511yescharacteristic
728C22C2×#57711yescharacteristic
729C22C2×#57811yescharacteristic
730C4—22no—
731C4—22no—
732C4—22no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#313514yes—
749C4#313714yes—
750C4#346012yes—
751C4#349412yes—
752C2—42no—
753C2—162no—
754C2#3526411yescharacteristic
755C2C2×#313511yescharacteristic
756C2C2×#313711yescharacteristic
757C2#6242511yescharacteristic
758C2#6242711yescharacteristic
759C2#6254311yescharacteristic
760C2#6254511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. #35294C2semidirect
3. #33287C2semidirect
4. #33286C2semidirect
5. #33232C2semidirect
6. #33230C2semidirect
7. #30220C2semidirect
8. #30219C2semidirect
9. #30205C2semidirect
10. #30204C2semidirect
11. #27774C2semidirect
12. #27754C2semidirect
13. #27267C2semidirect
14. #27265C2semidirect
15. #21864C2semidirect
17. #21400C2semidirect
19. #20473C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#352942
#332872
#332862
#332322
#332302
#302202
#302192
#302052
#302042
#277742
#277542
#272672
#272652
#218642
#218052
#214002
#211952
#204732