Finite Group Databasebeta

← #6444633

Subgroups — #6444633

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3083C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic, complemented
3#35640C211yesmaximal, characteristic, complemented
4#34265C211yesmaximal, characteristic
5#26479C214yesmaximal, complemented
6#26475C214yesmaximal, complemented
7#26468C214yesmaximal, complemented
8#26467C214yesmaximal, complemented
9#23612C211yesmaximal, characteristic, complemented
10#23609C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23453C211yesmaximal, characteristic, complemented
13#21263C211yesmaximal, characteristic
14#21206C211yesmaximal, characteristic
15#21174C211yesmaximal, characteristic, complemented
16#21147C211yesmaximal, characteristic, complemented
17#20882C211yesmaximal, characteristic, complemented
18#20873C211yesmaximal, characteristic, complemented
19#20455C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#3083—28no—
26C2×#479C2212yescomplemented
27C2×#477C2212yescomplemented
28C2×#478—28no—
29C2×#478C2214yes—
30C2×#478C2214yes—
31C2×#478C2212yescomplemented
32C2×#474C2212yescomplemented
33C2×#428C2211yescharacteristic
34C2×#427C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C2×#418C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#416C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C22×#102C2211yescharacteristic
48C2×#405C2211yescharacteristic
49#2583—22no—
50#2552—22no—
51#2552C2214yescomplemented
52#2550—22no—
53#2550C2214yescomplemented
54#2502—22no—
55#2367C2214yes—
56#2366C2214yes—
57#2363—22no—
58#2361—22no—
59#2321—22no—
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2309C2211yescharacteristic
63C4×#120—22no—
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2274C2211yescharacteristic
68#2274C2211yescharacteristic
69#2068C2212yescomplemented
70#2067C2214yes—
71#2067C2212yescomplemented
72#2066C2212yescomplemented
73#2065C2214yes—
74#2065C2212yescomplemented
75#2026C2214yescomplemented
76#2025C2214yes—
77#2025C2214yescomplemented
78#2023C2214yes—
79#1983C2214yes—
80#1982C2214yes—
81#1961C2212yes—
82#1959C2212yes—
83#1958C2212yes—
84#1957C2212yes—
85#1949C2214yescomplemented
86#1948C2214yescomplemented
87#1940C2214yescomplemented
88#1939C2214yescomplemented
89#1865C2214yescomplemented
90#1864C2214yes—
91#1864C2214yescomplemented
92#1862C2214yes—
93#1776C2214yescomplemented
94C4×#107C2214yescomplemented
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110#535—24no—
111#535Q818yescomplemented
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×#127C2312yes—
115C2×#125—21no—
116C2×#125—24no—
117C2×#125—22no—
118C2×#125—42no—
119C2×#125C2314yes—
120C2×#125C2312yes—
121C2×#124C2312yes—
122C2×C22⋊Q8—22no—
123C2×#121—22no—
124C2×#120—22no—
125#520—24no—
126#520—42no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—24no—
129C2×C4×D4—22no—
130C2×C4×D4—24no—
131C2×C4×D4—22no—
132C2×C4×D4—42no—
133C2×C4×D4C2314yes—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C22⋊C4—21no—
148C22×C42C2311yescharacteristic
149#479—28no—
150#478—216no—
151#478—216no—
152#478—28no—
153#477—28no—
154#474—28no—
155#428—24no—
156#427—24no—
157C2×#107D418yescomplemented
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#106—24no—
163C2×#106C2314yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166#424—24no—
167#424—24no—
168C2×C4⋊C8—24no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2314yes—
174#418—24no—
175#418—42no—
176#417—24no—
177#417—42no—
178#416—24no—
179#416—24no—
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2311yescharacteristic
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#102—24no—
186C2×#102—24no—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2314yes—
190C2×#102C2311yescharacteristic
191C2×#102C2312yes—
192C2×#102C2314yes—
193C2×#102C2311yescharacteristic
194C2×#102C2312yes—
195C2×#102C2312yes—
196C2×#102C2312yes—
197C2×#102C2312yes—
198#405—24no—
199C2×#97C2312yes—
200C2×C4×C8C2312yes—
201#398—24no—
202#398—24no—
203#398—24no—
204#395—24no—
205#391—24no—
206#391C2312yes—
207#390—21no—
208#390—21no—
209#390—24no—
210#390C2314yes—
211#390C2312yes—
212#389—22no—
213#389—24no—
214#389—24no—
215#389D412yes—
216#389D412yes—
217#389C2314yes—
218#389C2312yes—
219#389C2312yes—
220#388—24no—
221#388—24no—
222#386—21no—
223#386—21no—
224#386—24no—
225#386—24no—
226#385—21no—
227#385—21no—
228#385—24no—
229#385—22no—
230#385—22no—
231#385D412yes—
232#385D412yes—
233#384—21no—
234#384—24no—
235#384C2312yes—
236#382—21no—
237#382C2312yes—
238#379—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4C2314yes—
243C4×C4⋊C4C2314yes—
244C4×C4⋊C4C2314yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—24no—
249#340C2314yes—
250#340C2312yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256C25—21no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41C2×Q814yes—
263C22×Q8—22no—
264C22×Q8C2411yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—24no—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4C2×Q814yes—
274C22×D4C2412yes—
275C22×D4C2411yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—24no—
281C23×C4—22no—
282C23×C4—44no—
283C23×C4—84no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8C2×D414yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C4⋊Q8—24no—
305C4⋊Q8—42no—
306C4⋊Q8—44no—
307#127—24no—
308#127C2×Q818yes—
309#125—28no—
310#125—24no—
311#125—44no—
312#125—42no—
313#125—48no—
314#125—44no—
315#124—24no—
316#124C2×Q818yes—
317C22⋊Q8—48no—
318#121—216no—
319#120—216no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×D4—28no—
323C4×D4—24no—
324C4×D4—44no—
325C4×D4—42no—
326C4×D4—48no—
327C4×D4—48no—
328C4×D4—88no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—44no—
337#117C2×Q814yes—
338#117#4114yes—
339#117#4114yes—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—44no—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C42—21no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—24no—
394C2×C42—24no—
395C2×C42C2×D412yes—
396C2×C42C2412yes—
397C2×C42C2412yes—
398C2×C42C2412yes—
399#107—24no—
400#107—24no—
401#107C2×D418yes—
402#107C2×D418yes—
403#106—28no—
404#106—24no—
405#106—24no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—28no—
411#103—22no—
412#103—22no—
413#103—24no—
414#103—24no—
415#103—42no—
416#102—24no—
417#102—24no—
418#102—22no—
419#102—22no—
420#102—28no—
421#102—24no—
422#102—24no—
423#102—28no—
424#102—42no—
425#102—42no—
426#102—42no—
427#97—24no—
428C4×C8—24no—
429#95—22no—
430#95—24no—
431#95—24no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—24no—
436#95—24no—
437#95C2×D414yes—
438#95C2×D414yes—
439C24—22no—
440C24—21no—
441C24—24no—
442C24—42no—
443C24—42no—
444C24—82no—
445C24—82no—
446C24C22×D411yescharacteristic
447#41—28no—
448#41—28no—
449#41—44no—
450#41—44no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—44no—
457C2×Q8—84no—
458C2×Q8C22×Q812yes—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—216no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—84no—
481C2×D4—84no—
482C2×D4C22×Q814yes—
483C2×D4C22×Q812yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—48no—
500C22×C4—44no—
501C22×C4—48no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4—816no—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D414yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C22×Q812yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4H(5,𝔽2)12yes—
522C22×C42−1+412yes—
523C22×C4C2511yescharacteristic
524M16—44no—
525M16—44no—
526C2×C8—28no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—44no—
541C2×C8C22×D412yes—
542C2×C8C22×D412yes—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—42no—
567C4⋊C4—84no—
568C4⋊C4—84no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4C22⋊Q818yes—
572C4⋊C4C22×D414yes—
573C4⋊C4C22×D414yes—
574C4⋊C4C2×#4114yes—
575C22⋊C4—24no—
576C22⋊C4—24no—
577C22⋊C4—216no—
578C22⋊C4—48no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—42no—
582C22⋊C4—42no—
583C22⋊C4—84no—
584C22⋊C4—82no—
585C22⋊C4—84no—
586C22⋊C4C22⋊Q818yes—
587C42—24no—
588C42—24no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—44no—
594C42—42no—
595C42—42no—
596C42—84no—
597C42—84no—
598C42C22×Q814yes—
599C42C2×#4114yes—
600C23—22no—
601C23—22no—
602C23—24no—
603C23—22no—
604C23—42no—
605C23—44no—
606C23—42no—
607C23—44no—
608C23—41no—
609C23—42no—
610C23—41no—
611C23—42no—
612C23—48no—
613C23—88no—
614C23—84no—
615C23—84no—
616C23—88no—
617C23C2×C22⋊Q812yes—
618C23D4212yes—
619C23#54812yes—
620C23C23×D411yescharacteristic
621Q8—22no—
622Q8—22no—
623Q8—41no—
624Q8—41no—
625Q8—164no—
626D4—24no—
627D4—22no—
628D4—24no—
629D4—22no—
630D4—48no—
631D4—42no—
632D4—41no—
633D4—42no—
634D4—41no—
635D4—48no—
636D4—816no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—48no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—816no—
674C2×C4—84no—
675C2×C4—164no—
676C2×C4C2×C22⋊Q818yes—
677C2×C4C2×C22⋊Q812yes—
678C2×C4#53214yes—
679C2×C4#54214yes—
680C2×C4D4212yes—
681C2×C4#54812yes—
682C2×C4D4×Q814yes—
683C2×C4#55414yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C23×Q811yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C8—22no—
691C8—22no—
692C8—42no—
693C8—42no—
694C8—42no—
695C22—24no—
696C22—22no—
697C22—22no—
698C22—24no—
699C22—44no—
700C22—48no—
701C22—42no—
702C22—44no—
703C22—816no—
704C22—81no—
705C22—82no—
706C22—82no—
707C22—81no—
708C22—82no—
709C22—84no—
710C22#245214yes—
711C22C22×C22⋊Q811yescharacteristic
712C22C2×D4211yescharacteristic
713C22C2×#54811yescharacteristic
714C22#358412yes—
715C22#358512yes—
716C4—28no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—44no—
723C4—44no—
724C4—44no—
725C4—81no—
726C4—82no—
727C4—81no—
728C4—82no—
729C4#245214yes—
730C4C2×#53211yescharacteristic
731C4C2×#54211yescharacteristic
732C4C2×D4×Q811yescharacteristic
733C4C2×#55411yescharacteristic
734C2—24no—
735C2—82no—
736C2—84no—
737C2C2×#245211yescharacteristic
738C2#6049612yes—
739C2#6105411yescharacteristic
740C2#6105811yescharacteristic
741C2#6134912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3083C2semidirect
2. C22×#478C2semidirect
3. #35640C2semidirect
5. #26479C2semidirect
6. #26475C2semidirect
7. #26468C2semidirect
8. #26467C2semidirect
9. #23612C2semidirect
10. #23609C2semidirect
11. #23456C2semidirect
12. #23453C2semidirect
15. #21174C2semidirect
16. #21147C2semidirect
17. #20882C2semidirect
18. #20873C2semidirect
19. #20455C2semidirect
26. C2×#479C22semidirect
27. C2×#477C22semidirect
31. C2×#478C22semidirect
32. C2×#474C22semidirect
51. #2552C22semidirect
53. #2550C22semidirect
69. #2068C22semidirect
71. #2067C22semidirect
72. #2066C22semidirect
74. #2065C22semidirect
75. #2026C22semidirect
77. #2025C22semidirect
85. #1949C22semidirect
86. #1948C22semidirect
87. #1940C22semidirect
88. #1939C22semidirect
89. #1865C22semidirect
91. #1864C22semidirect
93. #1776C22semidirect
94. C4×#107C22semidirect
111. #535Q8semidirect
157. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30832
C22×#4782
#356402
#342652
#264792
#264752
#264682
#264672
#236122
#236092
#234562
#234532
#212632
#212062
#211742
#211472
#208822
#208732
#204552