Finite Group Databasebeta

← #6457688

Subgroups — #6457688

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3137C212yesmaximal
2#35316C211yesmaximal, characteristic
3#35167C211yesmaximal, characteristic, complemented
4#27784C214yesmaximal, complemented
5#27782C214yesmaximal, complemented
6#27777C212yesmaximal, complemented
7#27776C212yesmaximal, complemented
8#27769C212yesmaximal, complemented
9#27768C212yesmaximal, complemented
10#23512C212yesmaximal
11#23457C212yesmaximal, complemented
12C2×#1981C211yesmaximal, characteristic, complemented
13#21202C211yesmaximal, characteristic, complemented
14#21088C212yesmaximal, complemented
15#20645C211yesmaximal, characteristic
16#20441C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic
18C2×#544C2212yes—
19C2×#536C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22#3137—216no—
23C2×#491C2214yes—
24C2×#489C2214yes—
25C2×#488C2214yes—
26C2×#487C2214yes—
27C22×C4⋊C8C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#419C2212yes—
31C2×#419C2212yes—
32C2×#416C2212yes—
33C22×#103C2212yes—
34C22×#103C2212yes—
35C2×#405C2212yes—
36C2×C4×M16C2211yescharacteristic
37#2857—21no—
38#2857—21no—
39#2857—22no—
40#2659—22no—
41#2621—22no—
42#2552—21no—
43#2552C2212yes—
44#2551—21no—
45#2551C2212yes—
46#2550—22no—
47#2550C2214yes—
48#2547C2212yes—
49#2547C2212yes—
50#2546C2212yes—
51#2546C2212yes—
52#2486—22no—
53#2486—22no—
54#2316C2212yes—
55#2310C2212yes—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2043C2214yes—
60#2043C2214yes—
61#2042C2214yes—
62#2042C2214yes—
63#2026C2214yes—
64#2026C2214yes—
65#2024C2214yes—
66#2024C2214yes—
67#1981C2212yes—
68#1981C2212yes—
69#1981C2214yes—
70#1960C2214yes—
71#1956C2212yes—
72#1956C2212yes—
73#1935C2214yes—
74#1935C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1805C2212yes—
78#1804C2212yes—
79#1803C2211yescharacteristic
80#1803C2211yescharacteristic
81#1802C2211yescharacteristic
82#1802C2211yescharacteristic
83#1771C2214yes—
84#1770C2212yes—
85C8×C4⋊C4C2212yes—
86#1763C2212yes—
87#1762C2212yes—
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2212yes—
93#1729C2212yes—
94C22×#41C2311yescharacteristic
95C23×Q8C2312yes—
96C22×M16C2311yescharacteristic
97C22×M16C2311yescharacteristic
98C23×C8C2312yes—
99#544—28no—
100#544—44no—
101#536—28no—
102#536—44no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2314yes—
113C2×#125—22no—
114C2×#125C2314yes—
115C2×#124C2314yes—
116C2×#123—24no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—24no—
120C4×#41—24no—
121C4×#41—24no—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2312yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#491—216no—
140#489—216no—
141#488—216no—
142#487—216no—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147#419—28no—
148#419—44no—
149#417—24no—
150#417D418yes—
151#416—28no—
152C2×#103—28no—
153C2×#103—28no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170#405—28no—
171C4×M16—24no—
172C2×#97C2312yes—
173C2×#97C2314yes—
174C2×C4×C8C2312yes—
175C2×C4×C8C2314yes—
176#398—24no—
177#398—24no—
178#398—24no—
179#398—24no—
180#397—24no—
181#397—24no—
182#395—24no—
183#395—24no—
184#391—22no—
185#391—22no—
186#391—24no—
187#391C2314yes—
188#391C2314yes—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—24no—
198#389C2314yes—
199#389C2314yes—
200#384—22no—
201#384D411yescharacteristic
202#384D411yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382D412yes—
206#382C2312yes—
207#381—24no—
208#380—21no—
209#380—21no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—22no—
214#380—24no—
215#380D412yes—
216#380D412yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C2×#95—21no—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41C2×D414yes—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8C2414yes—
251C22×Q8C2411yescharacteristic
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16C2×D414yes—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—48no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—84no—
288#125—28no—
289#125—44no—
290#125—48no—
291#124—28no—
292#124—44no—
293#123—416no—
294C22⋊Q8—416no—
295C22⋊Q8—48no—
296C22⋊Q8—48no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—44no—
306C4×D4—44no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117—22no—
311#117—22no—
312#117—24no—
313#117—44no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
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—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4#4114yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4C2×D414yes—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—44no—
365C2×C42—82no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42#4114yes—
369C2×C42#4114yes—
370C2×C42C2414yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C4⋊C8—22no—
374C4⋊C8—22no—
375C4⋊C8—24no—
376#103—24no—
377#103—24no—
378#103—28no—
379#103—28no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—24no—
385#103—44no—
386#103—44no—
387#103C2×D418yes—
388#102—24no—
389#102—24no—
390#102—24no—
391#102C2×D418yes—
392#97—28no—
393#97—24no—
394C4×C8—28no—
395C4×C8—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95C2×D414yes—
407#95C2×D414yes—
408C24—21no—
409C24C22×D411yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414C2×Q8—21no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—21no—
424C2×Q8—24no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—84no—
434C2×Q8—84no—
435C2×Q8C22×D412yes—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C22×D412yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4#12414yes—
476C22×C4#12414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491M16—28no—
492M16—44no—
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×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—48no—
505C2×C8C22×D414yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—81no—
538C4⋊C4—81no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4C22×D414yes—
542C4⋊C4C2×#4114yes—
543C22⋊C4—28no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C42—24no—
551C42—24no—
552C42—42no—
553C42—42no—
554C42—42no—
555C42—44no—
556C42—44no—
557C42—82no—
558C42#12414yes—
559C42#12414yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×#12412yes—
576C23D4212yes—
577C23#54812yes—
578C23C23×D411yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583Q8—44no—
584Q8—44no—
585Q8—44no—
586Q8—44no—
587Q8—88no—
588Q8—168no—
589D4—22no—
590D4—22no—
591D4—48no—
592D4—41no—
593D4—41no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4C2×#12412yes—
639C2×C4C2×#12412yes—
640C2×C4C2×#12412yes—
641C2×C4C2×#12412yes—
642C2×C4C2×#12412yes—
643C2×C4#53214yes—
644C2×C4#54214yes—
645C2×C4D4212yes—
646C2×C4#54714yes—
647C2×C4#54812yes—
648C2×C4#55514yes—
649C2×C4#57712yes—
650C2×C4#57812yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C8—24no—
658C8—42no—
659C8—42no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22#249114yes—
675C22C22×#12411yescharacteristic
676C22C2×D4211yescharacteristic
677C22C2×#54811yescharacteristic
678C22C2×#57711yescharacteristic
679C22C2×#57811yescharacteristic
680C22#358611yescharacteristic
681C22#358611yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—81no—
693C4—84no—
694C4—81no—
695C4#249114yes—
696C4#313714yescomplemented
697C4C2×#53211yescharacteristic
698C4C2×#54211yescharacteristic
699C4C2×#54711yescharacteristic
700C4C2×#55511yescharacteristic
701C2—24no—
702C2—82no—
703C2C2×#249111yescharacteristic
704C2C2×#313711yescharacteristic
705C2#6062011yescharacteristic
706C2#6109411yescharacteristic
707C2#6109711yescharacteristic
708C2#6134011yescharacteristic
709C2#6134211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35167C2semidirect
4. #27784C2semidirect
5. #27782C2semidirect
6. #27777C2semidirect
7. #27776C2semidirect
8. #27769C2semidirect
9. #27768C2semidirect
11. #23457C2semidirect
12. C2×#1981C2semidirect
13. #21202C2semidirect
14. #21088C2semidirect
16. #20441C2semidirect
696. C4#3137semidirect

Maximal subgroups

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

subgroupindex
C2×#31372
#353162
#351672
#277842
#277822
#277772
#277762
#277692
#277682
#235122
#234572
C2×#19812
#212022
#210882
#206452
#204412
C4×#4172