Finite Group Databasebeta

← #6852593

Subgroups — #6852593

Order 512.

1925 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#32342C214yesmaximal, complemented
5#31518C214yesmaximal, complemented
6#29980C212yesmaximal, complemented
7#29254C212yesmaximal, complemented
8#29208C212yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28527C212yesmaximal, complemented
11#28279C211yesmaximal, characteristic, complemented
12#27813C212yesmaximal, complemented
13#25838C212yesmaximal, complemented
14#24326C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22145C212yesmaximal
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#533—21no—
22C2×#500C2212yes—
23C2×#474C2212yes—
24C2×#474C2212yes—
25C2×#459C2211yescharacteristic
26C2×C4×D8C2212yes—
27#2868C2211yescharacteristic
28#2868C2212yes—
29#2556—22no—
30#2551C2211yescharacteristic
31#2546C2211yescharacteristic
32#2542—22no—
33#2542—21no—
34#2542—22no—
35#2542—21no—
36#2542C2214yes—
37#2542C2211yescharacteristic
38#2542C2212yes—
39#2538C2211yescharacteristic
40#2506C2212yes—
41#2492—21no—
42#2452C2212yes—
43#2429—21no—
44#2396—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#127C2212yes—
48C4×#127C2211yescharacteristic
49C4×C22⋊Q8—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2099C2214yes—
54#2093C2214yes—
55#2093C2214yes—
56#2083C2214yes—
57#2067C2214yes—
58#2057C2214yes—
59#2057C2214yes—
60#2057C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#1990C2212yes—
64#1990C2214yes—
65#1988C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1970C2214yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1934C2214yes—
77#1925C2212yes—
78#1916C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1862C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1845C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C23×Q8—22no—
96C22×D8C2312yes—
97D4×Q8—44no—
98D4×Q8—44no—
99D4×Q8—48no—
100D4×Q8—48no—
101#533—48no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×#127C2311yescharacteristic
113C2×#125—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—24no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×C4×D4C2314yes—
138C2×#117—22no—
139C22×C42—21no—
140#500—44no—
141#474—44no—
142#474—44no—
143#459Q818yescomplemented
144C4×D8—44no—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2314yes—
164#400—22no—
165#399—22no—
166#399—24no—
167#399—24no—
168#399—24no—
169#398—22no—
170#398—22no—
171#396—22no—
172#396—22no—
173#395—22no—
174#393—22no—
175#391—21no—
176#391—21no—
177#391—22no—
178#390—22no—
179#390—21no—
180#390—21no—
181#390—22no—
182#390C2312yes—
183#390C2314yes—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—22no—
188#389—21no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#387—22no—
197#385—22no—
198#384—21no—
199#384—22no—
200#384—21no—
201#384—22no—
202#384C2312yes—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2312yes—
207#384C2314yes—
208#380—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—48no—
243C22×Q8—48no—
244C22×Q8—48no—
245C22×Q8C2×D411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C2×D8—42no—
256C2×D8—42no—
257C2×D8C2×Q818yescomplemented
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275#127—22no—
276#127C2×Q814yes—
277#125—48no—
278C22⋊Q8—48no—
279C22⋊Q8—416no—
280C22⋊Q8—416no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284C22⋊Q8—416no—
285C22⋊Q8—48no—
286C22⋊Q8—44no—
287C22⋊Q8—48no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—48no—
292C4×Q8—44no—
293C4×Q8D818yescomplemented
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—41no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—48no—
305C4×D4—44no—
306C4×D4—48no—
307C4×D4C2×Q814yescomplemented
308#117—48no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—42no—
375#107—44no—
376#107—44no—
377#107—44no—
378#107—44no—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—24no—
382C4⋊C8—42no—
383C4⋊C8C2×Q814yescomplemented
384#102—24no—
385#102—44no—
386#102—42no—
387#102—42no—
388#102—44no—
389#102—44no—
390#102C2×Q818yescomplemented
391C4×C8—24no—
392C4×C8—44no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—21no—
406C24—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—48no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—82no—
421C2×Q8—832no—
422C2×Q8C2×D814yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C22×Q814yes—
440C2×D4C22×Q812yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—81no—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2511yescharacteristic
474D8—48no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8C22×Q814yes—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—81no—
520C4⋊C4C2×D818yes—
521C4⋊C4C2×D814yes—
522C4⋊C4AGL(1,ℤ/8)14yes—
523C4⋊C4C22×Q812yes—
524C4⋊C42−1+412yes—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—41no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—44no—
554C42C2×D814yes—
555C42C2×D818yes—
556C42C22×Q812yes—
557C42H(5,𝔽2)12yes—
558C42H(5,𝔽2)12yes—
559C422−1+412yes—
560C23—22no—
561C23—24no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23C23×D411yescharacteristic
575Q8—28no—
576Q8—88no—
577Q8—816no—
578Q8—84no—
579Q8—88no—
580D4—22no—
581D4—24no—
582D4—22no—
583D4—24no—
584D4—48no—
585D4—48no—
586D4—41no—
587D4—42no—
588D4—41no—
589D4—42no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—84no—
631C2×C4—816no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4#53512yes—
635C2×C4#53512yes—
636C2×C4D4×Q812yes—
637C2×C4D4×Q812yes—
638C2×C4C22×D812yes—
639C2×C4C22×D814yes—
640C2×C4C22×D814yes—
641C2×C4C22×D812yes—
642C2×C4C22×D812yes—
643C2×C4C2×AGL(1,ℤ/8)14yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4#57812yes—
646C2×C4C23×Q811yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C2×C4#58511yescharacteristic
652C8—24no—
653C8—42no—
654C8—44no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—82no—
662C22—81no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—81no—
668C22—82no—
669C22C2×#53511yescharacteristic
670C22C2×D4×Q811yescharacteristic
671C22#348911yescharacteristic
672C22C23×D811yescharacteristic
673C22C22×AGL(1,ℤ/8)11yescharacteristic
674C22C2×#57811yescharacteristic
675C22#358611yescharacteristic
676C4—24no—
677C4—22no—
678C4—24no—
679C4—22no—
680C4—24no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—41no—
686C4—81no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#320514yes—
692C4Q8×D814yes—
693C4#350111yescharacteristic
694C4#350311yescharacteristic
695C4#355011yescharacteristic
696C4#355411yescharacteristic
697C2—82no—
698C2—84no—
699C2#3959611yescharacteristic
700C2C2×#320511yescharacteristic
701C2#6083711yescharacteristic
702C2C2×Q8×D811yescharacteristic
703C2Q8×AGL(1,ℤ/8)11yescharacteristic
704C2#6209511yescharacteristic
705C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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
1. Q8×#127C2semidirect
2. #34929C2semidirect
3. #33195C2semidirect
4. #32342C2semidirect
5. #31518C2semidirect
6. #29980C2semidirect
7. #29254C2semidirect
8. #29208C2semidirect
9. #28689C2semidirect
10. #28527C2semidirect
11. #28279C2semidirect
12. #27813C2semidirect
13. #25838C2semidirect
14. #24326C2semidirect
15. #22821C2semidirect
143. #459Q8semidirect
257. C2×D8C2×Q8semidirect
293. C4×Q8D8semidirect
307. C4×D4C2×Q8semidirect
383. C4⋊C8C2×Q8semidirect
390. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#349292
#331952
#323422
#315182
#299802
#292542
#292082
#286892
#285272
#282792
#278132
#258382
#243262
#228212
#221452
C4×#4592