Finite Group Databasebeta

← #6852594

Subgroups — #6852594

Order 512.

1617 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32346C214yesmaximal, complemented
5#31522C214yesmaximal
6#29981C212yesmaximal
7#29255C212yesmaximal, complemented
8#29206C212yesmaximal, complemented
9#28687C211yesmaximal, characteristic, complemented
10#28530C212yesmaximal
11#28283C211yesmaximal, characteristic, complemented
12#27815C212yesmaximal, complemented
13#25837C212yesmaximal, complemented
14#24329C212yesmaximal
15#22821C211yesmaximal, characteristic, complemented
16#22146C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#533—21no—
20C2×#498C2212yes—
21C2×#475C2212yes—
22C2×#476C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2868C2212yes—
27#2567C2214yes—
28#2567C2212yes—
29#2556—22no—
30#2551—22no—
31#2551—21no—
32#2551—22no—
33#2551—21no—
34#2551C2211yescharacteristic
35#2546C2211yescharacteristic
36#2542C2211yescharacteristic
37#2538C2211yescharacteristic
38#2506C2212yes—
39#2492—21no—
40#2452C2212yes—
41#2429—21no—
42#2396—22no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2096C2214yes—
55#2094C2214yes—
56#2084C2214yes—
57#2066C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2058C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#1990C2212yes—
64#1990C2214yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1981C2212yes—
69#1968C2214yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1933C2214yes—
77#1927C2212yes—
78#1916C2214yes—
79#1880C2214yes—
80#1877C2212yes—
81#1865C2214yes—
82#1865C2214yes—
83#1865C2214yes—
84#1844C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×SD16C2312yes—
96Q82—44no—
97Q82—44no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×#125—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C22×C42—21no—
130#498—44no—
131#476—44no—
132#475—44no—
133#461Q818yescomplemented
134C4×SD16—44no—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#396—22no—
159#396—22no—
160#395—24no—
161#395—24no—
162#395—24no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#391C2314yes—
170#390C2312yes—
171#390C2312yes—
172#389—24no—
173#389—24no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#387—22no—
191#385—22no—
192#384—21no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#384C2314yes—
202#380—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2412yes—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C2×SD16—42no—
247C2×SD16—42no—
248C2×SD16C2×Q818yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—416no—
262C4⋊Q8—48no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—84no—
266C4⋊Q8—82no—
267C4⋊Q8—82no—
268C4⋊Q8—84no—
269C4⋊Q8C2×Q814yes—
270#125—48no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×Q8—48no—
284C4×Q8—44no—
285C4×Q8SD1618yescomplemented
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—41no—
291C4×D4C2×Q814yes—
292#117—48no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#106—42no—
359#106—44no—
360#106—44no—
361#106—44no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—42no—
367C4⋊C8C2×Q814yes—
368#103—24no—
369#103—44no—
370#103—42no—
371#103—44no—
372#103C2×Q818yes—
373#102—42no—
374#102—44no—
375C4×C8—24no—
376C4×C8—44no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—48no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—84no—
412C2×Q8—84no—
413C2×Q8C2×SD1614yes—
414C2×Q8C22×Q814yes—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4C22×Q812yes—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8C22×Q814yes—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4C2×SD1618yes—
500C4⋊C4C2×SD1614yes—
501C4⋊C4#13714yes—
502C4⋊C4C22×Q812yes—
503C4⋊C42−1+412yes—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—82no—
511C42—24no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—41no—
522C42—42no—
523C42—48no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42C2×SD1614yes—
530C42C2×SD1618yes—
531C42C22×Q812yes—
532C42H(5,𝔽2)12yes—
533C42H(5,𝔽2)12yes—
534C422−1+412yes—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C23×D411yescharacteristic
543Q8—28no—
544Q8—24no—
545Q8—24no—
546Q8—42no—
547Q8—42no—
548Q8—88no—
549Q8—84no—
550Q8—164no—
551Q8—168no—
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4#53512yes—
602C2×C4#53512yes—
603C2×C4D4×Q812yes—
604C2×C4D4×Q812yes—
605C2×C4C22×SD1612yes—
606C2×C4C22×SD1614yes—
607C2×C4C22×SD1614yes—
608C2×C4C22×SD1612yes—
609C2×C4C22×SD1612yes—
610C2×C4C2×#13714yes—
611C2×C4C2×#13712yes—
612C2×C4#57712yes—
613C2×C4C23×Q811yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4#58511yescharacteristic
619C8—24no—
620C8—42no—
621C8—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#53511yescharacteristic
630C22C2×D4×Q811yescharacteristic
631C22#348911yescharacteristic
632C22C23×SD1611yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57711yescharacteristic
635C22#358511yescharacteristic
636C4—24no—
637C4—22no—
638C4—24no—
639C4—22no—
640C4—24no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—41no—
647C4—81no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4#320614yes—
653C4Q8×SD1614yes—
654C4#350111yescharacteristic
655C4#350311yescharacteristic
656C4#355011yescharacteristic
657C4#355411yescharacteristic
658C2—82no—
659C2#3959611yescharacteristic
660C2C2×#320611yescharacteristic
661C2#6083311yescharacteristic
662C2C2×Q8×SD1611yescharacteristic
663C2Q8×#13711yescharacteristic
664C2#6209611yescharacteristic
665C2#6213111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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×C4⋊Q8C2semidirect
4. #32346C2semidirect
7. #29255C2semidirect
8. #29206C2semidirect
9. #28687C2semidirect
11. #28283C2semidirect
12. #27815C2semidirect
13. #25837C2semidirect
15. #22821C2semidirect
133. #461Q8semidirect
285. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349292
#331972
#323462
#315222
#299812
#292552
#292062
#286872
#285302
#282832
#278152
#258372
#243292
#228212
#221462
C4×#4612