Finite Group Databasebeta

← #6852644

Subgroups — #6852644

Order 512.

1503 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32360C214yesmaximal, complemented
5#31534C214yesmaximal
6#29980C212yesmaximal
7#29257C212yesmaximal, complemented
8#29213C212yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28539C212yesmaximal
11#28284C211yesmaximal, characteristic, complemented
12#27824C212yesmaximal, complemented
13#25841C212yesmaximal, complemented
14#24329C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#500C2212yes—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2868C2212yes—
26#2574—22no—
27#2567C2211yescharacteristic
28#2567C2214yes—
29#2567C2212yes—
30#2563C2211yescharacteristic
31#2552C2211yescharacteristic
32#2551—22no—
33#2551—22no—
34#2551—21no—
35#2551—21no—
36#2538C2211yescharacteristic
37#2508C2212yes—
38#2497—21no—
39#2454C2212yes—
40#2423—21no—
41#2396—22no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2212yes—
48C4×C22⋊Q8—21no—
49#2293C2212yes—
50#2285—21no—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2099C2214yes—
54#2097C2214yes—
55#2093C2214yes—
56#2083C2214yes—
57#2068C2214yes—
58#2058C2214yes—
59#2057C2214yes—
60#2057C2214yes—
61#2026C2214yes—
62#2023C2214yes—
63#1992C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1981C2212yes—
69#1973C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1935C2214yes—
77#1927C2212yes—
78#1917C2214yes—
79#1880C2214yes—
80#1877C2212yes—
81#1865C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96Q82—44no—
97Q82—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#125C2312yes—
109C2×C22⋊Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42—21no—
122#500—44no—
123#477—44no—
124#476—44no—
125#461Q818yescomplemented
126C4×SD16—44no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#396—22no—
152#395—22no—
153#395—24no—
154#395—24no—
155#395—24no—
156#393—22no—
157#391—21no—
158#391—21no—
159#391C2312yes—
160#391C2314yes—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—24no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384—22no—
187#384—21no—
188#384—22no—
189#384—21no—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2314yes—
195#382—22no—
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C43C2312yes—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C2×SD16—42no—
245C2×SD16—42no—
246C2×SD16C2×Q818yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—416no—
258C4⋊Q8—48no—
259C4⋊Q8—84no—
260C4⋊Q8—82no—
261C4⋊Q8—82no—
262C4⋊Q8—84no—
263C4⋊Q8C2×Q814yes—
264#125—24no—
265#125—42no—
266C22⋊Q8—44no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8—82no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—41no—
282C4×D4C2×Q814yes—
283#117—84no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—44no—
349#107—44no—
350#107—44no—
351#106—42no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—24no—
356C4⋊C8—42no—
357C4⋊C8C2×Q814yes—
358#103—24no—
359#103—44no—
360#103—42no—
361#103—44no—
362#103C2×Q818yes—
363#102—42no—
364#102—44no—
365#97—44no—
366C4×C8—24no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—81no—
397C2×Q8—81no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×Q8#13714yes—
403C2×Q8C22×Q814yes—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C22×Q812yes—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438SD16—48no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8C22×Q814yes—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4C22×Q812yes—
491C4⋊C42−1+412yes—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—24no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—41no—
511C42—42no—
512C42—48no—
513C42—44no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42#13714yes—
519C42C22×Q812yes—
520C42H(5,𝔽2)12yes—
521C42H(5,𝔽2)12yes—
522C422−1+412yes—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C23×D411yescharacteristic
531Q8—28no—
532Q8—24no—
533Q8—24no—
534Q8—42no—
535Q8—42no—
536Q8—84no—
537Q8—164no—
538Q8—168no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#53512yes—
591C2×C4#53512yes—
592C2×C4D4×Q812yes—
593C2×C4D4×Q812yes—
594C2×C4C2×#13714yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57612yes—
600C2×C4C23×Q811yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C2×C4#58511yescharacteristic
606C8—24no—
607C8—42no—
608C8—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#53511yescharacteristic
617C22C2×D4×Q811yescharacteristic
618C22#348911yescharacteristic
619C22C22×#13711yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57611yescharacteristic
622C22#358511yescharacteristic
623C4—24no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—81no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—82no—
639C4#321814yes—
640C4#338514yes—
641C4#350111yescharacteristic
642C4#350311yescharacteristic
643C4#355011yescharacteristic
644C4#355411yescharacteristic
645C2—82no—
646C2#3959611yescharacteristic
647C2C2×#321811yescharacteristic
648C2#6083211yescharacteristic
649C2C2×#338511yescharacteristic
650C2Q8×#13711yescharacteristic
651C2#6210811yescharacteristic
652C2#6214911yescharacteristic
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. #32360C2semidirect
7. #29257C2semidirect
8. #29213C2semidirect
9. #28689C2semidirect
11. #28284C2semidirect
12. #27824C2semidirect
13. #25841C2semidirect
15. #22825C2semidirect
125. #461Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349412
#331972
#323602
#315342
#299802
#292572
#292132
#286892
#285392
#282842
#278242
#258412
#243292
#228252
#221522
C4×#4612