Finite Group Databasebeta

← #6560666

Subgroups — #6560666

Order 512.

1609 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#36534C211yesmaximal, characteristic
3#30216C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#29352C212yesmaximal, complemented
6#29347C211yesmaximal, characteristic, complemented
7#28726C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic
9#27827C214yesmaximal, complemented
10#27821C212yesmaximal, complemented
11#27783C214yesmaximal
12#27776C212yesmaximal, complemented
13#23821C212yesmaximal, complemented
14#23815C211yesmaximal, characteristic
15#22544C212yesmaximal, complemented
16#22522C212yesmaximal, complemented
17#20570C211yesmaximal, characteristic, complemented
18C2×Q82—21no—
19C2×#556C2211yescharacteristic
20C2×#554C2211yescharacteristic
21C2×#489C2214yes—
22C2×#486C2212yes—
23C2×#446C2212yes—
24#2868C2212yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2861C2212yes—
28#2567—22no—
29#2567—21no—
30#2565C2212yes—
31#2563C2211yescharacteristic
32#2560—22no—
33#2560—22no—
34#2551—22no—
35#2551—21no—
36#2551C2212yes—
37#2551C2214yes—
38#2547C2211yescharacteristic
39#2544C2212yes—
40#2536—21no—
41#2536—21no—
42#2352—21no—
43#2352—21no—
44Q8×C4⋊C4—21no—
45Q8×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49C4×#124C2212yes—
50C42×Q8C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2072C2212yes—
54#2072C2212yes—
55#2072C2212yes—
56#2071C2211yescharacteristic
57#2071C2211yescharacteristic
58#2071C2212yes—
59#2071C2212yes—
60#2042C2214yes—
61#2041C2212yes—
62#2024C2214yes—
63#2024C2212yes—
64#2024C2214yes—
65#2023C2211yescharacteristic
66#2023C2211yescharacteristic
67#2023C2214yes—
68#1993C2214yes—
69#1992C2214yes—
70#1990C2212yes—
71#1990C2212yes—
72#1989C2214yes—
73#1987C2212yes—
74#1983C2214yes—
75#1981C2212yes—
76#1959C2214yes—
77#1956C2212yes—
78#1936C2214yes—
79#1935C2214yes—
80#1933C2214yes—
81#1803C2214yes—
82#1802C2212yes—
83#1802C2212yes—
84#1802C2214yes—
85#1771C2214yes—
86#1770C2212yes—
87#1767C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89#1763C2214yes—
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95Q82—48no—
96#556—28no—
97#554—28no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#127C2311yescharacteristic
110C2×#126—22no—
111C2×#125C2312yes—
112C2×#124C2312yes—
113C2×#124C2312yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131#489—48no—
132#486—44no—
133#446—28no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#97C2314yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150#398—22no—
151#398—22no—
152#397—22no—
153#395—24no—
154#395—24no—
155#395—24no—
156#394—22no—
157#394—22no—
158#393—22no—
159#391—21no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2311yescharacteristic
169#390C2311yescharacteristic
170#390C2312yes—
171#389—22no—
172#389—24no—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—21no—
177#389C2312yes—
178#389C2314yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#386—22no—
188#386—22no—
189#384—22no—
190#384—21no—
191#384—22no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2314yes—
198#382—21no—
199#382—21no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—44no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—416no—
259C4⋊Q8—48no—
260C4⋊Q8—44no—
261C4⋊Q8—84no—
262C4⋊Q8—82no—
263#127—22no—
264#127—41no—
265#126—48no—
266#125—28no—
267#124—24no—
268#124—28no—
269#124—42no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—28no—
276C4×Q8—48no—
277C4×Q8—48no—
278C4×Q8—44no—
279C4×Q8#4118yes—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—44no—
284C4×D4—44no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
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—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C4⋊C8—24no—
350C4⋊C8—28no—
351#103—28no—
352#103—44no—
353#103—44no—
354#103—44no—
355#102—28no—
356#102—24no—
357#102—24no—
358#102—24no—
359#102—42no—
360#102—44no—
361#97—28no—
362#97—44no—
363C4×C8—24no—
364C4×C8—42no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C24—21no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—48no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—82no—
395C2×Q8—84no—
396C2×Q8—82no—
397C2×Q8—84no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8#12414yes—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4H(5,𝔽2)12yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4#12414yes—
485C4⋊C4#12418yes—
486C4⋊C42−1+414yes—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C42—24no—
501C42—24no—
502C42—21no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—42no—
508C42—41no—
509C42—42no—
510C42—41no—
511C42—48no—
512C42—44no—
513C42—44no—
514C42AGL(1,ℤ/8)14yes—
515C42#13714yes—
516C42C2×#4114yes—
517C42C2×#4118yes—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—48no—
535Q8—44no—
536Q8—44no—
537Q8—164no—
538Q8—162no—
539Q8—168no—
540D4—24no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545D4—84no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—164no—
590C2×C4C2×#12414yes—
591C2×C4C2×#12414yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#12412yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×2−1+412yes—
603C2×C4#58511yescharacteristic
604C8—42no—
605C8—42no—
606C8—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C22×#12411yescharacteristic
622C22#348611yescharacteristic
623C22#348811yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22C22×#13711yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—24no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—84no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4#314114yes—
645C4#351812yes—
646C4#352212yes—
647C2—82no—
648C2—161no—
649C2#3594811yescharacteristic
650C2C2×#314111yescharacteristic
651C2#6062511yescharacteristic
652C2#6182711yescharacteristic
653C2#6183011yescharacteristic
654C2#6195911yescharacteristic
655C2#6196011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
3. #30216C2semidirect
4. #30205C2semidirect
5. #29352C2semidirect
6. #29347C2semidirect
7. #28726C2semidirect
9. #27827C2semidirect
10. #27821C2semidirect
12. #27776C2semidirect
13. #23821C2semidirect
15. #22544C2semidirect
16. #22522C2semidirect
17. #20570C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#365342
#302162
#302052
#293522
#293472
#287262
#287162
#278272
#278212
#277832
#277762
#238212
#238152
#225442
#225222
#205702