Finite Group Databasebeta

← #6560657

Subgroups — #6560657

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36496C211yesmaximal, characteristic
3#30207C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29352C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#28722C212yesmaximal, complemented
9#27824C212yesmaximal, complemented
10#27819C214yesmaximal, complemented
11#27775C212yesmaximal
12#27773C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23815C212yesmaximal, complemented
15#22543C212yesmaximal, complemented
16#22519C212yesmaximal, complemented
17#20572C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#490C2214yes—
21C2×#489C2212yes—
22C2×C8×Q8C2212yes—
23#2868C2211yescharacteristic
24#2861C2212yes—
25#2859C2212yes—
26#2858C2211yescharacteristic
27#2582—21no—
28#2567—21no—
29#2567C2211yescharacteristic
30#2566C2212yes—
31#2552C2212yes—
32#2550C2214yes—
33#2544C2212yes—
34#2541C2211yescharacteristic
35#2507—22no—
36#2461—22no—
37#2461—22no—
38#2451—22no—
39#2426—21no—
40#2426—21no—
41#2369—21no—
42#2361—21no—
43#2361—21no—
44#2355C2212yes—
45Q8×C4⋊C4—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2212yes—
49C4×#124C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2101C2212yes—
52#2099C2214yes—
53#2072C2211yescharacteristic
54#2072C2212yes—
55#2071C2212yes—
56#2071C2212yes—
57#2071C2211yescharacteristic
58#2071C2212yes—
59#2071C2212yes—
60#2042C2212yes—
61#2041C2214yes—
62#2026C2214yes—
63#2025C2214yes—
64#2024C2211yescharacteristic
65#2023C2211yescharacteristic
66#2023C2212yes—
67#2023C2214yes—
68#1993C2212yes—
69#1992C2214yes—
70#1991C2214yes—
71#1990C2212yes—
72#1989C2214yes—
73#1987C2212yes—
74#1983C2214yes—
75#1982C2212yes—
76#1958C2214yes—
77#1957C2212yes—
78#1936C2214yes—
79#1936C2214yes—
80#1935C2214yes—
81#1804C2214yes—
82#1804C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1770C2214yes—
86#1770C2212yes—
87#1767C2212yes—
88#1767C2211yescharacteristic
89C4×#103C2212yes—
90C4×#102C2214yes—
91C4×#102C2214yes—
92C4×#102C2212yes—
93#1744C2212yes—
94#1744C2211yescharacteristic
95#559—28no—
96#554—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#127C2312yes—
104C2×#125C2312yes—
105C2×#125C2314yes—
106C2×#124—21no—
107C2×#124—21no—
108C2×#124—21no—
109C2×#124C2312yes—
110C2×#124C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×#121—21no—
115C2×#121—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4—21no—
129#490—48no—
130#489—44no—
131C8×Q8—28no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#97C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#398—24no—
149#398—24no—
150#396—22no—
151#396—22no—
152#395—24no—
153#395—22no—
154#394—22no—
155#394—22no—
156#392—22no—
157#391—22no—
158#391—22no—
159#391—21no—
160#391—24no—
161#391C2312yes—
162#391C2311yescharacteristic
163#390—22no—
164#390—22no—
165#390—21no—
166#390—21no—
167#390C2311yescharacteristic
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—24no—
181#389C2314yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—21no—
188#388—21no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382C2311yescharacteristic
202#382C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C43C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256#127—24no—
257#127—24no—
258#127—42no—
259#125—28no—
260#125—44no—
261#125—44no—
262#124—22no—
263#124—28no—
264#124—41no—
265#124—44no—
266#124—82no—
267#124—82no—
268#123—48no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271#121—44no—
272#121—44no—
273C4×Q8—24no—
274C4×Q8—42no—
275C4×Q8—42no—
276C4×Q8—44no—
277C4×Q8—84no—
278C4×Q8#4118yes—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—44no—
283C4×D4—84no—
284C4×D4—84no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C4⋊C8—28no—
348C4⋊C8—24no—
349#103—24no—
350#103—42no—
351#103—44no—
352#102—28no—
353#102—24no—
354#102—24no—
355#102—28no—
356#102—44no—
357#102—44no—
358#102—44no—
359#97—42no—
360#97—44no—
361C4×C8—28no—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8#12414yes—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4H(5,𝔽2)12yes—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4#12414yes—
483C4⋊C4#12418yes—
484C4⋊C42−1+414yes—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—21no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—44no—
515C42—84no—
516C42C2×#4118yes—
517C42C2×#4114yes—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—48no—
535Q8—42no—
536Q8—42no—
537Q8—162no—
538Q8—164no—
539D4—24no—
540D4—42no—
541D4—44no—
542D4—44no—
543D4—88no—
544D4—88no—
545D4—84no—
546D4—162no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—164no—
592C2×C4—164no—
593C2×C4C2×#12414yes—
594C2×C4C2×#12414yes—
595C2×C4C2×#12412yes—
596C2×C4C2×#12412yes—
597C2×C4#57512yes—
598C2×C4#57512yes—
599C2×C4#57612yes—
600C2×C4#57712yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+412yes—
604C2×C4#58511yescharacteristic
605C8—42no—
606C8—42no—
607C8—44no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C22×#12411yescharacteristic
623C22#348611yescharacteristic
624C22#348811yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57611yescharacteristic
628C22C2×#57711yescharacteristic
629C4—22no—
630C4—22no—
631C4—24no—
632C4—24no—
633C4—22no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—44no—
639C4—84no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4#313514yes—
646C4#351812yes—
647C4#352212yes—
648C2—82no—
649C2—162no—
650C2#3594811yescharacteristic
651C2C2×#313511yescharacteristic
652C2#6062211yescharacteristic
653C2#6183511yescharacteristic
654C2#6183611yescharacteristic
655C2#6199511yescharacteristic
656C2#6199611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #36563C2semidirect
3. #30207C2semidirect
4. #30203C2semidirect
5. #29354C2semidirect
6. #29352C2semidirect
7. #28726C2semidirect
8. #28722C2semidirect
9. #27824C2semidirect
10. #27819C2semidirect
12. #27773C2semidirect
13. #23817C2semidirect
14. #23815C2semidirect
15. #22543C2semidirect
16. #22519C2semidirect
17. #20572C2semidirect

Maximal subgroups

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

subgroupindex
#365632
#364962
#302072
#302032
#293542
#293522
#287262
#287222
#278242
#278192
#277752
#277732
#238172
#238152
#225432
#225192
#205722