Finite Group Databasebeta

← #6541829

Subgroups — #6541829

Order 512.

1789 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#30109C212yesmaximal, complemented
4#29798C211yesmaximal, characteristic, complemented
5#29259C211yesmaximal, characteristic, complemented
6#28440C211yesmaximal, characteristic, complemented
7#28317C212yesmaximal, complemented
8#26952C214yesmaximal, complemented
9#26233C212yesmaximal, complemented
10#25758C212yesmaximal
11#25171C214yesmaximal, complemented
12#23935C212yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23187C212yesmaximal, complemented
16#23100C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C2×#533C2212yes—
20C2×#494C2212yes—
21C2×#485C2214yes—
22C2×#462C2214yes—
23C2×#461C2212yes—
24C2×#452C2214yes—
25C2×#452C2214yes—
26C2×#441C2214yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2552—21no—
30#2542C2212yes—
31#2538—21no—
32#2441—22no—
33#2429C2214yes—
34#2423—22no—
35#2374—22no—
36#2365C2211yescharacteristic
37#2365C2212yes—
38#2363—21no—
39#2362C2212yes—
40#2354—21no—
41#2354C2211yescharacteristic
42#2354C2211yescharacteristic
43Q8×C4⋊C4—21no—
44#2345—21no—
45#2341C2212yes—
46#2335—22no—
47#2324—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2297C2212yes—
52#2293C2211yescharacteristic
53#2274—21no—
54C42×D4C2211yescharacteristic
55#2100C2211yescharacteristic
56#2095C2212yes—
57#2094C2211yescharacteristic
58#2088C2212yes—
59#2065C2212yes—
60#2058C2211yescharacteristic
61#2043C2212yes—
62#2043C2212yes—
63#2036C2214yes—
64#2035C2214yes—
65#2024C2212yes—
66#2023C2211yescharacteristic
67#2016C2212yes—
68#2016C2212yes—
69#2013C2212yes—
70#2013C2212yes—
71#2001C2214yes—
72#2001C2214yes—
73#1981C2212yes—
74#1981C2212yes—
75#1973C2214yes—
76#1971C2214yes—
77#1956C2212yes—
78#1954C2212yes—
79#1947C2212yes—
80#1928C2214yes—
81#1924C2214yes—
82#1880C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1873C2214yes—
86#1863C2214yes—
87#1831C2211yescharacteristic
88#1817C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99#547—24no—
100#547—24no—
101#533—48no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—22no—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2314yes—
113C2×#121—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C42C2312yes—
129#494—28no—
130#485—48no—
131#462—48no—
132#461—28no—
133#452—48no—
134#452—48no—
135#441—48no—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#399—24no—
155#398—24no—
156#396—24no—
157#391—21no—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#390—21no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390C2311yescharacteristic
166#390C2312yes—
167#390C2311yescharacteristic
168#389—21no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389C2312yes—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#388—22no—
180#388—22no—
181#388—22no—
182#388—24no—
183#388—21no—
184#388—21no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#380—22no—
201#380—22no—
202#380—21no—
203#380—22no—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43C2311yescharacteristic
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—44no—
241C2×#41—44no—
242C22×Q8—21no—
243C22×Q8C2414yes—
244C22×Q8C2412yes—
245C22×D4—22no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—44no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×SD16—28no—
260C2×SD16—44no—
261C2×SD16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8#4114yes—
272#125—48no—
273#123—216no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—28no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—28no—
282C4×Q8—28no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—48no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—42no—
293C4×D4#4114yes—
294#117—44no—
295#117—48no—
296#117—44no—
297#117—44no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—44no—
364#106—24no—
365#106—24no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368#103—28no—
369#103—28no—
370#103—28no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—24no—
376#102—24no—
377C22⋊C8—28no—
378C22⋊C8—44no—
379#97—44no—
380C4×C8—24no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24C22×D412yes—
399#41—88no—
400#41—88no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8C2×#4114yes—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4C2×#4112yes—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2511yescharacteristic
458Q16—416no—
459Q16—88no—
460SD16—88no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8H(5,𝔽2)14yes—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C2×#4112yes—
503C4⋊C4H(5,𝔽2)14yes—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42—48no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—82no—
532C42—82no—
533C42—82no—
534C42#13714yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C23×D411yescharacteristic
552Q8—28no—
553Q8—48no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559D4—24no—
560D4—44no—
561D4—42no—
562D4—88no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4#52512yes—
614C2×C4#52512yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4C2×#13712yes—
618C2×C4#57512yes—
619C2×C4#57812yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×H(5,𝔽2)12yes—
624C8—24no—
625C8—44no—
626C8—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22C2×#52511yescharacteristic
642C22#348411yescharacteristic
643C22#349811yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57511yescharacteristic
646C22C2×#57811yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—41no—
659C4—81no—
660C4—84no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4#311614yes—
665C4#350812yes—
666C4#351912yes—
667C2—82no—
668C2—82no—
669C2#3574111yescharacteristic
670C2C2×#311611yescharacteristic
671C2#6059411yescharacteristic
672C2#6158411yescharacteristic
673C2#6159511yescharacteristic
674C2#6242411yescharacteristic
675C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36134C2semidirect
3. #30109C2semidirect
4. #29798C2semidirect
5. #29259C2semidirect
6. #28440C2semidirect
7. #28317C2semidirect
8. #26952C2semidirect
9. #26233C2semidirect
11. #25171C2semidirect
12. #23935C2semidirect
13. #23775C2semidirect
14. #23577C2semidirect
15. #23187C2semidirect
16. #23100C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343162
#301092
#297982
#292592
#284402
#283172
#269522
#262332
#257582
#251712
#239352
#237752
#235772
#231872
#231002
#228392