Finite Group Databasebeta

← #6798099

Subgroups — #6798099

Order 512.

1712 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#36180C211yesmaximal, characteristic, complemented
2#36115C211yesmaximal, characteristic
3#33230C211yesmaximal, characteristic, complemented
4#30657C214yesmaximal, complemented
5#30653C214yesmaximal, complemented
6#29853C212yesmaximal, complemented
7#29213C211yesmaximal, characteristic, complemented
8#28754C211yesmaximal, characteristic, complemented
9#27914C212yesmaximal, complemented
10#26532C212yesmaximal, complemented
11#26530C212yesmaximal, complemented
12#26234C212yesmaximal, complemented
13#26230C212yesmaximal
14#25679C212yesmaximal, complemented
15#21867C212yesmaximal, complemented
16C4×#501C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#529C2212yes—
20C2×#501C2211yescharacteristic
21C2×#489C2212yes—
22C2×#489C2214yes—
23C2×#483C2214yes—
24C2×#480C2214yes—
25C2×#432C2212yes—
26#2868C2211yescharacteristic
27#2858C2211yescharacteristic
28#2857C2212yes—
29#2850C2212yes—
30#2567—21no—
31#2552—21no—
32#2552C2212yes—
33#2542C2211yescharacteristic
34#2541C2211yescharacteristic
35#2512—21no—
36#2502—22no—
37#2471—22no—
38#2467—22no—
39#2445—21no—
40#2440—22no—
41#2440C2214yes—
42#2427C2212yes—
43#2426C2212yes—
44#2366—21no—
45#2366C2211yescharacteristic
46#2365C2211yescharacteristic
47#2354—21no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#125C2211yescharacteristic
52C2×#384C2212yes—
53C42×D4C2211yescharacteristic
54#2095C2214yes—
55#2094C2214yes—
56#2068C2214yes—
57#2065C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2042C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2033C2214yes—
64#2033C2214yes—
65#1993C2211yescharacteristic
66#1992C2212yes—
67#1992C2211yescharacteristic
68#1982C2212yes—
69#1981C2212yes—
70#1959C2212yes—
71#1959C2212yes—
72#1959C2212yes—
73#1957C2212yes—
74#1956C2212yes—
75#1954C2212yes—
76#1948C2212yes—
77#1947C2212yes—
78#1944C2212yes—
79#1943C2212yes—
80#1939C2212yes—
81#1924C2214yes—
82#1874C2214yes—
83#1872C2214yes—
84#1864C2214yes—
85#1863C2214yes—
86#1847C2211yescharacteristic
87#1847C2212yes—
88#1803C2214yes—
89#1802C2214yes—
90C4×#107C2212yes—
91C4×#106C2212yes—
92#1767C2211yescharacteristic
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127—21no—
102C2×#127C2311yescharacteristic
103#529—28no—
104#529—44no—
105C2×#126—22no—
106C2×#125—22no—
107C2×#125C2311yescharacteristic
108C2×#124C2312yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121C2312yes—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131#501—24no—
132#489—48no—
133#489—44no—
134#483—48no—
135#480—48no—
136#432—44no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2312yes—
156C2×#97C2311yescharacteristic
157C2×#97C2312yes—
158C2×#97C2311yescharacteristic
159C2×C4×C8C2312yes—
160#399—22no—
161#398—24no—
162#398—24no—
163#397—22no—
164#396—22no—
165#395—24no—
166#394—22no—
167#394—22no—
168#392—22no—
169#391—21no—
170#391—22no—
171#391C2312yes—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390C2312yes—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2314yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—21no—
194#387—21no—
195#387—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#385—24no—
201#385—24no—
202#384—21no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384—22no—
207#384C2311yescharacteristic
208#384C2314yes—
209#384C2312yes—
210#384C2312yes—
211#382C2311yescharacteristic
212#380—22no—
213#380—22no—
214#380C2314yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2412yes—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#127—24no—
261#127—44no—
262#126—48no—
263#125—22no—
264#125—48no—
265#125#4114yes—
266#124—24no—
267#124—24no—
268#124—44no—
269#123—48no—
270C22⋊Q8—24no—
271C22⋊Q8—416no—
272C22⋊Q8—42no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275#121—24no—
276#121—42no—
277#121—44no—
278C4×Q8—24no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—44no—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—24no—
359#107—44no—
360#106—24no—
361#106—24no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—28no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—44no—
371#102—44no—
372#102—44no—
373C22⋊C8—28no—
374C22⋊C8—44no—
375#97—24no—
376#97—22no—
377#97—22no—
378#97—24no—
379C4×C8—24no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—22no—
396C24C22×D412yes—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4AGL(1,ℤ/8)14yes—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8H(5,𝔽2)14yes—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#13714yes—
504C4⋊C4C2×#4112yes—
505C4⋊C4C2×#4112yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—41no—
529C42—41no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—42no—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C422−1+414yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C2×AGL(1,ℤ/8)14yes—
551C23C23×D411yescharacteristic
552Q8—44no—
553Q8—44no—
554Q8—84no—
555Q8—164no—
556D4—28no—
557D4—44no—
558D4—44no—
559D4—88no—
560D4—84no—
561D4—162no—
562D4—164no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—88no—
612C2×C4—168no—
613C2×C4#53612yes—
614C2×C4#53612yes—
615C2×C4#54612yes—
616C2×C4#54612yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×#13714yes—
619C2×C4C2×#13712yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)12yes—
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+412yes—
624C8—24no—
625C8—42no—
626C8—44no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22C2×#53611yescharacteristic
642C22C2×#54611yescharacteristic
643C22#349811yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C22×#13711yescharacteristic
646C22#358511yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—81no—
663C4—81no—
664C4#323814yes—
665C4#352212yes—
666C4#354812yes—
667C2—44no—
668C2—162no—
669C2#3895211yescharacteristic
670C2C2×#323811yescharacteristic
671C2#6088511yescharacteristic
672C2#6116111yescharacteristic
673C2#6116211yescharacteristic
674C2#6244711yescharacteristic
675C2#6245411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #36180C2semidirect
3. #33230C2semidirect
4. #30657C2semidirect
5. #30653C2semidirect
6. #29853C2semidirect
7. #29213C2semidirect
8. #28754C2semidirect
9. #27914C2semidirect
10. #26532C2semidirect
11. #26530C2semidirect
12. #26234C2semidirect
14. #25679C2semidirect
15. #21867C2semidirect
16. C4×#501C2semidirect
17. #21270C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#361152
#332302
#306572
#306532
#298532
#292132
#287542
#279142
#265322
#265302
#262342
#262302
#256792
#218672
C4×#5012
#212702