Finite Group Databasebeta

← #6434368

Subgroups — #6434368

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic
3#30088C214yesmaximal, complemented
4#30087C214yesmaximal, complemented
5#30083C212yesmaximal, complemented
6#30082C212yesmaximal, complemented
7#30079C212yesmaximal, complemented
8#30078C212yesmaximal, complemented
9#23732C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic
11#23662C212yesmaximal, complemented
12#23630C211yesmaximal, characteristic
13C2×#2072C211yesmaximal, characteristic, complemented
14#23510C212yesmaximal
15#20880C212yesmaximal, complemented
16#20878C212yesmaximal
17#20774C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#519C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#417C2212yes—
24C2×#419C2212yes—
25C2×#416C2212yes—
26C22×#103C2212yes—
27#2859C2212yes—
28#2859C2212yes—
29#2647C2214yes—
30#2571—21no—
31#2571—21no—
32#2552—21no—
33#2552—21no—
34#2442—22no—
35#2442—22no—
36#2362—21no—
37#2362—21no—
38#2360—21no—
39#2360—21no—
40#2359—22no—
41#2359—22no—
42#2327C2212yes—
43#2326C2212yes—
44#2317C2212yes—
45#2317C2211yescharacteristic
46#2316C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49#2297C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2097C2212yes—
55#2097C2214yes—
56#2096C2211yescharacteristic
57#2096C2212yes—
58#2095C2211yescharacteristic
59#2095C2212yes—
60#2094C2211yescharacteristic
61#2093C2211yescharacteristic
62#2092C2212yes—
63#2085C2214yes—
64#2085C2214yes—
65#2084C2214yes—
66#2083C2214yes—
67#2072C2212yes—
68#2072C2212yes—
69#2072C2214yes—
70#2072C2212yes—
71#2072C2212yes—
72#2071C2212yes—
73#2071C2212yes—
74#2043C2214yes—
75#2043C2214yes—
76#2042C2214yes—
77#2042C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1864C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1863C2214yes—
85#1862C2214yes—
86#1840C2214yes—
87#1839C2212yes—
88#1839C2212yes—
89#1736C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—24no—
100#536—24no—
101C2×C4⋊Q8C2312yes—
102C2×#125—22no—
103C2×#125—22no—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108#520—44no—
109#519—44no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×#117—21no—
118C2×#117—22no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#428—44no—
130#427—44no—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137#419—44no—
138#417—28no—
139#416—44no—
140C2×#103—28no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151#400—24no—
152#400—24no—
153#398—24no—
154#398—24no—
155#391—21no—
156#391—21no—
157#391C2311yescharacteristic
158#391C2312yes—
159#391C2312yes—
160#391C2311yescharacteristic
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—21no—
186#386—21no—
187#385—22no—
188#385—24no—
189#385—22no—
190#385—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—22no—
200#382—22no—
201#382—22no—
202#382C2312yes—
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380—24no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221#376C2312yes—
222C2×#95—21no—
223C2×#95—21no—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41#4114yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8#4114yes—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270#125—84no—
271#125—84no—
272#124—24no—
273#124—24no—
274#123—216no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—44no—
283C4×D4—48no—
284#117—22no—
285#117—22no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—48no—
291#117—44no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—42no—
357#107—42no—
358#107—44no—
359#106—42no—
360#106—42no—
361#106—44no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—42no—
366#103—42no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—42no—
371#102—42no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24C22×D411yescharacteristic
394#41—216no—
395#41—48no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—216no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—48no—
407C2×Q8C2×#4114yes—
408C2×Q8C2×#4112yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×#4112yes—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C42−1+414yes—
458C22×C4C2511yescharacteristic
459M16—48no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—48no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—44no—
529C42—44no—
530C42—42no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23#52512yes—
543C23#53512yes—
544C23#53612yes—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—24no—
548Q8—48no—
549Q8—44no—
550Q8—42no—
551D4—24no—
552D4—48no—
553D4—42no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—164no—
605C2×C4—162no—
606C2×C4—164no—
607C2×C4#52512yes—
608C2×C4#53512yes—
609C2×C4#53612yes—
610C2×C4#53918yes—
611C2×C4#53918yes—
612C2×C4#57712yes—
613C2×C4#57812yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4C2×2−1+412yes—
618C8—42no—
619C8—42no—
620C8—44no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#263314yes—
635C22C2×#52511yescharacteristic
636C22C2×#53511yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#57711yescharacteristic
639C22C2×#57811yescharacteristic
640C22#358511yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4#263314yes—
659C4C2×#53912yes—
660C4C2×#53912yes—
661C2—24no—
662C2—82no—
663C2C2×#263311yescharacteristic
664C2#6059311yescharacteristic
665C2#6059411yescharacteristic
666C2#6082411yescharacteristic
667C2#6082911yescharacteristic
668C2#6087211yescharacteristic
669C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #30088C2semidirect
4. #30087C2semidirect
5. #30083C2semidirect
6. #30082C2semidirect
7. #30079C2semidirect
8. #30078C2semidirect
9. #23732C2semidirect
11. #23662C2semidirect
13. C2×#2072C2semidirect
15. #20880C2semidirect
17. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#342762
#300882
#300872
#300832
#300822
#300792
#300782
#237322
#237302
#236622
#236302
C2×#20722
#235102
#208802
#208782
#207742