Finite Group Databasebeta

← #6434764

Subgroups — #6434764

Order 512.

1673 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30070C214yesmaximal, complemented
4#30068C214yesmaximal, complemented
5#30064C212yesmaximal, complemented
6#30063C212yesmaximal, complemented
7#30061C212yesmaximal, complemented
8#30060C212yesmaximal, complemented
9#23732C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic
11#23557C212yesmaximal, complemented
12#23512C212yesmaximal
13#23452C211yesmaximal, characteristic
14C2×#2024C211yesmaximal, characteristic, complemented
15#21066C212yesmaximal, complemented
16#21062C212yesmaximal
17#20774C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#536C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#417C2212yes—
24C2×#419C2212yes—
25C2×#416C2212yes—
26C22×#103C2212yes—
27#2862C2212yes—
28#2861C2212yes—
29#2642C2212yes—
30#2640C2212yes—
31#2492—21no—
32#2492—21no—
33#2489—21no—
34#2489—21no—
35#2488—22no—
36#2488—22no—
37#2369—21no—
38#2369—21no—
39#2365—21no—
40#2365—21no—
41#2345—22no—
42Q8×C22⋊C4—22no—
43#2331C2214yes—
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2310C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2212yes—
49#2297C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285C2211yescharacteristic
52#2274C2212yes—
53C4×#117C2211yescharacteristic
54#2097C2214yes—
55#2097C2212yes—
56#2096C2211yescharacteristic
57#2096C2212yes—
58#2095C2212yes—
59#2095C2211yescharacteristic
60#2094C2211yescharacteristic
61#2093C2211yescharacteristic
62#2092C2212yes—
63#2059C2214yes—
64#2059C2214yes—
65#2058C2214yes—
66#2057C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2024C2212yes—
72#2024C2212yes—
73#2024C2212yes—
74#2024C2214yes—
75#2024C2212yes—
76#2023C2212yes—
77#2023C2212yes—
78#1928C2214yes—
79#1928C2214yes—
80#1928C2214yes—
81#1927C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1926C2214yes—
85#1925C2214yes—
86#1840C2214yes—
87#1839C2212yes—
88#1839C2212yes—
89#1736C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—44no—
101#536—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127#428—44no—
128#427—44no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135#419—44no—
136#417—28no—
137#416—44no—
138C2×#103—28no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#397—22no—
154#397—22no—
155#393—22no—
156#393—22no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—21no—
164#391—21no—
165#391—24no—
166#391—24no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2311yescharacteristic
172#391C2312yes—
173#391C2311yescharacteristic
174#390C2312yes—
175#390C2311yescharacteristic
176#390C2311yescharacteristic
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#387—24no—
189#386—21no—
190#386—21no—
191#386—22no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382—22no—
196#382C2312yes—
197#381—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—24no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—24no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376C2312yes—
221C2×#95—21no—
222C2×#95—21no—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C22×Q8—28no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—82no—
251C22×Q8—82no—
252C22×Q8C2412yes—
253C22×Q8C2411yescharacteristic
254C22×D4—21no—
255C22×D4C2411yescharacteristic
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C4⋊Q8—28no—
273C4⋊Q8—28no—
274C4⋊Q8—28no—
275C4⋊Q8—28no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279#125—44no—
280#125—44no—
281#124—44no—
282#123—48no—
283C22⋊Q8—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C4×Q8—84no—
287C4×Q8—84no—
288#117—24no—
289#117—22no—
290#117—22no—
291#117—24no—
292#117—24no—
293#117—24no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4#4118yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—42no—
351#107—42no—
352#107—44no—
353#106—42no—
354#106—42no—
355#106—44no—
356#103—28no—
357#103—28no—
358#103—28no—
359#103—42no—
360#103—42no—
361#103—44no—
362#103—44no—
363#102—28no—
364#102—42no—
365#102—42no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C24C22×D411yescharacteristic
392#41—48no—
393#41—48no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—816no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
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×C4—81no—
449C22×C4—81no—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464M16—48no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—48no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C2×#4118yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C22⋊C4—81no—
513C22⋊C4—82no—
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—42no—
528C42—84no—
529C42—84no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23#52512yes—
542C23#53512yes—
543C23#53612yes—
544C23C23×D411yescharacteristic
545Q8—416no—
546Q8—44no—
547Q8—44no—
548Q8—42no—
549Q8—42no—
550Q8—88no—
551Q8—84no—
552Q8—164no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—162no—
608C2×C4#52512yes—
609C2×C4#53512yes—
610C2×C4#53612yes—
611C2×C4#54218yes—
612C2×C4#54218yes—
613C2×C4#57712yes—
614C2×C4#57812yes—
615C2×C4C22×#4112yes—
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C8—42no—
620C8—42no—
621C8—44no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#263214yes—
636C22C2×#52511yescharacteristic
637C22C2×#53511yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358511yescharacteristic
642C22#358611yescharacteristic
643C4—22no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—48no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4#263214yes—
660C4C2×#54212yes—
661C4C2×#54212yes—
662C2—24no—
663C2—161no—
664C2C2×#263211yescharacteristic
665C2#6057911yescharacteristic
666C2#6058311yescharacteristic
667C2#6082411yescharacteristic
668C2#6082911yescharacteristic
669C2#6088911yescharacteristic
670C2#6089311yescharacteristic
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
2. #34216C2semidirect
3. #30070C2semidirect
4. #30068C2semidirect
5. #30064C2semidirect
6. #30063C2semidirect
7. #30061C2semidirect
8. #30060C2semidirect
9. #23732C2semidirect
11. #23557C2semidirect
14. C2×#2024C2semidirect
15. #21066C2semidirect
17. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#342162
#300702
#300682
#300642
#300632
#300612
#300602
#237322
#237302
#235572
#235122
#234522
C2×#20242
#210662
#210622
#207742