Finite Group Databasebeta

← #6461730

Subgroups — #6461730

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34253C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29407C214yesmaximal, complemented
4#29404C214yesmaximal, complemented
5#29395C212yesmaximal, complemented
6#29394C212yesmaximal, complemented
7#29389C212yesmaximal, complemented
8#29388C212yesmaximal, complemented
9C2×#2083C211yesmaximal, characteristic, complemented
10#23630C211yesmaximal, characteristic
11#23554C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic
13#21062C212yesmaximal
14#21037C212yesmaximal, complemented
15#21035C212yesmaximal, complemented
16#20881C212yesmaximal
17#20774C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#428C2212yes—
21C22×#107C2212yes—
22C2×#417C2212yes—
23C2×#416C2212yes—
24#2360—22no—
25#2360—22no—
26#2355—21no—
27#2355—21no—
28#2353—22no—
29#2352—22no—
30Q8×C4⋊C4—21no—
31Q8×C4⋊C4—21no—
32#2347—21no—
33#2346—21no—
34#2345—21no—
35#2344—21no—
36#2330C2214yes—
37#2320C2212yes—
38#2319C2212yes—
39#2317C2212yes—
40#2316C2211yescharacteristic
41#2311C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44#2297C2211yescharacteristic
45#2295C2212yes—
46#2293C2212yes—
47C2×#389C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2274C2211yescharacteristic
50#2274C2212yes—
51C4×#117C2211yescharacteristic
52#2083C2212yes—
53#2083C2212yes—
54#2083C2214yes—
55#2072C2212yes—
56#2072C2212yes—
57#2071C2212yes—
58#2071C2212yes—
59#2059C2214yes—
60#2058C2212yes—
61#2057C2212yes—
62#2026C2212yes—
63#2025C2212yes—
64#2024C2211yescharacteristic
65#2024C2211yescharacteristic
66#2023C2211yescharacteristic
67#2023C2211yescharacteristic
68#1928C2214yes—
69#1927C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1921C2214yes—
73#1921C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1918C2214yes—
78#1918C2214yes—
79#1918C2214yes—
80#1863C2214yes—
81#1863C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1840C2214yes—
85#1839C2212yes—
86#1839C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#533—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104#520—24no—
105#520—24no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C2×#117—22no—
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#428—28no—
128C2×#107—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133#417—28no—
134#416—28no—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141#391—22no—
142#391—22no—
143#391C2312yes—
144#391C2311yescharacteristic
145#391C2311yescharacteristic
146#390C2312yes—
147#390C2311yescharacteristic
148#390C2311yescharacteristic
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—24no—
156#389—24no—
157#389C2314yes—
158#389C2314yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—24no—
172#387—24no—
173#386—21no—
174#386—21no—
175#386—22no—
176#385—24no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2312yes—
182#383—22no—
183#382—22no—
184#382—22no—
185#381—22no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212#376—22no—
213#376C2312yes—
214C2×#95—21no—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#336C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41#4114yes—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263#125—24no—
264#125—24no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—84no—
272C4×Q8—84no—
273C4×D4—216no—
274C4×D4—24no—
275C4×D4—24no—
276#117—28no—
277#117—22no—
278#117—22no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—84no—
283#117#4118yes—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4#4114yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—28no—
352#107—28no—
353#107—28no—
354#106—28no—
355#103—28no—
356#103—24no—
357#103—24no—
358#102—28no—
359#102—24no—
360#102—24no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24C22×D411yescharacteristic
383#41—216no—
384#41—48no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8C2×#4112yes—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4C2×#4112yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4—81no—
429C22×C4—82no—
430C22×C4—81no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4112yes—
443C22×C42−1+414yes—
444C22×C4C2511yescharacteristic
445M16—48no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—48no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4C2×#4114yes—
487C4⋊C4C2×#4114yes—
488C4⋊C4C2×#4114yes—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—82no—
498C22⋊C4—81no—
499C22⋊C4—82no—
500C42—28no—
501C42—22no—
502C42—22no—
503C42—28no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—82no—
515C42—82no—
516C42—84no—
517C42—84no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23#52512yes—
530C23#52512yes—
531C23#53612yes—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—48no—
535Q8—48no—
536Q8—42no—
537D4—24no—
538D4—48no—
539D4—42no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4—162no—
592C2×C4—164no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#53612yes—
596C2×C4#55218yes—
597C2×C4#55618yes—
598C2×C4#57612yes—
599C2×C4#57812yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4112yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×2−1+412yes—
604C8—44no—
605C8—42no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22#259914yes—
621C22C2×#52511yescharacteristic
622C22C2×#52511yescharacteristic
623C22C2×#53611yescharacteristic
624C22C2×#57611yescharacteristic
625C22C2×#57811yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—42no—
631C4—41no—
632C4—44no—
633C4—41no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—82no—
641C4—84no—
642C4—81no—
643C4—81no—
644C4#259914yes—
645C4C2×#55212yes—
646C4C2×#55612yes—
647C2—24no—
648C2—82no—
649C2C2×#259911yescharacteristic
650C2#6058411yescharacteristic
651C2#6058911yescharacteristic
652C2#6059011yescharacteristic
653C2#6059411yescharacteristic
654C2#6088011yescharacteristic
655C2#6088711yescharacteristic
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. #34214C2semidirect
3. #29407C2semidirect
4. #29404C2semidirect
5. #29395C2semidirect
6. #29394C2semidirect
7. #29389C2semidirect
8. #29388C2semidirect
9. C2×#2083C2semidirect
11. #23554C2semidirect
14. #21037C2semidirect
15. #21035C2semidirect
17. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#342532
#342142
#294072
#294042
#293952
#293942
#293892
#293882
C2×#20832
#236302
#235542
#234512
#210622
#210372
#210352
#208812
#207742