Finite Group Databasebeta

← #6458458

Subgroups — #6458458

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#35181C211yesmaximal, characteristic, complemented
3#27744C214yesmaximal, complemented
4#27742C214yesmaximal, complemented
5#27738C212yesmaximal, complemented
6#27735C212yesmaximal, complemented
7#27729C212yesmaximal, complemented
8#27728C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23634C212yesmaximal
11#21324C211yesmaximal, characteristic, complemented
12C2×#1987C211yesmaximal, characteristic, complemented
13#21087C212yesmaximal
14#20763C212yesmaximal, complemented
15#20645C211yesmaximal, characteristic
16#20441C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#417C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#416C2212yes—
24C2×#405C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2857—21no—
27#2857—21no—
28#2851—22no—
29#2646—22no—
30#2628—22no—
31#2571—21no—
32#2571C2212yes—
33#2570—22no—
34#2570C2214yes—
35#2568C2212yes—
36#2567C2212yes—
37#2563C2212yes—
38#2552—21no—
39#2552C2212yes—
40#2542C2212yes—
41#2487—22no—
42#2409—22no—
43#2317C2212yes—
44#2317C2212yes—
45#2317C2212yes—
46#2311C2212yes—
47#2285C2211yescharacteristic
48#2285C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2074C2214yes—
55#2073C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#1993C2214yes—
59#1990C2212yes—
60#1990C2212yes—
61#1987C2212yes—
62#1987C2212yes—
63#1987C2214yes—
64#1936C2214yes—
65#1935C2214yes—
66#1934C2214yes—
67#1933C2214yes—
68#1836C2214yes—
69#1836C2214yes—
70#1835C2214yes—
71#1834C2214yes—
72#1805C2212yes—
73#1804C2212yes—
74#1803C2211yescharacteristic
75#1803C2211yescharacteristic
76#1802C2211yescharacteristic
77#1802C2211yescharacteristic
78#1771C2214yes—
79#1770C2212yes—
80C8×C4⋊C4C2212yes—
81#1763C2212yes—
82#1762C2212yes—
83C4×#103C2212yes—
84C4×#102C2212yes—
85#1736C2211yescharacteristic
86#1736C2212yes—
87#1736C2212yes—
88#1736C2211yescharacteristic
89#1735C2212yes—
90C2×#340C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C22×#41C2311yescharacteristic
94C23×Q8—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C4×#41—24no—
108C4×#41—24no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C2×C4⋊C8—24no—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130#417—24no—
131#417—24no—
132#416—28no—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141#405—28no—
142C4×M16—24no—
143C2×#97C2314yes—
144C2×#97C2312yes—
145C2×C4×C8C2314yes—
146C2×C4×C8C2312yes—
147#400—24no—
148#400—24no—
149#400—24no—
150#399—24no—
151#398—24no—
152#398—24no—
153#395—24no—
154#393—24no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#387—24no—
170#387—24no—
171#386—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384—22no—
176#384D411yescharacteristic
177#384D411yescharacteristic
178#384C2314yes—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2314yes—
182#382—22no—
183#382—22no—
184#382D412yes—
185#382C2314yes—
186#382C2312yes—
187#382C2314yes—
188#380—21no—
189#380—21no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380D412yes—
195#380D412yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C2×#95—21no—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#336C2314yes—
222#336C2314yes—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41#4114yes—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—44no—
232C22×Q8—82no—
233C22×Q8—82no—
234C22×Q8C2411yescharacteristic
235C22×D4—21no—
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—22no—
251C2×M16#4114yes—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#125—84no—
266C22⋊Q8—84no—
267C22⋊Q8—84no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—44no—
277C4×D4—44no—
278#117—24no—
279#117—24no—
280#117—22no—
281#117—22no—
282#117—24no—
283#117—24no—
284#117—44no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D414yes—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42—82no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42#4114yes—
340C2×C42#4114yes—
341C2×C42C2414yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—24no—
347#103—24no—
348#103—24no—
349#103—24no—
350#103—24no—
351#102—24no—
352#102—24no—
353#102—24no—
354#102—24no—
355#97—28no—
356#97—24no—
357C4×C8—28no—
358C4×C8—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—21no—
374C24C22×D411yescharacteristic
375#41—28no—
376#41—28no—
377#41—44no—
378#41—44no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—816no—
391C2×Q8C2×#4112yes—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4C2×#4112yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#12414yes—
432C22×C4#12414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447M16—28no—
448M16—44no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—48no—
461C2×C8C2×#4114yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C22⋊C4—28no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—24no—
507C42—24no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—44no—
513C42—82no—
514C42#12414yes—
515C42#12414yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C42C2×#4112yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×#12412yes—
532C23#54712yes—
533C23#54812yes—
534C23C23×D411yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—88no—
540Q8—88no—
541D4—22no—
542D4—22no—
543D4—48no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—84no—
590C2×C4C2×#12412yes—
591C2×C4C2×#12412yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#12412yes—
594C2×C4C2×#12412yes—
595C2×C4#52914yes—
596C2×C4#54114yes—
597C2×C4#54712yes—
598C2×C4#54812yes—
599C2×C4#55214yes—
600C2×C4#55214yes—
601C2×C4#57712yes—
602C2×C4#57812yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C8—24no—
610C8—42no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22#248914yes—
627C22C22×#12411yescharacteristic
628C22C2×#54711yescharacteristic
629C22C2×#54811yescharacteristic
630C22C2×#57711yescharacteristic
631C22C2×#57811yescharacteristic
632C22#358511yescharacteristic
633C22#358511yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—81no—
645C4—81no—
646C4—84no—
647C4#248914yes—
648C4#313714yes—
649C4C2×#52911yescharacteristic
650C4C2×#54111yescharacteristic
651C4C2×#55211yescharacteristic
652C4C2×#55211yescharacteristic
653C2—24no—
654C2—82no—
655C2C2×#248911yescharacteristic
656C2C2×#313711yescharacteristic
657C2#6061911yescharacteristic
658C2#6127911yescharacteristic
659C2#6128211yescharacteristic
660C2#6134711yescharacteristic
661C2#6134811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35181C2semidirect
3. #27744C2semidirect
4. #27742C2semidirect
5. #27738C2semidirect
6. #27735C2semidirect
7. #27729C2semidirect
8. #27728C2semidirect
9. #23748C2semidirect
11. #21324C2semidirect
12. C2×#1987C2semidirect
14. #20763C2semidirect
16. #20441C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#351812
#277442
#277422
#277382
#277352
#277292
#277282
#237482
#236342
#213242
C2×#19872
#210872
#207632
#206452
#204412
C4×#4172