Finite Group Databasebeta

← #6800344

Subgroups — #6800344

Order 512.

1591 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#32310C214yesmaximal, complemented
5#31454C214yesmaximal, complemented
6#29966C212yesmaximal, complemented
7#28789C212yesmaximal, complemented
8#27960C211yesmaximal, characteristic, complemented
9#27960C212yesmaximal, complemented
10#27727C212yesmaximal, complemented
11#27644C212yesmaximal
12#27644C211yesmaximal, characteristic
13#26506C212yesmaximal, complemented
14#26106C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22075C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#554—21no—
19C22×#125—21no—
20C2×#480C2212yes—
21C2×#480C2211yescharacteristic
22#2857C2212yes—
23#2857C2211yescharacteristic
24#2564—22no—
25#2556—22no—
26#2552—21no—
27#2552—21no—
28#2552—22no—
29#2552C2214yes—
30#2552C2212yes—
31#2552C2212yes—
32#2551C2211yescharacteristic
33#2542C2211yescharacteristic
34#2506C2212yes—
35#2492—21no—
36#2452C2212yes—
37#2417—21no—
38C2×#398—22no—
39#2365C2211yescharacteristic
40#2360—21no—
41D4×C4⋊C4C2211yescharacteristic
42C4×#121C2212yes—
43C4×#121C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2099C2214yes—
51#2099C2214yes—
52#2099C2214yes—
53#2099C2214yes—
54#2086C2214yes—
55#2086C2214yes—
56#2083C2212yes—
57#2083C2214yes—
58#2071C2214yes—
59#2071C2212yes—
60#2057C2214yes—
61#1990C2212yes—
62#1990C2214yes—
63#1988C2212yes—
64#1947C2214yes—
65#1947C2212yes—
66#1947C2212yes—
67#1942C2214yes—
68#1939C2212yes—
69#1934C2214yes—
70#1925C2212yes—
71#1918C2212yes—
72#1918C2214yes—
73#1918C2214yes—
74#1916C2214yes—
75#1874C2211yescharacteristic
76#1862C2214yes—
77#1852C2211yescharacteristic
78#1852C2212yes—
79#1834C2214yes—
80#1834C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1823C2212yes—
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1802C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×#125—24no—
101C2×#125—22no—
102C2×#125—22no—
103C2×#125—48no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×#121C2311yescharacteristic
110C2×C4×Q8—21no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#480—28no—
130#480—24no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2311yescharacteristic
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#399—22no—
147#398—22no—
148#398—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#396—22no—
153#395—22no—
154#393—22no—
155#391—21no—
156#391—21no—
157#391—22no—
158#390C2311yescharacteristic
159#390C2312yes—
160#389—21no—
161#389—22no—
162#389—24no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#388—22no—
176#388—21no—
177#387—22no—
178#387—22no—
179#386—21no—
180#386—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—22no—
186#384—22no—
187#384C2312yes—
188#384C2314yes—
189#384C2314yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#380—21no—
194#380—22no—
195#380—21no—
196#380C2312yes—
197#380C2311yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C2×#95—21no—
213C2×#95—22no—
214C2×#95D412yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2314yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2411yescharacteristic
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252#125—48no—
253#125—416no—
254#125—48no—
255#125—48no—
256C22⋊Q8—48no—
257C22⋊Q8—44no—
258C22⋊Q8—48no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261#121—22no—
262#121#4114yes—
263C4×Q8—44no—
264C4×D4—24no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—44no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—84no—
304C2×C4⋊C4D818yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—28no—
344#107—24no—
345#107—42no—
346#107—44no—
347C4⋊C8—24no—
348C4⋊C8—44no—
349#102—28no—
350#102—24no—
351#102—42no—
352C22⋊C8—22no—
353C22⋊C8—24no—
354C22⋊C8—22no—
355C4×C8—24no—
356C4×C8—44no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C24C22×D411yescharacteristic
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—42no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4C2×#4112yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4C2×D814yes—
425C22×C4C2×D818yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4112yes—
436C22×C42−1+412yes—
437C22×C4C2511yescharacteristic
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—81no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C2×D814yes—
485C4⋊C4C2×D818yes—
486C4⋊C4C2×D814yes—
487C4⋊C4#13714yes—
488C4⋊C4C2×#4112yes—
489C22⋊C4—24no—
490C22⋊C4—44no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—81no—
497C22⋊C4—82no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—44no—
507C42—44no—
508C42—42no—
509C422−1+414yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23#54812yes—
522C23C22×D812yes—
523C23#57712yes—
524C23C23×D411yescharacteristic
525Q8—84no—
526Q8—164no—
527D4—24no—
528D4—48no—
529D4—42no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4—84no—
582C2×C4—164no—
583C2×C4#53612yes—
584C2×C4#53612yes—
585C2×C4#54812yes—
586C2×C4C22×D812yes—
587C2×C4C22×D814yes—
588C2×C4C22×D814yes—
589C2×C4C22×D812yes—
590C2×C4C2×#13714yes—
591C2×C4C2×#13712yes—
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×2−1+412yes—
594C2×C4C2×2−1+411yescharacteristic
595C2×C4#58512yes—
596C8—42no—
597C8—44no—
598C8—42no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22#335814yes—
613C22C2×#53611yescharacteristic
614C22C2×#54811yescharacteristic
615C22#348511yescharacteristic
616C22C23×D811yescharacteristic
617C22C22×#13711yescharacteristic
618C22C2×#57711yescharacteristic
619C22#358611yescharacteristic
620C4—24no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—82no—
630C4—84no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4#322614yes—
637C4#351512yes—
638C4#356312yes—
639C2—24no—
640C2—82no—
641C2#3912211yescharacteristic
642C2C2×#322611yescharacteristic
643C2#6087211yescharacteristic
644C2C2×#335811yescharacteristic
645C2#6132211yescharacteristic
646C2#6169411yescharacteristic
647C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #33146C2semidirect
4. #32310C2semidirect
5. #31454C2semidirect
6. #29966C2semidirect
7. #28789C2semidirect
8. #27960C2semidirect
9. #27960C2semidirect
10. #27727C2semidirect
13. #26506C2semidirect
14. #26106C2semidirect
15. #22625C2semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25522
#331462
#323102
#314542
#299662
#287892
#279602
#277272
#276442
#265062
#261062
#226252
#220752
C4×#4802