Finite Group Databasebeta

← #6798771

Subgroups — #6798771

Order 512.

1637 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36174C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#31760C214yesmaximal, complemented
5#31756C214yesmaximal, complemented
6#29854C212yesmaximal, complemented
7#29212C211yesmaximal, characteristic, complemented
8#28726C211yesmaximal, characteristic, complemented
9#27882C212yesmaximal, complemented
10#26022C212yesmaximal, complemented
11#26020C212yesmaximal, complemented
12#25908C212yesmaximal, complemented
13#24971C212yesmaximal, complemented
14#24967C212yesmaximal
15#21841C212yesmaximal, complemented
16C4×#489C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#489C2211yescharacteristic
21C2×#489C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2551—21no—
28#2551C2212yes—
29#2546—21no—
30#2538C2211yescharacteristic
31#2537C2211yescharacteristic
32#2526—22no—
33#2518—22no—
34#2495—21no—
35#2490C2212yes—
36#2489C2212yes—
37#2440—22no—
38#2440C2214yes—
39#2434—22no—
40#2420—21no—
41#2363—21no—
42#2355C2211yescharacteristic
43#2354C2211yescharacteristic
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46#2346—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#124C2211yescharacteristic
49#2283C2212yes—
50C42×D4C2211yescharacteristic
51#2077C2214yes—
52#2076C2214yes—
53#2072C2212yes—
54#2071C2212yes—
55#2067C2214yes—
56#2066C2214yes—
57#2059C2214yes—
58#2036C2214yes—
59#2034C2214yes—
60#2024C2214yes—
61#2023C2214yes—
62#1993C2212yes—
63#1993C2212yes—
64#1993C2212yes—
65#1993C2212yes—
66#1993C2211yescharacteristic
67#1992C2214yes—
68#1988C2214yes—
69#1988C2214yes—
70#1959C2211yescharacteristic
71#1954C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1936C2212yes—
76#1935C2212yes—
77#1924C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1874C2214yes—
81#1872C2214yes—
82#1847C2212yes—
83#1836C2211yescharacteristic
84#1834C2212yes—
85#1834C2212yes—
86#1824C2214yes—
87#1824C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8—22no—
96D4×Q8—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125C2312yes—
108C2×#124C2311yescharacteristic
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2314yes—
124C22×C42C2312yes—
125#489—24no—
126#489—44no—
127#432—44no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×#97C2311yescharacteristic
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#398—24no—
147#397—22no—
148#397—22no—
149#396—22no—
150#395—24no—
151#395—24no—
152#394—22no—
153#394—22no—
154#391—21no—
155#391—22no—
156#391C2312yes—
157#391C2311yescharacteristic
158#390—21no—
159#390—22no—
160#390C2312yes—
161#390C2311yescharacteristic
162#389—21no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389C2312yes—
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—21no—
178#388—22no—
179#388—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#385—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#383—22no—
193#382—21no—
194#382—22no—
195#382C2312yes—
196#382C2311yescharacteristic
197#381—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—24no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—48no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—82no—
262C4⋊Q8#4114yes—
263#125—44no—
264#124—22no—
265#124—22no—
266C22⋊Q8—416no—
267C22⋊Q8—48no—
268C22⋊Q8—44no—
269#121—44no—
270#121—44no—
271C4×Q8—24no—
272C4×Q8—44no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—48no—
278#117—44no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—44no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—24no—
348#103—42no—
349#103—44no—
350#102—24no—
351#102—42no—
352#102—44no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355#97—22no—
356#97—22no—
357#97—24no—
358#97—42no—
359C4×C8—24no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D412yes—
381C2×Q8—24no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—84no—
394C2×Q8—832no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4#13714yes—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—84no—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4AGL(1,ℤ/8)14yes—
480C4⋊C4C2×#4114yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—44no—
507C42—44no—
508C42—82no—
509C42C2×#4112yes—
510C42H(5,𝔽2)12yes—
511C422−1+414yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×#13714yes—
526C23C23×D411yescharacteristic
527Q8—48no—
528Q8—42no—
529Q8—42no—
530Q8—816no—
531Q8—88no—
532D4—28no—
533D4—42no—
534D4—42no—
535D4—164no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—816no—
585C2×C4—84no—
586C2×C4#53612yes—
587C2×C4#53612yes—
588C2×C4#54612yes—
589C2×C4#54612yes—
590C2×C4C2×AGL(1,ℤ/8)14yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×#13712yes—
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×2−1+412yes—
596C2×C4#58512yes—
597C8—42no—
598C8—42no—
599C8—44no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×#53611yescharacteristic
615C22C2×#54611yescharacteristic
616C22#350011yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C22×#13711yescharacteristic
619C22#358511yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—24no—
624C4—22no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—44no—
629C4—42no—
630C4—81no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4—84no—
637C4#323914yes—
638C4#351812yes—
639C4#354712yes—
640C2—44no—
641C2—161no—
642C2#3895811yescharacteristic
643C2C2×#323911yescharacteristic
644C2#6088711yescharacteristic
645C2#6115511yescharacteristic
646C2#6115711yescharacteristic
647C2#6258211yescharacteristic
648C2#6259511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Q8C2semidirect
3. #33297C2semidirect
4. #31760C2semidirect
5. #31756C2semidirect
6. #29854C2semidirect
7. #29212C2semidirect
8. #28726C2semidirect
9. #27882C2semidirect
10. #26022C2semidirect
11. #26020C2semidirect
12. #25908C2semidirect
13. #24971C2semidirect
15. #21841C2semidirect
16. C4×#489C2semidirect
17. #21270C2semidirect

Maximal subgroups

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

subgroupindex
#361742
D4×C4⋊Q82
#332972
#317602
#317562
#298542
#292122
#287262
#278822
#260222
#260202
#259082
#249712
#249672
#218412
C4×#4892
#212702