Finite Group Databasebeta

← #6624979

Subgroups — #6624979

Order 512.

1759 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#36564C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#32954C212yesmaximal
5#29111C212yesmaximal
6#28124C212yesmaximal
7#26911C214yesmaximal, complemented
8#26775C214yesmaximal, complemented
9#26462C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic
11#25324C212yesmaximal, complemented
12#24212C211yesmaximal, characteristic
13#23221C212yesmaximal, complemented
14#23004C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic
17#21455C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#483C2211yescharacteristic
22C2×#475C2214yes—
23C2×#462C2212yes—
24C2×#451C2212yes—
25C2×#441C2214yes—
26#2857C2211yescharacteristic
27#2857C2212yes—
28#2586—22no—
29#2567C2211yescharacteristic
30#2551C2211yescharacteristic
31#2549—22no—
32#2507—21no—
33#2507C2212yes—
34#2451C2212yes—
35#2440—21no—
36#2369—21no—
37#2369C2211yescharacteristic
38#2365—22no—
39#2365—21no—
40#2365—21no—
41#2365C2214yes—
42#2365C2212yes—
43#2365C2212yes—
44Q8×C4⋊C4C2211yescharacteristic
45#2345—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#387—22no—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2072C2214yes—
55#2066C2214yes—
56#2056C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2024C2214yes—
60#2016C2212yes—
61#1987C2212yes—
62#1981C2212yes—
63#1956C2212yes—
64#1948C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1935C2214yes—
68#1927C2212yes—
69#1927C2214yes—
70#1920C2214yes—
71#1920C2212yes—
72#1917C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1872C2211yescharacteristic
76#1865C2214yes—
77#1865C2214yes—
78#1865C2214yes—
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1823C2211yescharacteristic
82#1823C2212yes—
83#1803C2214yes—
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—48no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—22no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#487—28no—
127#483—24no—
128#475—48no—
129#462—28no—
130#451—28no—
131#441—48no—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#398—22no—
152#398—22no—
153#398—22no—
154#395—22no—
155#395—22no—
156#395—22no—
157#391—21no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2312yes—
164#391C2314yes—
165#391C2314yes—
166#391C2311yescharacteristic
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—24no—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2311yescharacteristic
183#387—22no—
184#387—22no—
185#387—21no—
186#387—22no—
187#387—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#385—22no—
192#385—21no—
193#384—21no—
194#384C2312yes—
195#384C2311yescharacteristic
196#380—21no—
197#380—22no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—44no—
243C22×Q8C2411yescharacteristic
244C22×Q8C2414yes—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—44no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8#4118yes—
267C22⋊Q8—22no—
268C22⋊Q8—84no—
269C22⋊Q8—82no—
270C22⋊Q8#4114yes—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—44no—
335C2×C42—48no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#106—24no—
346#106—28no—
347#106—24no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#103—28no—
351#103—24no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—24no—
356#103—44no—
357#103—44no—
358#103—44no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8—22no—
362#97—44no—
363C4×C8—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—28no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—416no—
398C2×Q8—416no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—81no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—81no—
407C2×Q8C2×#4118yes—
408C2×Q8C2×#4112yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4#13714yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4112yes—
451C22×C42−1+412yes—
452C22×C4C2511yescharacteristic
453Q16—416no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4#13714yes—
499C4⋊C4C2×#4112yes—
500C4⋊C4H(5,𝔽2)14yes—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—42no—
510C42—42no—
511C42—48no—
512C42—48no—
513C42—42no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—82no—
518C42—82no—
519C42C2×#4114yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#54812yes—
525C23C2×#13712yes—
526C23#57712yes—
527C23C23×D411yescharacteristic
528Q8—24no—
529Q8—28no—
530Q8—28no—
531Q8—416no—
532Q8—44no—
533Q8—42no—
534Q8—44no—
535Q8—164no—
536Q8—162no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—416no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#54812yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13714yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4C2×2−1+411yescharacteristic
604C8—42no—
605C8—44no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22#336914yes—
614C22C2×#52511yescharacteristic
615C22C2×#54811yescharacteristic
616C22#348411yescharacteristic
617C22C22×#13711yescharacteristic
618C22C22×#13711yescharacteristic
619C22C2×#57711yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—24no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—41no—
631C4—48no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4—81no—
637C4—82no—
638C4#311314yes—
639C4#348112yes—
640C4#352012yes—
641C2—24no—
642C2#3652811yescharacteristic
643C2C2×#311311yescharacteristic
644C2#6058211yescharacteristic
645C2C2×#336911yescharacteristic
646C2#6132211yescharacteristic
647C2#6158611yescharacteristic
648C2#6159311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
7. #26911C2semidirect
8. #26775C2semidirect
9. #26462C2semidirect
11. #25324C2semidirect
13. #23221C2semidirect
14. #23004C2semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#23652
#331482
#329542
#291112
#281242
#269112
#267752
#264622
#258562
#253242
#242122
#232212
#230042
#226292
C4×#4832
#214552