Finite Group Databasebeta

← #6557150

Subgroups — #6557150

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36407C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29332C212yesmaximal, complemented
6#29132C211yesmaximal, characteristic, complemented
7#28755C211yesmaximal, characteristic, complemented
8#27769C212yesmaximal, complemented
9#27749C212yesmaximal, complemented
10#27283C214yesmaximal, complemented
11#27277C214yesmaximal, complemented
12#25856C212yesmaximal, complemented
13#23820C211yesmaximal, characteristic, complemented
14#23507C212yesmaximal, complemented
15#22430C212yesmaximal
16#22430C212yesmaximal
17#20567C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#491C2212yes—
21C2×#487C2214yes—
22C2×#436C2212yes—
23C2×#419C2214yes—
24C2×#419C2214yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2857C2212yes—
29#2574—22no—
30#2556—22no—
31#2552C2211yescharacteristic
32#2551—21no—
33#2551—21no—
34#2551C2211yescharacteristic
35#2546C2212yes—
36#2454—21no—
37#2452—21no—
38#2441C2212yes—
39#2440—22no—
40#2440—22no—
41#2439C2212yes—
42#2434C2214yes—
43#2370—21no—
44#2369—21no—
45#2354C2211yescharacteristic
46#2354C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#125C2211yescharacteristic
51#2296C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2099C2212yes—
56#2074C2212yes—
57#2072C2214yes—
58#2043C2211yescharacteristic
59#2043C2212yes—
60#2043C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2043C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2214yes—
67#2043C2214yes—
68#2026C2212yes—
69#2024C2214yes—
70#1981C2212yes—
71#1981C2212yes—
72#1960C2212yes—
73#1956C2214yes—
74#1956C2212yes—
75#1956C2212yes—
76#1935C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1873C2214yes—
82#1873C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1787C2214yes—
86#1787C2214yes—
87C8×C4⋊C4C2212yes—
88#1767C2211yescharacteristic
89C4×#103C2212yes—
90C4×#103C2212yes—
91#1753C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C23×Q8C2314yes—
96C22×M16C2314yes—
97D4×Q8—28no—
98D4×Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125C2311yescharacteristic
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8C2314yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C42C2312yes—
126#491—44no—
127#487—48no—
128#436—28no—
129C2×C4⋊C8C2312yes—
130#419—48no—
131#419—48no—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#399—22no—
151#398—22no—
152#398—24no—
153#396—22no—
154#396—22no—
155#395—22no—
156#395—24no—
157#395—24no—
158#393—22no—
159#391—22no—
160#391—22no—
161#391C2312yes—
162#391C2314yes—
163#390—21no—
164#390—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#387—21no—
186#387—21no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#382—21no—
202#382C2311yescharacteristic
203#380—22no—
204#380—22no—
205#380—22no—
206#380C2312yes—
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2311yescharacteristic
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—28no—
239C22×Q8—28no—
240C22×Q8—28no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2414yes—
245C22×Q8C2412yes—
246C22×D4C2×D411yescharacteristic
247C23×C4—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×M16—28no—
252C2×M16—44no—
253C2×M16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270#125—22no—
271#125—41no—
272#125—44no—
273#125—82no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—416no—
277C22⋊Q8—84no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—48no—
283C4×D4#4118yes—
284#117—24no—
285#117—42no—
286#117—44no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
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—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C4⋊C8—24no—
353#103—24no—
354#103—28no—
355#103—24no—
356#103—24no—
357#103—28no—
358#103—44no—
359#103—42no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367C22⋊C8—28no—
368#97—42no—
369C4×C8—24no—
370C4×C8—44no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24C22×D412yes—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—48no—
403C2×Q8—416no—
404C2×Q8—416no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—84no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—88no—
411C2×Q8—88no—
412C2×Q8H(5,𝔽2)14yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4#12414yes—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—84no—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4118yes—
447C22×C4C2511yescharacteristic
448M16—88no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4#12414yes—
498C4⋊C4#13714yes—
499C4⋊C4#13714yes—
500C4⋊C4H(5,𝔽2)12yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4#12418yes—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—21no—
515C42—22no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—82no—
524C42C2×#4114yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23C2×#12414yes—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—24no—
535Q8—42no—
536Q8—42no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—816no—
542Q8—164no—
543D4—28no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—816no—
594C2×C4C2×#12412yes—
595C2×C4C2×#12414yes—
596C2×C4C2×#12412yes—
597C2×C4C2×#13714yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4#57712yes—
602C2×C4#57812yes—
603C2×C4C22×#4112yes—
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4#58511yescharacteristic
607C8—42no—
608C8—44no—
609C8—42no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C22×#12411yescharacteristic
618C22#348411yescharacteristic
619C22#348811yescharacteristic
620C22C22×#13711yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57711yescharacteristic
623C22C2×#57811yescharacteristic
624C4—24no—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—24no—
629C4—22no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—44no—
634C4—82no—
635C4—81no—
636C4—84no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4#313914yes—
642C4#351312yes—
643C4#351912yes—
644C2—44no—
645C2#3590311yescharacteristic
646C2C2×#313911yescharacteristic
647C2#6062311yescharacteristic
648C2#6159311yescharacteristic
649C2#6160211yescharacteristic
650C2#6200011yescharacteristic
651C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #30220C2semidirect
4. #30199C2semidirect
5. #29332C2semidirect
6. #29132C2semidirect
7. #28755C2semidirect
8. #27769C2semidirect
9. #27749C2semidirect
10. #27283C2semidirect
11. #27277C2semidirect
12. #25856C2semidirect
13. #23820C2semidirect
14. #23507C2semidirect

Maximal subgroups

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

subgroupindex
#364072
D4×C4⋊Q82
#302202
#301992
#293322
#291322
#287552
#277692
#277492
#272832
#272772
#258562
#238202
#235072
#224302
#205672