Finite Group Databasebeta

← #6616255

Subgroups — #6616255

Order 512.

1787 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33109C211yesmaximal, characteristic
4#33100C212yesmaximal
5#27017C214yesmaximal, complemented
6#26707C214yesmaximal, complemented
7#26440C212yesmaximal, complemented
8#25902C211yesmaximal, characteristic
9#25276C212yesmaximal, complemented
10#24606C212yesmaximal
11#24301C211yesmaximal, characteristic
12#24064C212yesmaximal
13#23222C212yesmaximal, complemented
14#22970C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic
16C4×#484C211yesmaximal, characteristic
17#21394C212yesmaximal
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8—21no—
21C2×#487C2212yes—
22C2×#484C2211yescharacteristic
23C2×#484C2212yes—
24C2×#475C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2551C2211yescharacteristic
29#2546C2211yescharacteristic
30#2506—21no—
31#2462—22no—
32#2451C2212yes—
33#2446C2212yes—
34#2440—21no—
35#2438—22no—
36#2365—21no—
37#2365—21no—
38#2365—22no—
39#2365—21no—
40#2365C2212yes—
41#2365C2214yes—
42#2365C2212yes—
43#2365C2211yescharacteristic
44Q8×C4⋊C4C2211yescharacteristic
45#2344—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#391—21no—
48C2×#389C2212yes—
49C2×#387—22no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2086C2214yes—
54#2066C2214yes—
55#2058C2212yes—
56#2057C2214yes—
57#2043C2214yes—
58#2024C2214yes—
59#2024C2212yes—
60#1990C2212yes—
61#1988C2214yes—
62#1988C2212yes—
63#1981C2214yes—
64#1981C2212yes—
65#1947C2212yes—
66#1944C2214yes—
67#1939C2212yes—
68#1935C2214yes—
69#1928C2212yes—
70#1928C2214yes—
71#1928C2214yes—
72#1927C2214yes—
73#1921C2212yes—
74#1919C2214yes—
75#1918C2214yes—
76#1917C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1863C2214yes—
83#1863C2214yes—
84#1853C2212yes—
85#1852C2211yescharacteristic
86#1824C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8—22no—
112C2×C4×Q8—24no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—48no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×#117—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—22no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#487—28no—
132#484—24no—
133#484—44no—
134#475—28no—
135#432—44no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#395—22no—
159#395—22no—
160#391—22no—
161#391—24no—
162#391—22no—
163#391—22no—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#391C2311yescharacteristic
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2314yes—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—21no—
187#387—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—22no—
192#387—22no—
193#385—22no—
194#385—21no—
195#384—21no—
196#384C2311yescharacteristic
197#381—22no—
198#380—21no—
199#380—21no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—28no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2412yes—
244C22×Q8C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8#4118yes—
267#125—48no—
268C22⋊Q8—22no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—48no—
272C22⋊Q8#4114yes—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8—48no—
279C4×Q8—48no—
280C4×Q8—416no—
281C4×Q8—48no—
282#117—48no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—44no—
341C2×C42—48no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—24no—
352#107—44no—
353#106—28no—
354C4⋊C8—24no—
355C4⋊C8—44no—
356#103—24no—
357#103—28no—
358#103—28no—
359#103—28no—
360#103—24no—
361#103—28no—
362#103—44no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8—24no—
366#97—44no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—432no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8C2×#4112yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4#13714yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4AGL(1,ℤ/8)14yes—
500C4⋊C4C2×#4118yes—
501C4⋊C4C2×#4112yes—
502C4⋊C42−1+414yes—
503C22⋊C4—24no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—44no—
516C42—44no—
517C42—44no—
518C42—48no—
519C42—42no—
520C42—48no—
521C42—44no—
522C42—42no—
523C42C2×#4114yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#54612yes—
529C23C2×#13712yes—
530C23#57812yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—416no—
534Q8—416no—
535Q8—42no—
536Q8—44no—
537Q8—44no—
538Q8—88no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—416no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4#52512yes—
596C2×C4#52512yes—
597C2×C4#54612yes—
598C2×C4C2×AGL(1,ℤ/8)14yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4112yes—
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+412yes—
606C8—42no—
607C8—42no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22#332814yes—
616C22C2×#52511yescharacteristic
617C22C2×#54611yescharacteristic
618C22#348611yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57811yescharacteristic
622C22#358611yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—42no—
628C4—48no—
629C4—44no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4—82no—
640C4#311014yes—
641C4#348212yes—
642C4#352212yes—
643C2—24no—
644C2#3642211yescharacteristic
645C2C2×#311011yescharacteristic
646C2#6058311yescharacteristic
647C2C2×#332811yescharacteristic
648C2#6116111yescharacteristic
649C2#6182911yescharacteristic
650C2#6183711yescharacteristic
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
5. #27017C2semidirect
6. #26707C2semidirect
7. #26440C2semidirect
9. #25276C2semidirect
13. #23222C2semidirect
14. #22970C2semidirect

Maximal subgroups

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

subgroupindex
#361332
C2×#23652
#331092
#331002
#270172
#267072
#264402
#259022
#252762
#246062
#243012
#240642
#232222
#229702
#226032
C4×#4842
#213942