Finite Group Databasebeta

← #6541355

Subgroups — #6541355

Order 512.

1671 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#36134C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29970C212yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#29206C211yesmaximal, characteristic, complemented
7#29156C212yesmaximal, complemented
8#26445C212yesmaximal, complemented
9#26347C214yesmaximal, complemented
10#26045C212yesmaximal, complemented
11#25999C214yesmaximal, complemented
12#24304C212yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23551C212yesmaximal, complemented
15#22696C211yesmaximal, characteristic, complemented
16#22657C212yesmaximal, complemented
17#22489C212yesmaximal
18C2×#555C2211yescharacteristic
19C2×#533C2212yes—
20C2×#478C2214yes—
21C2×C8×D4C2212yes—
22C2×#427C2214yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2552C2212yes—
26#2542—21no—
27#2538—21no—
28#2441C2214yes—
29#2429—22no—
30#2423—22no—
31#2369C2211yescharacteristic
32#2367—22no—
33#2365—21no—
34#2363C2212yes—
35#2361—21no—
36#2354C2211yescharacteristic
37#2354C2212yes—
38Q8×C4⋊C4—21no—
39D4×C4⋊C4C2211yescharacteristic
40#2347—21no—
41#2347C2212yes—
42#2337—22no—
43#2324—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#121C2212yes—
47#2296C2212yes—
48#2293C2211yescharacteristic
49#2274—21no—
50C42×D4C2211yescharacteristic
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2099C2212yes—
54#2094C2211yescharacteristic
55#2093C2212yes—
56#2088C2214yes—
57#2088C2212yes—
58#2086C2214yes—
59#2065C2212yes—
60#2059C2214yes—
61#2059C2214yes—
62#2058C2212yes—
63#2058C2212yes—
64#2058C2211yescharacteristic
65#2057C2212yes—
66#2057C2212yes—
67#2023C2212yes—
68#2023C2211yescharacteristic
69#1990C2212yes—
70#1989C2214yes—
71#1987C2212yes—
72#1948C2214yes—
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1921C2214yes—
77#1920C2214yes—
78#1893C2214yes—
79#1875C2214yes—
80#1864C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1850C2214yes—
84#1844C2212yes—
85#1831C2211yescharacteristic
86#1825C2214yes—
87#1807C2214yes—
88#1802C2212yes—
89#1797C2212yes—
90C4×#106C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C23×C8C2314yes—
96#555—24no—
97#555Q818yescomplemented
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125C2314yes—
105C2×#124—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×D4—22no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—21no—
123C2×#117C2314yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C42C2312yes—
128#478—48no—
129C8×D4—28no—
130#427—48no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C4×C8C2312yes—
145#399—24no—
146#398—24no—
147#396—24no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—22no—
152#390—21no—
153#390—22no—
154#390—22no—
155#390C2311yescharacteristic
156#390C2311yescharacteristic
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—24no—
163#389C2312yes—
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2311yescharacteristic
170#388—22no—
171#388—22no—
172#388—22no—
173#388—24no—
174#388—21no—
175#388—22no—
176#388—21no—
177#388—22no—
178#387—24no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#383—22no—
192#380—22no—
193#380—21no—
194#380C2312yes—
195#380C2314yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×D4—22no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C22×C8—28no—
246C22×C8—28no—
247C22×C8—44no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8C2×Q814yes—
257#125—28no—
258#125—44no—
259#124—216no—
260C22⋊Q8—48no—
261C22⋊Q8—48no—
262C22⋊Q8—84no—
263#121—28no—
264#121—24no—
265#121C2×Q818yes—
266C4×Q8—44no—
267C4×Q8—48no—
268C4×Q8—48no—
269C4×Q8—82no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—42no—
273C4×D4—44no—
274C4×D4—84no—
275C4×D4C2×Q814yes—
276C4×D4C2×Q814yes—
277C4×D4#4114yes—
278#117—24no—
279#117—28no—
280#117—28no—
281#117—44no—
282#117—44no—
283#117—44no—
284#117C2×Q818yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—28no—
351#107—44no—
352#106—24no—
353#106—24no—
354#106—44no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#102—24no—
360#102—24no—
361#102—44no—
362C22⋊C8—28no—
363C4×C8—24no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—24no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C24C22×D412yes—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4C22⋊Q814yes—
400C2×D4C22×Q812yes—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q814yes—
433C22×C4C2×#4114yes—
434C22×C4C2511yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—416no—
444C2×C82−1+414yes—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C22⋊Q814yes—
478C4⋊C4C22×Q812yes—
479C4⋊C4C22×Q814yes—
480C4⋊C4H(5,𝔽2)14yes—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—48no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—81no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C22⋊C4C22⋊Q818yes—
496C42—28no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—42no—
503C42—42no—
504C42—41no—
505C42—48no—
506C42—42no—
507C42—44no—
508C42—82no—
509C42—82no—
510C42—82no—
511C42#13514yes—
512C42C22×Q812yes—
513C42C2×#4112yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C2×C22⋊Q814yes—
528C23C23×D411yescharacteristic
529Q8—48no—
530Q8—84no—
531Q8—88no—
532D4—28no—
533D4—24no—
534D4—42no—
535D4—88no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—164no—
585C2×C4C2×C22⋊Q814yes—
586C2×C4C2×C22⋊Q812yes—
587C2×C4C2×C22⋊Q812yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13512yes—
591C2×C4#57512yes—
592C2×C4#57712yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4C2×2−1+412yes—
597C8—24no—
598C8—42no—
599C8—44no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C22×C22⋊Q811yescharacteristic
615C22#348411yescharacteristic
616C22#349911yescharacteristic
617C22C22×#13511yescharacteristic
618C22C2×#57511yescharacteristic
619C22C2×#57711yescharacteristic
620C22#358511yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—22no—
626C4—41no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—84no—
636C4—82no—
637C4#307614yes—
638C4#350312yes—
639C4#352012yes—
640C2—44no—
641C2—82no—
642C2#3573911yescharacteristic
643C2C2×#307611yescharacteristic
644C2#6049511yescharacteristic
645C2#6156611yescharacteristic
646C2#6160411yescharacteristic
647C2#6247011yescharacteristic
648C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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
1. #36134C2semidirect
3. #29980C2semidirect
4. #29970C2semidirect
5. #29277C2semidirect
6. #29206C2semidirect
7. #29156C2semidirect
8. #26445C2semidirect
9. #26347C2semidirect
10. #26045C2semidirect
11. #25999C2semidirect
12. #24304C2semidirect
13. #23775C2semidirect
14. #23551C2semidirect
15. #22696C2semidirect
16. #22657C2semidirect
97. #555Q8semidirect

Maximal subgroups

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

subgroupindex
#361342
#343212
#299802
#299702
#292772
#292062
#291562
#264452
#263472
#260452
#259992
#243042
#237752
#235512
#226962
#226572
#224892