Finite Group Databasebeta

← #6461728

Subgroups — #6461728

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34253C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#29403C214yesmaximal, complemented
4#29400C214yesmaximal, complemented
5#29395C212yesmaximal, complemented
6#29392C212yesmaximal, complemented
7#29390C212yesmaximal, complemented
8#29389C212yesmaximal, complemented
9#23657C211yesmaximal, characteristic, complemented
10#23627C211yesmaximal, characteristic
11C2×#2058C211yesmaximal, characteristic, complemented
12#23452C211yesmaximal, characteristic
13#21068C212yesmaximal
14#21038C212yesmaximal, complemented
15#21036C212yesmaximal, complemented
16#20878C212yesmaximal
17#20774C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#428C2212yes—
21C22×#106C2212yes—
22C2×#417C2212yes—
23C2×#416C2212yes—
24#2369—21no—
25#2369—21no—
26#2365—21no—
27#2365—21no—
28#2353—22no—
29#2352—22no—
30#2347—21no—
31#2346—21no—
32#2345—21no—
33#2345—22no—
34#2344—21no—
35Q8×C22⋊C4—22no—
36#2331C2214yes—
37#2320C2212yes—
38#2319C2212yes—
39#2316C2211yescharacteristic
40#2316C2212yes—
41#2311C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44#2297C2211yescharacteristic
45#2295C2212yes—
46#2293C2212yes—
47C2×#389C2211yescharacteristic
48#2285C2211yescharacteristic
49#2274C2212yes—
50#2274C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2085C2214yes—
53#2084C2212yes—
54#2083C2212yes—
55#2074C2212yes—
56#2073C2212yes—
57#2072C2211yescharacteristic
58#2072C2211yescharacteristic
59#2071C2211yescharacteristic
60#2071C2211yescharacteristic
61#2058C2212yes—
62#2058C2212yes—
63#2058C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2212yes—
67#2023C2212yes—
68#1927C2214yes—
69#1927C2214yes—
70#1926C2214yes—
71#1926C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1920C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1865C2214yes—
81#1864C2214yes—
82#1863C2214yes—
83#1862C2214yes—
84#1840C2214yes—
85#1839C2212yes—
86#1839C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104#520—24no—
105#520—24no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4C2312yes—
111C2×#117—21no—
112C2×#117—22no—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#428—28no—
125C2×#107C2314yes—
126C2×#106—28no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130#417—28no—
131#416—28no—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138#391—22no—
139#391—22no—
140#391—24no—
141#391—24no—
142#391C2312yes—
143#391C2311yescharacteristic
144#391C2311yescharacteristic
145#390C2312yes—
146#390C2311yescharacteristic
147#390C2311yescharacteristic
148#389—21no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389C2314yes—
157#389C2314yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—24no—
171#387—24no—
172#386—21no—
173#386—21no—
174#386—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#383—22no—
178#382—22no—
179#382C2312yes—
180#381—22no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—24no—
186#380—22no—
187#380—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212#376—22no—
213#376C2312yes—
214C2×#95—21no—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41#4114yes—
239C22×Q8—28no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—24no—
258C22×C8—24no—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—28no—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—84no—
271C4×Q8—84no—
272C4×D4—216no—
273C4×D4—24no—
274C4×D4—24no—
275#117—28no—
276#117—22no—
277#117—22no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—84no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4#4114yes—
310C2×C4⋊C4#4118yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#106—28no—
349#106—28no—
350#106—28no—
351#103—24no—
352#103—24no—
353#103—28no—
354#102—24no—
355#102—24no—
356#102—28no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D411yescharacteristic
381#41—216no—
382#41—48no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8—816no—
393C2×Q8C2×#4112yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4C2×#4112yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C42−1+414yes—
443C22×C4C2511yescharacteristic
444M16—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—48no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4C2×#4118yes—
485C4⋊C4C2×#4114yes—
486C22⋊C4—28no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—81no—
494C22⋊C4—82no—
495C22⋊C4—81no—
496C22⋊C4—82no—
497C42—28no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—24no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—82no—
512C42—82no—
513C42—84no—
514C42—84no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23#52512yes—
527C23#52512yes—
528C23#53612yes—
529C23C23×D411yescharacteristic
530Q8—24no—
531Q8—416no—
532Q8—42no—
533Q8—88no—
534D4—24no—
535D4—48no—
536D4—42no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—88no—
588C2×C4—88no—
589C2×C4—162no—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4#53612yes—
593C2×C4#55218yes—
594C2×C4#55618yes—
595C2×C4#57712yes—
596C2×C4#57712yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4112yes—
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×2−1+412yes—
601C8—42no—
602C8—42no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#259914yes—
618C22C2×#52511yescharacteristic
619C22C2×#52511yescharacteristic
620C22C2×#53611yescharacteristic
621C22C2×#57711yescharacteristic
622C22C2×#57711yescharacteristic
623C22#358511yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—42no—
628C4—41no—
629C4—48no—
630C4—41no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4#259914yes—
642C4C2×#55212yes—
643C4C2×#55612yes—
644C2—24no—
645C2—82no—
646C2C2×#259911yescharacteristic
647C2#6057911yescharacteristic
648C2#6058211yescharacteristic
649C2#6059711yescharacteristic
650C2#6059911yescharacteristic
651C2#6088111yescharacteristic
652C2#6088711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #29403C2semidirect
4. #29400C2semidirect
5. #29395C2semidirect
6. #29392C2semidirect
7. #29390C2semidirect
8. #29389C2semidirect
9. #23657C2semidirect
11. C2×#2058C2semidirect
14. #21038C2semidirect
15. #21036C2semidirect
17. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#342532
#342162
#294032
#294002
#293952
#293922
#293902
#293892
#236572
#236272
C2×#20582
#234522
#210682
#210382
#210362
#208782
#207742