Finite Group Databasebeta

← #6458923

Subgroups — #6458923

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34250C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27623C214yesmaximal, complemented
4#27622C214yesmaximal, complemented
5#27614C212yesmaximal, complemented
6#27613C212yesmaximal, complemented
7#27612C212yesmaximal, complemented
8#27612C212yesmaximal, complemented
9#23745C211yesmaximal, characteristic, complemented
10C2×#2100C211yesmaximal, characteristic, complemented
11#23657C211yesmaximal, characteristic, complemented
12#23656C211yesmaximal, characteristic, complemented
13#21039C212yesmaximal, complemented
14#21036C212yesmaximal, complemented
15#20849C212yesmaximal
16#20703C212yesmaximal
17#20684C211yesmaximal, characteristic
18C2×#520C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#408C2212yes—
24#2355—21no—
25#2355—21no—
26#2354—21no—
27#2354—21no—
28#2352—22no—
29#2352—22no—
30#2351—21no—
31Q8×C4⋊C4—21no—
32#2349—22no—
33#2349—21no—
34D4×C4⋊C4—22no—
35D4×C4⋊C4—21no—
36#2317C2211yescharacteristic
37#2317C2212yes—
38#2316C2211yescharacteristic
39#2316C2212yes—
40#2314C2214yes—
41#2311C2212yes—
42#2296C2211yescharacteristic
43#2296C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46#2285C2212yes—
47#2285C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2274C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2100C2214yes—
54#2100C2212yes—
55#2100C2212yes—
56#2099C2212yes—
57#2099C2212yes—
58#2085C2212yes—
59#2085C2212yes—
60#2085C2214yes—
61#2084C2211yescharacteristic
62#2084C2211yescharacteristic
63#2084C2212yes—
64#2083C2211yescharacteristic
65#2083C2211yescharacteristic
66#2083C2212yes—
67#1921C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1919C2214yes—
73#1918C2214yes—
74#1918C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1853C2214yes—
78#1852C2214yes—
79#1825C2214yes—
80#1825C2214yes—
81#1824C2214yes—
82#1823C2214yes—
83#1818C2214yes—
84#1817C2212yes—
85#1817C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90C2×#340C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#520—28no—
99#520—24no—
100#520—24no—
101C4×#41—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8D411yescharacteristic
105C2×C4×Q8D411yescharacteristic
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4D412yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4D411yescharacteristic
114C2×#117—22no—
115C2×#117—22no—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#428—28no—
135#427—28no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142#408—28no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145#391—22no—
146#390—22no—
147#389—21no—
148#389—22no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—22no—
154#389—21no—
155#389C2311yescharacteristic
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2311yescharacteristic
161#389C2312yes—
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2311yescharacteristic
165#388—24no—
166#388—24no—
167#387—24no—
168#387—24no—
169#386—24no—
170#385—21no—
171#385—22no—
172#385—21no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—22no—
179#384—21no—
180#384—21no—
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#382—21no—
186#382—22no—
187#382—22no—
188#382—22no—
189#382C2312yes—
190#380—21no—
191#380—21no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380C2314yes—
198#380C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C22⋊C4—24no—
212C2×#95—21no—
213C2×#95—22no—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—24no—
238C2×#41#4118yes—
239C22×Q8C2×D411yescharacteristic
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×D4—216no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—48no—
273C4×D4—44no—
274#117—28no—
275#117—24no—
276#117—22no—
277#117—24no—
278#117—24no—
279#117—28no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—84no—
284#117—84no—
285#117#4114yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4#4114yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—24no—
360#107—24no—
361#107—28no—
362#106—24no—
363#106—24no—
364#106—28no—
365C22⋊C8—28no—
366C22⋊C8—28no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—21no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385#41—216no—
386#41—216no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8#12314yes—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4#12318yes—
395C2×D4#12314yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—81no—
426C22×C4—82no—
427C22×C4—81no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—82no—
433C22×C4—81no—
434C22×C4—82no—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4118yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C42−1+414yes—
448C22×C4C2511yescharacteristic
449M16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—48no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—28no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4C2×#4112yes—
499C4⋊C4C2×#4114yes—
500C4⋊C4C2×#4112yes—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—28no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—24no—
519C42—22no—
520C42—44no—
521C42—42no—
522C42—44no—
523C42—84no—
524C42—84no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23C2×#12314yes—
536C23C2×#12312yes—
537C23#52512yes—
538C23#53612yes—
539C23C23×D411yescharacteristic
540Q8—28no—
541D4—216no—
542D4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—162no—
597C2×C4—164no—
598C2×C4—164no—
599C2×C4C2×#12314yes—
600C2×C4C2×#12312yes—
601C2×C4#52512yes—
602C2×C4#53612yes—
603C2×C4#55218yes—
604C2×C4#55218yes—
605C2×C4#57712yes—
606C2×C4#57812yes—
607C2×C4C22×#4112yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×2−1+412yes—
611C8—42no—
612C8—42no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22#248414yes—
628C22C22×#12311yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#57711yescharacteristic
632C22C2×#57811yescharacteristic
633C22#358511yescharacteristic
634C22#358511yescharacteristic
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4#248414yes—
652C4C2×#55212yes—
653C4C2×#55212yes—
654C2—24no—
655C2—44no—
656C2C2×#248411yescharacteristic
657C2#6052511yescharacteristic
658C2#6052611yescharacteristic
659C2#6059711yescharacteristic
660C2#6060011yescharacteristic
661C2#6087811yescharacteristic
662C2#6088311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27623C2semidirect
4. #27622C2semidirect
5. #27614C2semidirect
6. #27613C2semidirect
7. #27612C2semidirect
8. #27612C2semidirect
9. #23745C2semidirect
10. C2×#2100C2semidirect
11. #23657C2semidirect
12. #23656C2semidirect
13. #21039C2semidirect
14. #21036C2semidirect

Maximal subgroups

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

subgroupindex
#342502
#341882
#276232
#276222
#276142
#276132
#276122
#237452
C2×#21002
#236572
#236562
#210392
#210362
#208492
#207032
#206842