Finite Group Databasebeta

← #6458920

Subgroups — #6458920

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34250C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27621C214yesmaximal, complemented
4#27619C214yesmaximal, complemented
5#27615C212yesmaximal, complemented
6#27613C212yesmaximal, complemented
7#27612C212yesmaximal, complemented
8#27611C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23744C211yesmaximal, characteristic, complemented
11#23658C211yesmaximal, characteristic, complemented
12C2×#2084C211yesmaximal, characteristic, complemented
13#21038C212yesmaximal, complemented
14#21035C212yesmaximal, complemented
15#20848C212yesmaximal
16#20703C212yesmaximal
17#20684C211yesmaximal, characteristic
18C2×#520C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C22×#106C2212yes—
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#2101C2212yes—
52#2101C2212yes—
53#2101C2214yes—
54#2100C2211yescharacteristic
55#2100C2211yescharacteristic
56#2100C2212yes—
57#2099C2211yescharacteristic
58#2099C2211yescharacteristic
59#2099C2212yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2214yes—
63#2084C2212yes—
64#2084C2212yes—
65#2083C2212yes—
66#2083C2212yes—
67#1921C2214yes—
68#1920C2214yes—
69#1920C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1919C2214yes—
73#1919C2214yes—
74#1918C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1853C2214yes—
78#1853C2214yes—
79#1825C2214yes—
80#1825C2214yes—
81#1824C2214yes—
82#1823C2214yes—
83#1818C2214yes—
84#1817C2212yes—
85#1817C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
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—
135C2×#107C2314yes—
136C2×#106—28no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140#408—28no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143#391—22no—
144#390—22no—
145#389—21no—
146#389—22no—
147#389—21no—
148#389—22no—
149#389—22no—
150#389—21no—
151#389—22no—
152#389—21no—
153#389C2311yescharacteristic
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2311yescharacteristic
159#389C2312yes—
160#389C2311yescharacteristic
161#389C2312yes—
162#389C2311yescharacteristic
163#388—24no—
164#388—24no—
165#387—24no—
166#387—24no—
167#386—24no—
168#385—21no—
169#385—22no—
170#385—21no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#384—21no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382—21no—
184#382—22no—
185#382—22no—
186#382—22no—
187#382C2312yes—
188#380—21no—
189#380—21no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380C2314yes—
196#380C2314yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C22⋊C4—24no—
210C2×#95—21no—
211C2×#95—22no—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
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—28no—
282#117—84no—
283#117—84no—
284#117#4114yes—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
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—28no—
360#106—28no—
361#106—28no—
362#106—28no—
363C22⋊C8—28no—
364C22⋊C8—28no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—21no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383#41—216no—
384#41—216no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8#12314yes—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4#12318yes—
393C2×D4#12314yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—81no—
424C22×C4—82no—
425C22×C4—81no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—81no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—82no—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4118yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C42−1+414yes—
446C22×C4C2511yescharacteristic
447M16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—48no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C2×#4112yes—
496C4⋊C4C2×#4114yes—
497C4⋊C4C2×#4112yes—
498C22⋊C4—28no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—28no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—44no—
518C42—42no—
519C42—44no—
520C42—84no—
521C42—84no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23C2×#12314yes—
533C23C2×#12312yes—
534C23#52512yes—
535C23#53612yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538D4—216no—
539D4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—162no—
594C2×C4—164no—
595C2×C4—164no—
596C2×C4C2×#12314yes—
597C2×C4C2×#12312yes—
598C2×C4#52512yes—
599C2×C4#53612yes—
600C2×C4#55218yes—
601C2×C4#55218yes—
602C2×C4#57712yes—
603C2×C4#57712yes—
604C2×C4C22×#4112yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×2−1+412yes—
608C8—42no—
609C8—42no—
610C8—44no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22#248414yes—
625C22C22×#12311yescharacteristic
626C22C2×#52511yescharacteristic
627C22C2×#53611yescharacteristic
628C22C2×#57711yescharacteristic
629C22C2×#57711yescharacteristic
630C22#358511yescharacteristic
631C22#358611yescharacteristic
632C4—22no—
633C4—22no—
634C4—24no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4#248414yes—
649C4C2×#55212yes—
650C4C2×#55212yes—
651C2—24no—
652C2—44no—
653C2C2×#248411yescharacteristic
654C2#6052211yescharacteristic
655C2#6052511yescharacteristic
656C2#6059111yescharacteristic
657C2#6059311yescharacteristic
658C2#6088111yescharacteristic
659C2#6088611yescharacteristic
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. #27621C2semidirect
4. #27619C2semidirect
5. #27615C2semidirect
6. #27613C2semidirect
7. #27612C2semidirect
8. #27611C2semidirect
9. #23747C2semidirect
10. #23744C2semidirect
11. #23658C2semidirect
12. C2×#2084C2semidirect
13. #21038C2semidirect
14. #21035C2semidirect

Maximal subgroups

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

subgroupindex
#342502
#341882
#276212
#276192
#276152
#276132
#276122
#276112
#237472
#237442
#236582
C2×#20842
#210382
#210352
#208482
#207032
#206842