Finite Group Databasebeta

← #6435879

Subgroups — #6435879

Order 512.

1637 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#30216C212yesmaximal, complemented
4#30214C212yesmaximal, complemented
5#30211C214yesmaximal, complemented
6#30210C214yesmaximal, complemented
7#30207C212yesmaximal, complemented
8#30205C212yesmaximal, complemented
9#23630C211yesmaximal, characteristic
10#23629C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic
12C2×#2072C211yesmaximal, characteristic, complemented
13#23458C212yesmaximal
14#23457C212yesmaximal, complemented
15#21326C212yesmaximal, complemented
16#21205C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#544C2212yes—
20C2×#536C2211yescharacteristic
21C2×#417C2212yes—
22C2×#419C2212yes—
23C2×#416C2212yes—
24C22×#103C2212yes—
25C2×#405C2212yes—
26#2868C2212yes—
27#2862C2212yes—
28#2861C2212yes—
29#2859C2212yes—
30#2660C2214yes—
31#2642C2212yes—
32#2640C2212yes—
33#2552—21no—
34#2552—21no—
35#2551—21no—
36#2551—21no—
37#2492—21no—
38#2492—21no—
39#2489—21no—
40#2489—21no—
41#2488—22no—
42#2488—22no—
43#2440—22no—
44#2440—22no—
45#2317C2212yes—
46#2311C2211yescharacteristic
47#2310C2211yescharacteristic
48#2297C2211yescharacteristic
49C2×#391C2211yescharacteristic
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2074C2212yes—
54#2074C2214yes—
55#2073C2212yes—
56#2072C2212yes—
57#2072C2214yes—
58#2072C2211yescharacteristic
59#2072C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2211yescharacteristic
65#2071C2212yes—
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2211yescharacteristic
69#2026C2214yes—
70#2026C2214yes—
71#2026C2214yes—
72#2025C2214yes—
73#2024C2214yes—
74#2024C2214yes—
75#2024C2214yes—
76#2023C2214yes—
77#1993C2214yes—
78#1992C2214yes—
79#1991C2214yes—
80#1990C2214yes—
81#1960C2214yes—
82#1959C2214yes—
83#1958C2214yes—
84#1956C2214yes—
85#1746C2214yes—
86#1745C2212yes—
87#1744C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#544—44no—
101#544—44no—
102#536—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2314yes—
110C2×#125—21no—
111C2×#125—22no—
112C2×#125C2312yes—
113C2×#125C2314yes—
114C2×#124C2312yes—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—22no—
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129#419—44no—
130#417—44no—
131#416—44no—
132C2×#103—44no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143#405—28no—
144C2×#97C2314yes—
145C2×C4×C8C2314yes—
146#400—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#398—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#395—24no—
155#395—24no—
156#393—22no—
157#393—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#391C2312yes—
170#391C2311yescharacteristic
171#390C2312yes—
172#390C2311yescharacteristic
173#390C2311yescharacteristic
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#386—21no—
185#386—21no—
186#386—22no—
187#385—24no—
188#384—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2314yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382C2314yes—
200#382C2312yes—
201#381—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216#376C2312yes—
217C2×#95—21no—
218C2×#95—21no—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—24no—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4—21no—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C4⋊Q8—28no—
269C4⋊Q8—28no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273C4⋊Q8—44no—
274C4⋊Q8—84no—
275C4⋊Q8—84no—
276#125—28no—
277#125—44no—
278#125—44no—
279#125—48no—
280#125—44no—
281#124—28no—
282#123—48no—
283C22⋊Q8—416no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287#117—22no—
288#117—22no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117#4118yes—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353#103—44no—
354#103—44no—
355#103—42no—
356#103—42no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—42no—
361#102—42no—
362#102—44no—
363#97—28no—
364C4×C8—28no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D411yescharacteristic
389#41—48no—
390#41—48no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—48no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C42−1+414yes—
456C22×C4C2511yescharacteristic
457M16—48no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—48no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4#12414yes—
500C4⋊C4#12414yes—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—82no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C22⋊C4—81no—
512C22⋊C4#12418yes—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42C2×#4118yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×#12412yes—
538C23#53512yes—
539C23#53612yes—
540C23C23×D411yescharacteristic
541Q8—44no—
542Q8—44no—
543Q8—42no—
544Q8—42no—
545Q8—84no—
546Q8—164no—
547Q8—168no—
548D4—42no—
549D4—42no—
550D4—84no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—164no—
600C2×C4—162no—
601C2×C4C2×#12414yes—
602C2×C4C2×#12414yes—
603C2×C4C2×#12412yes—
604C2×C4#53512yes—
605C2×C4#53612yes—
606C2×C4#54218yes—
607C2×C4#54218yes—
608C2×C4#57712yes—
609C2×C4#57812yes—
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+412yes—
614C8—42no—
615C8—42no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22#264014yes—
631C22C22×#12411yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53611yescharacteristic
634C22C2×#57711yescharacteristic
635C22C2×#57811yescharacteristic
636C22#358511yescharacteristic
637C22#358611yescharacteristic
638C4—22no—
639C4—22no—
640C4—24no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—41no—
646C4—81no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4#264014yes—
655C4C2×#54212yes—
656C4C2×#54212yes—
657C2—24no—
658C2—161no—
659C2C2×#264011yescharacteristic
660C2#6062311yescharacteristic
661C2#6062511yescharacteristic
662C2#6083311yescharacteristic
663C2#6083711yescharacteristic
664C2#6087211yescharacteristic
665C2#6087411yescharacteristic
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. #35275C2semidirect
3. #30216C2semidirect
4. #30214C2semidirect
5. #30211C2semidirect
6. #30210C2semidirect
7. #30207C2semidirect
8. #30205C2semidirect
10. #23629C2semidirect
12. C2×#2072C2semidirect
14. #23457C2semidirect
15. #21326C2semidirect
17. #20368C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#352752
#302162
#302142
#302112
#302102
#302072
#302052
#236302
#236292
#236272
C2×#20722
#234582
#234572
#213262
#212052
#203682