Finite Group Databasebeta

← #6432565

Subgroups — #6432565

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34253C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#23792C214yesmaximal, complemented
4#23790C214yesmaximal, complemented
5#23783C212yesmaximal, complemented
6#23781C212yesmaximal, complemented
7#23776C212yesmaximal, complemented
8#23774C212yesmaximal, complemented
9C2×#2099C211yesmaximal, characteristic, complemented
10#23734C211yesmaximal, characteristic
11#23671C212yesmaximal, complemented
12#23616C212yesmaximal
13#23554C211yesmaximal, characteristic, complemented
14#23451C211yesmaximal, characteristic
15#20453C212yesmaximal, complemented
16#20430C211yesmaximal, characteristic, complemented
17#20415C212yesmaximal
18C2×#533C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#424C2212yes—
23C2×#417C2212yes—
24C2×#416C2212yes—
25C2×#405C2212yes—
26#2360—22no—
27#2360—22no—
28#2355—21no—
29#2355—21no—
30#2353—22no—
31#2352—22no—
32Q8×C4⋊C4—21no—
33Q8×C4⋊C4—21no—
34#2347—21no—
35#2346—21no—
36#2345—21no—
37#2344—21no—
38#2330C2214yes—
39#2320C2212yes—
40#2319C2212yes—
41#2317C2212yes—
42#2316C2211yescharacteristic
43#2311C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46#2297C2211yescharacteristic
47#2295C2212yes—
48#2293C2212yes—
49C2×#389C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2212yes—
52#2274C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2099C2212yes—
55#2099C2212yes—
56#2099C2214yes—
57#2096C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2088C2214yes—
62#2088C2214yes—
63#2087C2214yes—
64#2086C2214yes—
65#2068C2214yes—
66#2067C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2059C2214yes—
70#2058C2212yes—
71#2057C2212yes—
72#2026C2212yes—
73#2025C2212yes—
74#2024C2211yescharacteristic
75#2024C2211yescharacteristic
76#2023C2211yescharacteristic
77#2023C2211yescharacteristic
78#1776C2214yes—
79#1776C2214yes—
80C4×#107C2214yes—
81C4×#106C2214yes—
82#1768C2214yes—
83#1767C2212yes—
84C4×C4⋊C8C2212yes—
85#1763C2214yes—
86#1762C2214yes—
87C4×#103C2214yes—
88C4×#102C2214yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#533—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104#520—24no—
105#520Q818yescomplemented
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C2×#117—22no—
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#428—28no—
128#428—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133#424—28no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136#417—28no—
137#416—28no—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144#405—28no—
145C2×#97C2314yes—
146C2×C4×C8C2314yes—
147#391—22no—
148#391—22no—
149#391C2312yes—
150#391C2311yescharacteristic
151#391C2311yescharacteristic
152#390C2312yes—
153#390C2311yescharacteristic
154#390C2311yescharacteristic
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—24no—
162#389—24no—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#386—21no—
180#386—21no—
181#386—22no—
182#385—24no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#383—22no—
189#382—22no—
190#382—22no—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—22no—
195#380—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376—22no—
219#376C2312yes—
220C2×#95—21no—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2312yes—
237C2×#41—22no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41C2×Q814yes—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8C2411yescharacteristic
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—24no—
256C2×M16#4118yes—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8—24no—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265#125—24no—
266#125—24no—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—84no—
273C4×Q8—84no—
274C4×Q8C2×Q818yes—
275C4×D4—216no—
276C4×D4—24no—
277C4×D4C2×Q818yes—
278#117—24no—
279#117—24no—
280#117—28no—
281#117—24no—
282#117—84no—
283#117C2×Q814yes—
284#117#4118yes—
285#117#4114yes—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×Q814yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
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#107—28no—
354#107—28no—
355#106—28no—
356#106—28no—
357C4⋊C8—28no—
358C4⋊C8—28no—
359#103—28no—
360#103—24no—
361#103—24no—
362#102—28no—
363#102—24no—
364#102—24no—
365#97—28no—
366C4×C8—28no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—21no—
379#95—21no—
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—216no—
390#41—48no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8C22×Q812yes—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C22×Q812yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—81no—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×Q812yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2511yescharacteristic
451M16—216no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—48no—
461C2×C8C2×#4118yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4C22⋊Q814yes—
491C4⋊C4C22⋊Q814yes—
492C4⋊C4C22×Q814yes—
493C4⋊C4C2×#4114yes—
494C4⋊C4C2×#4114yes—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—82no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—82no—
505C22⋊C4C22⋊Q818yes—
506C42—28no—
507C42—22no—
508C42—22no—
509C42—28no—
510C42—24no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—82no—
519C42—82no—
520C42—84no—
521C42—84no—
522C42C22×Q814yes—
523C42C2×#4114yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×C22⋊Q812yes—
536C23#52512yes—
537C23#53212yes—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—48no—
541Q8—48no—
542Q8—42no—
543D4—24no—
544D4—48no—
545D4—42no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4—164no—
596C2×C4—162no—
597C2×C4C2×C22⋊Q814yes—
598C2×C4C2×C22⋊Q814yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4#52512yes—
601C2×C4#52918yes—
602C2×C4#53212yes—
603C2×C4#53318yes—
604C2×C4#57612yes—
605C2×C4#57712yes—
606C2×C4C23×Q811yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4112yes—
609C2×C4C22×#4112yes—
610C8—28no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22#239614yes—
627C22C22×C22⋊Q811yescharacteristic
628C22C2×#52511yescharacteristic
629C22C2×#53211yescharacteristic
630C22C2×#57611yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358611yescharacteristic
633C22#358611yescharacteristic
634C4—24no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—41no—
642C4—44no—
643C4—41no—
644C4—44no—
645C4—42no—
646C4—81no—
647C4—84no—
648C4—81no—
649C4—82no—
650C4#239614yes—
651C4C2×#52912yes—
652C4C2×#53312yes—
653C2—24no—
654C2—82no—
655C2C2×#239611yescharacteristic
656C2#6049411yescharacteristic
657C2#6049711yescharacteristic
658C2#6058411yescharacteristic
659C2#6058511yescharacteristic
660C2#6072211yescharacteristic
661C2#6072611yescharacteristic
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. #34214C2semidirect
3. #23792C2semidirect
4. #23790C2semidirect
5. #23783C2semidirect
6. #23781C2semidirect
7. #23776C2semidirect
8. #23774C2semidirect
9. C2×#2099C2semidirect
11. #23671C2semidirect
13. #23554C2semidirect
15. #20453C2semidirect
16. #20430C2semidirect
105. #520Q8semidirect

Maximal subgroups

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

subgroupindex
#342532
#342142
#237922
#237902
#237832
#237812
#237762
#237742
C2×#20992
#237342
#236712
#236162
#235542
#234512
#204532
#204302
#204152