Finite Group Databasebeta

← #6432679

Subgroups — #6432679

Order 512.

1872 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#23832C214yesmaximal, complemented
4#23831C214yesmaximal, complemented
5#23823C212yesmaximal, complemented
6#23821C212yesmaximal, complemented
7#23817C212yesmaximal, complemented
8#23816C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23630C211yesmaximal, characteristic
11C2×#2072C211yesmaximal, characteristic, complemented
12#23510C212yesmaximal
13#23455C211yesmaximal, characteristic, complemented
14#23451C211yesmaximal, characteristic
15#20417C212yesmaximal, complemented
16#20415C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#417C2212yes—
23C2×#419C2212yes—
24C2×#416C2212yes—
25C22×#103C2212yes—
26C2×#405C2212yes—
27#2362—21no—
28#2362—21no—
29#2360—21no—
30#2360—21no—
31#2360—22no—
32#2360—22no—
33#2359—22no—
34#2359—22no—
35#2355—21no—
36#2355—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—21no—
39#2330C2214yes—
40#2327C2212yes—
41#2326C2212yes—
42#2317C2212yes—
43#2317C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#124C2212yes—
48#2297C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2072C2212yes—
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2072C2214yes—
62#2071C2212yes—
63#2071C2212yes—
64#2043C2214yes—
65#2043C2214yes—
66#2042C2214yes—
67#2042C2214yes—
68#2026C2212yes—
69#2026C2214yes—
70#2025C2212yes—
71#2024C2211yescharacteristic
72#2024C2212yes—
73#2024C2212yes—
74#2024C2211yescharacteristic
75#2023C2211yescharacteristic
76#2023C2211yescharacteristic
77#1763C2214yes—
78#1763C2214yes—
79#1763C2214yes—
80#1762C2214yes—
81C4×#103C2214yes—
82C4×#103C2214yes—
83C4×#103C2214yes—
84C4×#102C2214yes—
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#536—24no—
100#536—24no—
101#533—28no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112#520—28no—
113#519—28no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117—22no—
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#419—28no—
138#417—28no—
139#416—28no—
140C2×#103—28no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151#405—28no—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#391—22no—
155#391—22no—
156#391—21no—
157#391—21no—
158#391C2311yescharacteristic
159#391C2312yes—
160#391C2312yes—
161#391C2311yescharacteristic
162#390C2311yescharacteristic
163#390C2311yescharacteristic
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—24no—
172#389C2314yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#386—21no—
185#386—21no—
186#385—22no—
187#385—24no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C22⋊C4—24no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217#376C2312yes—
218C2×#95—21no—
219C2×#95—21no—
220C43C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41#4114yes—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8#4114yes—
249C22×Q8C2412yes—
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274#125—24no—
275#125—24no—
276#124—24no—
277#124—24no—
278#123—216no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—84no—
291C4×Q8—84no—
292C4×D4—28no—
293C4×D4—48no—
294#117—24no—
295#117—22no—
296#117—22no—
297#117—24no—
298#117—44no—
299#117—48no—
300#117#4118yes—
301#117#4118yes—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—24no—
370#103—24no—
371#103—28no—
372#103—28no—
373#102—28no—
374#102—24no—
375#102—24no—
376#97—28no—
377C4×C8—28no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D411yescharacteristic
398#41—216no—
399#41—48no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—216no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—48no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8C2×#4114yes—
418C2×Q8C2×#4112yes—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4C2×#4112yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4114yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4114yes—
468C22×C4C2511yescharacteristic
469M16—48no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—48no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4#12414yes—
509C4⋊C4#12414yes—
510C4⋊C4C2×#4114yes—
511C4⋊C4C2×#4114yes—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4#12418yes—
523C42—22no—
524C42—22no—
525C42—28no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42—84no—
539C42—84no—
540C42C2×#4118yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23#52512yes—
553C23#52512yes—
554C23C2×#12412yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—24no—
558Q8—48no—
559Q8—48no—
560Q8—48no—
561Q8—44no—
562Q8—42no—
563D4—24no—
564D4—48no—
565D4—42no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—164no—
615C2×C4—164no—
616C2×C4—162no—
617C2×C4#52512yes—
618C2×C4#52512yes—
619C2×C4C2×#12414yes—
620C2×C4C2×#12414yes—
621C2×C4C2×#12412yes—
622C2×C4#52918yes—
623C2×C4#52918yes—
624C2×C4#57712yes—
625C2×C4#57812yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4112yes—
629C2×C4C22×#4112yes—
630C8—42no—
631C8—42no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22#239814yes—
647C22C2×#52511yescharacteristic
648C22C2×#52511yescharacteristic
649C22C22×#12411yescharacteristic
650C22C2×#57711yescharacteristic
651C22C2×#57811yescharacteristic
652C22#358511yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—22no—
656C4—24no—
657C4—24no—
658C4—41no—
659C4—44no—
660C4—41no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—84no—
667C4—81no—
668C4—81no—
669C4—82no—
670C4#239814yes—
671C4C2×#52912yes—
672C4C2×#52912yes—
673C2—24no—
674C2—82no—
675C2C2×#239811yescharacteristic
676C2#6058511yescharacteristic
677C2#6058911yescharacteristic
678C2#6059311yescharacteristic
679C2#6059411yescharacteristic
680C2#6062311yescharacteristic
681C2#6062511yescharacteristic
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. #23832C2semidirect
4. #23831C2semidirect
5. #23823C2semidirect
6. #23821C2semidirect
7. #23817C2semidirect
8. #23816C2semidirect
9. #23748C2semidirect
11. C2×#2072C2semidirect
13. #23455C2semidirect
15. #20417C2semidirect
17. #20368C2semidirect

Maximal subgroups

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

subgroupindex
#342762
#342142
#238322
#238312
#238232
#238212
#238172
#238162
#237482
#236302
C2×#20722
#235102
#234552
#234512
#204172
#204152
#203682