Finite Group Databasebeta

← #6468358

Subgroups — #6468358

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#29220C211yesmaximal, characteristic, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29191C211yesmaximal, characteristic, complemented
6#29191C211yesmaximal, characteristic, complemented
7#29179C211yesmaximal, characteristic, complemented
8#29176C211yesmaximal, characteristic, complemented
9#29170C214yesmaximal
10#29141C211yesmaximal, characteristic, complemented
11#28515C211yesmaximal, characteristic, complemented
12#24854C214yesmaximal
13#23789C214yesmaximal, complemented
14#21622C214yesmaximal, complemented
15#21409C211yesmaximal, characteristic
16#21400C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic
18#20816C211yesmaximal, characteristic
19C4×#424C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#479C2214yes—
22C2×#424C2212yes—
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2629C2212yes—
30#2628C2212yes—
31#2570—21no—
32#2570—21no—
33#2550—21no—
34#2550—21no—
35#2441—21no—
36#2441—21no—
37#2439—21no—
38#2439—21no—
39#2370—21no—
40#2370—21no—
41#2362—21no—
42#2362—21no—
43#2356—21no—
44#2351—21no—
45#2347—21no—
46#2346—21no—
47#2331C2212yes—
48#2330C2212yes—
49#2323C2212yes—
50#2323C2212yes—
51#2320C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2276C2212yes—
60C4×#117C2211yescharacteristic
61#2101C2214yes—
62#2097C2214yes—
63#2088C2214yes—
64#2087C2214yes—
65#2066C2214yes—
66#2066C2214yes—
67#2058C2214yes—
68#2024C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1948C2214yes—
79#1921C2214yes—
80#1920C2214yes—
81#1894C2214yes—
82#1892C2214yes—
83#1865C2214yes—
84#1865C2214yes—
85#1845C2211yescharacteristic
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1809C2214yes—
94#1807C2214yes—
95#1803C2214yes—
96#1776C2214yes—
97C4×#106C2214yes—
98#1768C2212yes—
99#1768C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1763C2214yes—
105C4×#103C2214yes—
106#1729C2212yes—
107C22×M16C2312yes—
108C23×C8C2312yes—
109#533—24no—
110#533Q818yescomplemented
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#125—21no—
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×#117—21no—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#479—216no—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140#424—28no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C4×M16—24no—
150C2×#97C2312yes—
151C2×C4×C8—24no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#383—22no—
217#382—21no—
218#382—21no—
219#382—21no—
220#382—21no—
221#382—21no—
222#382—22no—
223#382C2312yes—
224#380—21no—
225#380—21no—
226#380—21no—
227#380C2312yes—
228#380C2312yes—
229#380C2312yes—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255#376—22no—
256#376C2311yescharacteristic
257#376C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8C2412yes—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—42no—
281C22×C8—24no—
282C22×C8—24no—
283C22×C8—42no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—44no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8C2×Q814yes—
294C4⋊Q8C2×Q814yes—
295#125—28no—
296#125—28no—
297#125—82no—
298#125—82no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8C2×Q818yes—
302C4×Q8—28no—
303C4×Q8—24no—
304C4×Q8—28no—
305C4×Q8—44no—
306C4×Q8—82no—
307C4×Q8—82no—
308C4×Q8C2×Q818yes—
309#117—28no—
310#117—24no—
311#117—42no—
312#117—82no—
313#117—82no—
314#117—82no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×Q814yes—
383C2×C42#4114yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—28no—
391#106—28no—
392#106—44no—
393#106—44no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8#4118yes—
398#103—28no—
399#103—28no—
400#103—44no—
401#103—44no—
402#97—24no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—24no—
406C4×C8—22no—
407C4×C8—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436C24C22×D411yescharacteristic
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8C22×Q814yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4C22⋊Q814yes—
462C22×C4C22⋊Q814yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479M16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—48no—
491C2×C8C2×#4114yes—
492C2×C8C2×#4114yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4C22×Q814yes—
535C22⋊C4—28no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—28no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42—82no—
558C42—82no—
559C42—82no—
560C42—82no—
561C42—82no—
562C42C22⋊Q814yes—
563C42C22⋊Q814yes—
564C42C22×Q812yes—
565C42C22×Q812yes—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C42C2×#4114yes—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C2×C22⋊Q812yes—
576C23C23×D411yescharacteristic
577Q8—28no—
578Q8—48no—
579Q8—44no—
580Q8—164no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4C2×C22⋊Q812yes—
638C2×C4C2×C22⋊Q812yes—
639C2×C4C2×C22⋊Q812yes—
640C2×C4C2×C22⋊Q812yes—
641C2×C4C2×C22⋊Q812yes—
642C2×C4#53212yes—
643C2×C4#53212yes—
644C2×C4#57712yes—
645C2×C4#57712yes—
646C2×C4#57712yes—
647C2×C4#57812yes—
648C2×C4C23×Q811yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4112yes—
651C2×C4#58512yes—
652C8—24no—
653C8—24no—
654C8—42no—
655C8—42no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22C22×C22⋊Q811yescharacteristic
664C22C2×#53211yescharacteristic
665C22#344911yescharacteristic
666C22C2×#57711yescharacteristic
667C22C2×#57711yescharacteristic
668C22C2×#57711yescharacteristic
669C22C2×#57811yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—41no—
676C4—41no—
677C4—44no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#308214yes—
690C4#308414yes—
691C4#347512yes—
692C4Q8×#4112yes—
693C2—42no—
694C2#3494711yescharacteristic
695C2C2×#308211yescharacteristic
696C2C2×#308411yescharacteristic
697C2#6071611yescharacteristic
698C2#6072311yescharacteristic
699C2#6100811yescharacteristic
700C2#6101211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29220C2semidirect
4. #29206C2semidirect
5. #29191C2semidirect
6. #29191C2semidirect
7. #29179C2semidirect
8. #29176C2semidirect
10. #29141C2semidirect
11. #28515C2semidirect
13. #23789C2semidirect
14. #21622C2semidirect
110. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342252
#292202
#292062
#291912
#291792
#291762
#291702
#291412
#285152
#248542
#237892
#216222
#214092
#214002
#208242
#208162
C4×#4242