Finite Group Databasebeta

← #6595477

Subgroups — #6595477

Order 512.

1517 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33212C211yesmaximal, characteristic, complemented
5#33209C211yesmaximal, characteristic, complemented
6#29220C211yesmaximal, characteristic, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28714C214yesmaximal
9#27878C211yesmaximal, characteristic, complemented
10#27877C211yesmaximal, characteristic, complemented
11#27876C211yesmaximal, characteristic, complemented
12#27876C211yesmaximal, characteristic, complemented
13#27846C214yesmaximal, complemented
14#27842C214yesmaximal, complemented
15#23168C214yesmaximal, complemented
16#21726C211yesmaximal, characteristic, complemented
17#20824C211yesmaximal, characteristic, complemented
18#20816C211yesmaximal, characteristic, complemented
19C4×#435C211yesmaximal, characteristic
20C2×#556—21no—
21C2×#501C2214yes—
22C2×#500C2211yescharacteristic
23C2×#435C2212yes—
24C2×C4×M16C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2568—21no—
33#2562C2212yes—
34#2547—21no—
35#2547C2212yes—
36#2514—21no—
37#2509—21no—
38#2499—21no—
39#2499—21no—
40#2496—21no—
41#2496—21no—
42#2496C2212yes—
43#2496C2212yes—
44#2495C2212yes—
45#2494—21no—
46#2494C2212yes—
47#2485—21no—
48#2421—21no—
49#2415—21no—
50#2370C2212yes—
51#2365—21no—
52#2365C2212yes—
53#2362—21no—
54#2347—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2101C2214yes—
63#2100C2214yes—
64#2077C2214yes—
65#2077C2214yes—
66#2076C2214yes—
67#2075C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1989C2214yes—
75#1987C2214yes—
76#1944C2214yes—
77#1943C2214yes—
78#1942C2214yes—
79#1921C2214yes—
80#1919C2214yes—
81#1919C2214yes—
82#1918C2214yes—
83#1847C2214yes—
84#1845C2211yescharacteristic
85#1845C2211yescharacteristic
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1809C2214yes—
94#1809C2214yes—
95#1808C2214yes—
96#1807C2214yes—
97#1776C2214yes—
98#1776C2214yes—
99C4×#106C2214yes—
100#1771C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102#1753C2212yes—
103#1752C2212yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×M16C2312yes—
108C23×C8C2312yes—
109#556—48no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#126—22no—
116C2×#126—22no—
117C2×#125C2312yes—
118C2×#124—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#501—216no—
136#500—24no—
137#435—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C4×M16—24no—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×#97C2312yes—
151C2×C4×C8—24no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156C2×C4×C8C2311yescharacteristic
157#401—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#397—22no—
163#397—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#394—22no—
169#391—22no—
170#391—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#389—22no—
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—21no—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2312yes—
214#383—22no—
215#382—21no—
216#382—21no—
217#382—21no—
218#382—21no—
219#382C2312yes—
220#380—21no—
221#380—21no—
222#380—21no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380C2312yes—
229#380C2312yes—
230#380C2312yes—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234#380C2312yes—
235#380C2312yes—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257C4×C22⋊C4C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×D4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C23×C4C2411yescharacteristic
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—24no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—28no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299#126—48no—
300#126—48no—
301#125—28no—
302#124—48no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C22⋊Q8—82no—
307C4×Q8—44no—
308C4×Q8—48no—
309C4×Q8—82no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—42no—
313#117—44no—
314#117—44no—
315#117—44no—
316#117—82no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—28no—
391#107—28no—
392#107—28no—
393#106—28no—
394#106—28no—
395#106—28no—
396C4⋊C8—24no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399#97—24no—
400#97—24no—
401#97#4118yes—
402C4×C8—24no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8#4114yes—
407#95—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#95—22no—
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435C24C22×D411yescharacteristic
436C24C22×D411yescharacteristic
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—41no—
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×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4AGL(1,ℤ/8)14yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2511yescharacteristic
490M16—48no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—48no—
500C2×C8C2×#4114yes—
501C2×C8C2×#4114yes—
502C2×C8C2×#4114yes—
503C2×C82−1+414yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—41no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—44no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—82no—
574C42—82no—
575C42C2×#4114yes—
576C42C2×#4112yes—
577C422−1+412yes—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23C2×AGL(1,ℤ/8)12yes—
592C23C23×D411yescharacteristic
593Q8—88no—
594Q8—84no—
595Q8—84no—
596D4—44no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
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—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
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×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4#53212yes—
657C2×C4#53212yes—
658C2×C4#54712yes—
659C2×C4#54712yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4#57512yes—
663C2×C4#57712yes—
664C2×C4#57812yes—
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4112yes—
667C2×C4C2×2−1+412yes—
668C2×C4C2×2−1+411yescharacteristic
669C8—24no—
670C8—42no—
671C8—42no—
672C8#55618yescomplemented
673C22—22no—
674C22—22no—
675C22—22no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22C2×#53211yescharacteristic
688C22C2×#54711yescharacteristic
689C22#349911yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C2×#57511yescharacteristic
692C22C2×#57711yescharacteristic
693C22C2×#57811yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#334514yes—
713C4#334914yes—
714C4C2×#55612yes—
715C4#350912yes—
716C2—42no—
717C2—81no—
718C2#3618811yescharacteristic
719C2#6070011yescharacteristic
720C2#6072311yescharacteristic
721C2C2×#334511yescharacteristic
722C2C2×#334911yescharacteristic
723C2#6247911yescharacteristic
724C2#6248311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33212C2semidirect
5. #33209C2semidirect
6. #29220C2semidirect
7. #29218C2semidirect
9. #27878C2semidirect
10. #27877C2semidirect
11. #27876C2semidirect
12. #27876C2semidirect
13. #27846C2semidirect
14. #27842C2semidirect
15. #23168C2semidirect
16. #21726C2semidirect
17. #20824C2semidirect
18. #20816C2semidirect
672. C8#556semidirect

Maximal subgroups

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

subgroupindex
#362102
#361882
#333162
#332122
#332092
#292202
#292182
#287142
#278782
#278772
#278762
#278462
#278422
#231682
#217262
#208242
#208162
C4×#4352