Finite Group Databasebeta

← #165222

Subgroups — #165222

Order 512.

799 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36302C211yesmaximal, characteristic
2#22591C211yesmaximal, characteristic, complemented
3#20813C211yesmaximal, characteristic, complemented
4#7724C211yesmaximal, characteristic
5#7716C211yesmaximal, characteristic, complemented
6#7696C211yesmaximal, characteristic, complemented
7#7696C211yesmaximal, characteristic, complemented
8#7676C211yesmaximal, characteristic, complemented
9#7664C211yesmaximal, characteristic
10#7648C211yesmaximal, characteristic, complemented
11#7628C211yesmaximal, characteristic, complemented
12#7614C211yesmaximal, characteristic, complemented
13#7594C211yesmaximal, characteristic, complemented
14#7589C211yesmaximal, characteristic, complemented
15#7574C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2861C2211yescharacteristic
18#2861C2211yescharacteristic
19#2856C2211yescharacteristic
20#2552C411yescharacteristic
21#2548C411yescharacteristic
22#2531—21no—
23#2531—21no—
24#2513—21no—
25#2500—21no—
26#2490C411yescharacteristic
27#2490C411yescharacteristic
28#2476—21no—
29#2476—21no—
30#2427C411yescharacteristic
31#2427C411yescharacteristic
32#2366C411yescharacteristic
33#2363—21no—
34#2355C411yescharacteristic
35#2354—21no—
36C4×C22⋊Q8C2211yescharacteristic
37#2293C2211yescharacteristic
38#2293C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1917—21no—
41#1917—21no—
42#1850—21no—
43#1843—21no—
44#1839C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1798—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1768—21no—
53#1767C2211yescharacteristic
54#1767C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56#1752—21no—
57#1745C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C2×#127C2×C411yescharacteristic
77C2×#125C2×C411yescharacteristic
78C2×#125C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×#121—21no—
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×#117—21no—
91C2×#117—21no—
92C2×#117D411yescharacteristic
93C2×#117D411yescharacteristic
94C22×C42C2311yescharacteristic
95#433—44no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111#398—21no—
112#398—21no—
113#397—21no—
114#397—21no—
115#397—21no—
116#397—21no—
117#394—21no—
118#394—21no—
119#394—21no—
120#394—21no—
121#394—21no—
122#394—21no—
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2311yescharacteristic
126#390—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2311yescharacteristic
138#390C2311yescharacteristic
139#389—21no—
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2311yescharacteristic
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
152#388—21no—
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#386—21no—
161#386—21no—
162#386—21no—
163#386—21no—
164#385—21no—
165#384—21no—
166#384C2×C411yescharacteristic
167#382—21no—
168#380—21no—
169#380—21no—
170#380C2×C411yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187#376D411yescharacteristic
188#376D411yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8C22×C411yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×C411yescharacteristic
213C22×D4C22×C411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C22×C411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8—81no—
230#127—42no—
231#125—81no—
232#125—81no—
233#124—81no—
234#124—81no—
235C22⋊Q8—22no—
236C22⋊Q8—41no—
237#121—42no—
238#121—42no—
239#121—81no—
240#121—81no—
241#121—81no—
242#121—81no—
243C4×Q8—42no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—44no—
249#117—22no—
250#117—22no—
251#117—42no—
252#117—81no—
253#117—81no—
254#117—81no—
255#117—81no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2411yescharacteristic
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C4⋊C8—81no—
331C22⋊C8—42no—
332C22⋊C8—42no—
333C22⋊C8—42no—
334#97—42no—
335#97—42no—
336#97—81no—
337C4×C8—42no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359C24—21no—
360C24—21no—
361C24C2×C22⋊C411yescharacteristic
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4AGL(1,ℤ/8)12yes—
420C22×C4#13712yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C2×C8—22no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—84no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4#13112yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—42no—
524C42—42no—
525C42—81no—
526C42—81no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23#41712yes—
548C23C22×C22⋊C411yescharacteristic
549Q8—81no—
550Q8—81no—
551D4—81no—
552D4—81no—
553D4—81no—
554D4—81no—
555D4—84no—
556D4—162no—
557D4—162no—
558D4—162no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
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—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#41012yes—
622C2×C4#41712yes—
623C2×C4#42112yes—
624C2×C4C2×#13111yescharacteristic
625C2×C4#56811yescharacteristic
626C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
627C2×C4C2×#13711yescharacteristic
628C2×C4#57611yescharacteristic
629C2×C4#57711yescharacteristic
630C8—42no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8—81no—
635C8—81no—
636C8—81no—
637C8—81no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22#288011yescharacteristic
660C22C2×#41011yescharacteristic
661C22#288811yescharacteristic
662C22C2×#41711yescharacteristic
663C22#289711yescharacteristic
664C22C2×#42111yescharacteristic
665C22#290111yescharacteristic
666C4—21no—
667C4—21no—
668C4—22no—
669C4—21no—
670C4—21no—
671C4—22no—
672C4—21no—
673C4—21no—
674C4—22no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C2—42no—
690C2—161no—
691C2—161no—
692C2#1741811yescharacteristic
693C2#1751511yescharacteristic
694C2#1752111yescharacteristic
695C2#1758111yescharacteristic
696C2#1760411yescharacteristic
697C2#1768411yescharacteristic
698C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22591C2semidirect
3. #20813C2semidirect
5. #7716C2semidirect
6. #7696C2semidirect
7. #7696C2semidirect
8. #7676C2semidirect
10. #7648C2semidirect
11. #7628C2semidirect
12. #7614C2semidirect
13. #7594C2semidirect
14. #7589C2semidirect
15. #7574C2semidirect

Maximal subgroups

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

subgroupindex
#363022
#225912
#208132
#77242
#77162
#76962
#76762
#76642
#76482
#76282
#76142
#75942
#75892
#75742