Finite Group Databasebeta

← #162403

Subgroups — #162403

Order 512.

841 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35109C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic, complemented
3#21996C211yesmaximal, characteristic, complemented
4#7730C211yesmaximal, characteristic, complemented
5#7714C211yesmaximal, characteristic, complemented
6#7710C211yesmaximal, characteristic, complemented
7#7694C211yesmaximal, characteristic, complemented
8#7684C211yesmaximal, characteristic, complemented
9#7681C211yesmaximal, characteristic, complemented
10#7664C211yesmaximal, characteristic, complemented
11#7661C211yesmaximal, characteristic, complemented
12#7601C211yesmaximal, characteristic, complemented
13#7594C211yesmaximal, characteristic, complemented
14#7584C211yesmaximal, characteristic, complemented
15#7574C211yesmaximal, characteristic, complemented
16C2×#436—21no—
17#2856C2211yescharacteristic
18#2855C2211yescharacteristic
19#2854C2211yescharacteristic
20#2850C2211yescharacteristic
21#2651—21no—
22#2635—21no—
23#2570C411yescharacteristic
24#2544C411yescharacteristic
25#2540C411yescharacteristic
26#2535C411yescharacteristic
27#2516—21no—
28#2476—21no—
29#2471—21no—
30#2464—21no—
31#2428C411yescharacteristic
32#2426C411yescharacteristic
33#2423C411yescharacteristic
34#2420C411yescharacteristic
35#2333—21no—
36#2333—21no—
37C4×#124C2211yescharacteristic
38#2293C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1939—21no—
41#1923—21no—
42#1916—21no—
43#1869—21no—
44#1851—21no—
45#1851—21no—
46#1850—21no—
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1831C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
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
75C22×M16—21no—
76C22×M16—21no—
77C23×C8—21no—
78C2×#127C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×#123—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121—21no—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121C2×C411yescharacteristic
88C2×#121C2×C411yescharacteristic
89C2×#121C2×C411yescharacteristic
90C2×#121C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×D4C2×C411yescharacteristic
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4C2×C411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×#117—21no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#436—44no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#400—21no—
115#399—21no—
116#399—21no—
117#398—21no—
118#398—21no—
119#397—21no—
120#397—21no—
121#396—21no—
122#394—21no—
123#394—21no—
124#394—21no—
125#394—21no—
126#394—21no—
127#392—21no—
128#392—21no—
129#392—21no—
130#391C2×C411yescharacteristic
131#391C2311yescharacteristic
132#390—21no—
133#390—21no—
134#390—21no—
135#390—21no—
136#390—21no—
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2311yescharacteristic
140#390C2311yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2311yescharacteristic
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#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#386—21no—
161#386—21no—
162#386—21no—
163#386—21no—
164#386—21no—
165#386—21no—
166#386—21no—
167#386—21no—
168#386—21no—
169#385—21no—
170#384C2×C411yescharacteristic
171#382C2×C411yescharacteristic
172#382C2×C411yescharacteristic
173#381—21no—
174#381—21no—
175#380C2×C411yescharacteristic
176#380C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
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×D4—21no—
213C22×D4—21no—
214C22×D4C22⋊C411yescharacteristic
215C22×D4C22×C411yescharacteristic
216C22×D4C22×C411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C2×M16—42no—
226C2×M16—42no—
227C2×M16—42no—
228C2×M16—42no—
229C2×M16—81no—
230C2×M16—81no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—42no—
240C22×C8—81no—
241C22×C8—81no—
242#127—81no—
243#124—24no—
244#124—81no—
245#123—81no—
246#123—81no—
247C22⋊Q8—81no—
248C22⋊Q8—81no—
249#121—44no—
250#121—42no—
251#121—42no—
252#121—81no—
253#121—81no—
254#121—81no—
255#121—81no—
256#121—81no—
257#121—81no—
258C4×Q8—81no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—42no—
264C4×D4—81no—
265#117—81no—
266#117—81no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22⋊C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—81no—
315C2×C42—81no—
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2411yescharacteristic
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—81no—
332C4⋊C8—81no—
333C4⋊C8—81no—
334C4⋊C8—81no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338C22⋊C8—81no—
339C22⋊C8—81no—
340C4×C8—42no—
341C4×C8—81no—
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#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359#95C22⋊C411yescharacteristic
360#95C22⋊C411yescharacteristic
361#95C22⋊C411yescharacteristic
362#95C22⋊C411yescharacteristic
363C24—21no—
364C24—21no—
365C24C2×C22⋊C411yescharacteristic
366C24C2×C22⋊C411yescharacteristic
367C2×Q8—22no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4—81no—
389C2×D4—81no—
390C2×D4—81no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—81no—
395C2×D4—81no—
396C2×D4—81no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4AGL(1,ℤ/8)12yes—
440C22×C4#13712yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444M16—84no—
445M16—84no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—84no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—41no—
495C22⋊C4—42no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—81no—
536C42—81no—
537C42#13114yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23#41712yes—
566C23C22×C22⋊C411yescharacteristic
567Q8—82no—
568D4—44no—
569D4—82no—
570D4—82no—
571D4—162no—
572D4—162no—
573D4—162no—
574D4—162no—
575D4—162no—
576D4—162no—
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—21no—
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—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—84no—
638C2×C4—162no—
639C2×C4#40812yes—
640C2×C4#40812yes—
641C2×C4#41712yes—
642C2×C4#42112yes—
643C2×C4C2×#13112yes—
644C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
645C2×C4C2×#13711yescharacteristic
646C2×C4#57611yescharacteristic
647C2×C4#57711yescharacteristic
648C8—42no—
649C8—81no—
650C8—81no—
651C8—81no—
652C8—81no—
653C8—81no—
654C8—81no—
655C8—81no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22C2×#40811yescharacteristic
685C22#288311yescharacteristic
686C22#288811yescharacteristic
687C22C2×#41711yescharacteristic
688C22#289711yescharacteristic
689C22C2×#42111yescharacteristic
690C22#290111yescharacteristic
691C4—22no—
692C4—22no—
693C4—21no—
694C4—21no—
695C4—22no—
696C4—21no—
697C4—21no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C2—42no—
713C2—81no—
714C2—161no—
715C2—161no—
716C2#1738911yescharacteristic
717C2#1746811yescharacteristic
718C2#1748111yescharacteristic
719C2#1759711yescharacteristic
720C2#1762211yescharacteristic
721C2#1768111yescharacteristic
722C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35109C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22700C2semidirect
3. #21996C2semidirect
4. #7730C2semidirect
5. #7714C2semidirect
6. #7710C2semidirect
7. #7694C2semidirect
8. #7684C2semidirect
9. #7681C2semidirect
10. #7664C2semidirect
11. #7661C2semidirect
12. #7601C2semidirect
13. #7594C2semidirect
14. #7584C2semidirect
15. #7574C2semidirect

Maximal subgroups

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

subgroupindex
#351092
#227002
#219962
#77302
#77142
#77102
#76942
#76842
#76812
#76642
#76612
#76012
#75942
#75842
#75742