Finite Group Databasebeta

← #163586

Subgroups — #163586

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35044C211yesmaximal, characteristic
2#22837C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7730C211yesmaximal, characteristic, complemented
5#7718C211yesmaximal, characteristic, complemented
6#7689C211yesmaximal, characteristic, complemented
7#7679C211yesmaximal, characteristic, complemented
8#7670C211yesmaximal, characteristic, complemented
9#7669C211yesmaximal, characteristic, complemented
10#7659C211yesmaximal, characteristic, complemented
11#7658C211yesmaximal, characteristic, complemented
12#7647C211yesmaximal, characteristic, complemented
13#7645C211yesmaximal, characteristic, complemented
14#7620C211yesmaximal, characteristic, complemented
15#7612C211yesmaximal, characteristic, complemented
16#2863C2211yescharacteristic
17#2858C2211yescharacteristic
18#2854C2211yescharacteristic
19#2850C2211yescharacteristic
20#2648—21no—
21#2631—21no—
22#2597—21no—
23#2557—21no—
24#2513C411yescharacteristic
25#2492C411yescharacteristic
26#2459—21no—
27#2456—21no—
28#2447C411yescharacteristic
29#2439C411yescharacteristic
30#2431C411yescharacteristic
31#2428C411yescharacteristic
32#2414C411yescharacteristic
33#2407C411yescharacteristic
34#2337—21no—
35#2333—21no—
36C4×#121C2211yescharacteristic
37#2297C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1955—21no—
40#1954—21no—
41#1939—21no—
42#1924—21no—
43#1923—21no—
44#1923—21no—
45#1923—21no—
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1771—21no—
54#1767C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56#1749C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
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×#41C2×C411yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
78C2×#124—21no—
79C2×#124—21no—
80C2×#124C2×C411yescharacteristic
81C2×#123—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113#400—21no—
114#397—21no—
115#397—21no—
116#397—21no—
117#397—21no—
118#396—21no—
119#394—21no—
120#394—21no—
121#394—21no—
122#393—21no—
123#393—21no—
124#392—21no—
125#392—21no—
126#392—21no—
127#392—21no—
128#392—21no—
129#391—21no—
130#391—21no—
131#391C2×C411yescharacteristic
132#391C2×C411yescharacteristic
133#391C2×C411yescharacteristic
134#391C2311yescharacteristic
135#390—21no—
136#390—21no—
137#390—21no—
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#390C2311yescharacteristic
143#390C2311yescharacteristic
144#389C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2311yescharacteristic
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#387—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#386—21no—
170#386—21no—
171#385—21no—
172#385—21no—
173#383—21no—
174#382C2×C411yescharacteristic
175#381—21no—
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C43C2311yescharacteristic
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C2×#41—42no—
207C2×#41—81no—
208C2×#41—81no—
209C22×Q8—21no—
210C22×Q8C22×C411yescharacteristic
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4C22⋊C411yescharacteristic
218C22×D4C22×C411yescharacteristic
219C22×D4C22×C411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C4⋊Q8—81no—
237#127—81no—
238#124—42no—
239#124—42no—
240#124—81no—
241#124—81no—
242#124—81no—
243#123—81no—
244#123—81no—
245C22⋊Q8—81no—
246#121—44no—
247#121—42no—
248#121—81no—
249#121—81no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—81no—
255C4×D4—81no—
256C4×D4—81no—
257#117—81no—
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⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—81no—
303C2×C42—81no—
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2411yescharacteristic
317C4⋊C8—42no—
318C4⋊C8—42no—
319C4⋊C8—42no—
320C4⋊C8—81no—
321C4⋊C8—81no—
322C22⋊C8—42no—
323C22⋊C8—42no—
324C22⋊C8—81no—
325C22⋊C8—81no—
326C22⋊C8—81no—
327C22⋊C8—81no—
328C22⋊C8—81no—
329C22⋊C8—81no—
330#97—42no—
331#97—81no—
332C4×C8—42no—
333C4×C8—81no—
334C4×C8—81no—
335#95—21no—
336#95—21no—
337#95—21no—
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#95—21no—
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358C24—21no—
359C24—21no—
360C24C2×C22⋊C411yescharacteristic
361C24C2×C22⋊C411yescharacteristic
362#41—88no—
363C2×Q8—22no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—42no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—81no—
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—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
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—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C2×C8—22no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
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—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—22no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—81no—
528C42—81no—
529C42#13512yes—
530C42#13512yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C22×C22⋊C411yescharacteristic
560Q8—82no—
561Q8—162no—
562D4—44no—
563D4—82no—
564D4—82no—
565D4—84no—
566D4—84no—
567D4—162no—
568D4—162no—
569D4—162no—
570D4—162no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
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—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—44no—
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—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—82no—
632C2×C4—84no—
633C2×C4—162no—
634C2×C4#41612yes—
635C2×C4#41612yes—
636C2×C4#42112yes—
637C2×C4#56812yes—
638C2×C4C2×#13511yescharacteristic
639C2×C4C2×#13511yescharacteristic
640C2×C4#57611yescharacteristic
641C2×C4#57711yescharacteristic
642C8—42no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C8—81no—
649C8—81no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22#287911yescharacteristic
679C22#288311yescharacteristic
680C22#288811yescharacteristic
681C22C2×#41611yescharacteristic
682C22#289711yescharacteristic
683C22C2×#42111yescharacteristic
684C22#290111yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—81no—
706C2—81no—
707C2—81no—
708C2—161no—
709C2—161no—
710C2#1740511yescharacteristic
711C2#1748911yescharacteristic
712C2#1750511yescharacteristic
713C2#1759911yescharacteristic
714C2#1761711yescharacteristic
715C2#1767911yescharacteristic
716C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35044C2
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. #22837C2semidirect
3. #22008C2semidirect
4. #7730C2semidirect
5. #7718C2semidirect
6. #7689C2semidirect
7. #7679C2semidirect
8. #7670C2semidirect
9. #7669C2semidirect
10. #7659C2semidirect
11. #7658C2semidirect
12. #7647C2semidirect
13. #7645C2semidirect
14. #7620C2semidirect
15. #7612C2semidirect

Maximal subgroups

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

subgroupindex
#350442
#228372
#220082
#77302
#77182
#76892
#76792
#76702
#76692
#76592
#76582
#76472
#76452
#76202
#76122