Finite Group Databasebeta

← #163553

Subgroups — #163553

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35008C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic, complemented
3#21996C211yesmaximal, characteristic, complemented
4#7732C211yesmaximal, characteristic, complemented
5#7723C211yesmaximal, characteristic, complemented
6#7685C211yesmaximal, characteristic, complemented
7#7678C211yesmaximal, characteristic, complemented
8#7672C211yesmaximal, characteristic, complemented
9#7665C211yesmaximal, characteristic, complemented
10#7663C211yesmaximal, characteristic, complemented
11#7658C211yesmaximal, characteristic, complemented
12#7647C211yesmaximal, characteristic, complemented
13#7641C211yesmaximal, characteristic, complemented
14#7616C211yesmaximal, characteristic, complemented
15#7610C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C2×#436—21no—
18#2860C2211yescharacteristic
19#2859C2211yescharacteristic
20#2857C2211yescharacteristic
21#2854C2211yescharacteristic
22#2664—21no—
23#2663—21no—
24#2556—21no—
25#2553—21no—
26#2506C411yescharacteristic
27#2501C411yescharacteristic
28#2496C411yescharacteristic
29#2488—21no—
30#2486C411yescharacteristic
31#2452C411yescharacteristic
32#2440C411yescharacteristic
33#2425—21no—
34#2422C411yescharacteristic
35#2416C411yescharacteristic
36#2322—21no—
37C4×#121C2211yescharacteristic
38C2×#389C2211yescharacteristic
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#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×M16—21no—
76C22×M16—21no—
77C23×C8—21no—
78#539—44no—
79#539—44no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#126—21no—
82C2×#126—21no—
83C2×#125—21no—
84C2×#125—21no—
85C2×#123—21no—
86C2×#123—21no—
87C2×#123—21no—
88C2×#123—21no—
89C2×#123—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×#121—21no—
94C2×#121C2×C411yescharacteristic
95C2×#121C2×C411yescharacteristic
96C2×C4×D4—21no—
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D411yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#436—44no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120#400—21no—
121#400—21no—
122#399—21no—
123#399—21no—
124#398—21no—
125#398—21no—
126#398—21no—
127#397—21no—
128#397—21no—
129#397—21no—
130#396—21no—
131#396—21no—
132#395—21no—
133#394—21no—
134#394—21no—
135#393—21no—
136#391—21no—
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2311yescharacteristic
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—21no—
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#389C2×C411yescharacteristic
152#389C2×C411yescharacteristic
153#389C2311yescharacteristic
154#389C2311yescharacteristic
155#389C2311yescharacteristic
156#388—21no—
157#388—21no—
158#388—21no—
159#388—21no—
160#388—21no—
161#388—21no—
162#387—21no—
163#387—21no—
164#386—21no—
165#386—21no—
166#386—21no—
167#385—21no—
168#385—21no—
169#385—21no—
170#384C2×C411yescharacteristic
171#384C2×C411yescharacteristic
172#384C2×C411yescharacteristic
173#382—21no—
174#381—21no—
175#380—21no—
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178#380C2×C411yescharacteristic
179#380C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C22⋊C4—21no—
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C2×#95—21no—
192C43C2311yescharacteristic
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C22×Q8—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22⋊C411yescharacteristic
213C22×D4C22×C411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C2×M16—42no—
224C2×M16—42no—
225C2×M16—42no—
226C2×M16—42no—
227C2×M16—81no—
228C2×M16—81no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—42no—
238C22×C8—81no—
239C22×C8—81no—
240C4⋊Q8—81no—
241#126—44no—
242#126—42no—
243#126—42no—
244#125—44no—
245#125—42no—
246#125—42no—
247#123—44no—
248#123—44no—
249#123—42no—
250#123—42no—
251#123—44no—
252#123—44no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—42no—
258#121—44no—
259#121—81no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—81no—
265C4×D4D814yes—
266C4×D4SD1614yes—
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⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—81no—
290C2×C4⋊C4C22⋊C411yescharacteristic
291C2×C4⋊C4C22⋊C411yescharacteristic
292C2×C4⋊C4C22⋊C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C42—21no—
322C2×C42—81no—
323C2×C42—81no—
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2411yescharacteristic
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—81no—
340C4⋊C8—81no—
341C4⋊C8—81no—
342C4⋊C8—81no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—81no—
347C22⋊C8—81no—
348C4×C8—42no—
349C4×C8—81no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371C24—21no—
372C24C2×C22⋊C411yescharacteristic
373C24C2×C22⋊C411yescharacteristic
374C2×Q8—42no—
375C2×Q8—81no—
376C2×Q8—81no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4#10214yes—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×D812yes—
434C22×C4C2×D812yes—
435C22×C4C2×SD1612yes—
436C22×C4C2×SD1612yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440M16—84no—
441M16—84no—
442C2×C8—22no—
443C2×C8—41no—
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—84no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4#10214yes—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4#10214yes—
521C22⋊C4#10214yes—
522C42—22no—
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—42no—
536C42C2×D812yes—
537C42C2×SD1612yes—
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—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×#10212yes—
558C23C2×#10212yes—
559C23C22×C22⋊C411yescharacteristic
560Q8—162no—
561D4—44no—
562D4—82no—
563D4—84no—
564D4—162no—
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—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—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
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—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4C2×#10212yes—
631C2×C4C2×#10212yes—
632C2×C4C2×#10212yes—
633C2×C4C2×#10212yes—
634C2×C4#42112yes—
635C2×C4#56812yes—
636C2×C4C22×D811yescharacteristic
637C2×C4C22×SD1611yescharacteristic
638C2×C4#57711yescharacteristic
639C2×C4#57811yescharacteristic
640C8—42no—
641C8—81no—
642C8—81no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22#287911yescharacteristic
670C22#288311yescharacteristic
671C22#288911yescharacteristic
672C22C22×#10211yescharacteristic
673C22#289811yescharacteristic
674C22C2×#42111yescharacteristic
675C22#290111yescharacteristic
676C4—22no—
677C4—22no—
678C4—21no—
679C4—21no—
680C4—22no—
681C4—21no—
682C4—21no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C2—42no—
699C2—42no—
700C2—161no—
701C2#1740611yescharacteristic
702C2#1748511yescharacteristic
703C2#1750611yescharacteristic
704C2#1759811yescharacteristic
705C2#1761311yescharacteristic
706C2#1768211yescharacteristic
707C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2416C4
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. #7732C2semidirect
5. #7723C2semidirect
6. #7685C2semidirect
7. #7678C2semidirect
8. #7672C2semidirect
9. #7665C2semidirect
10. #7663C2semidirect
11. #7658C2semidirect
12. #7647C2semidirect
13. #7641C2semidirect
14. #7616C2semidirect
15. #7610C2semidirect

Maximal subgroups

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

subgroupindex
#350082
#227002
#219962
#77322
#77232
#76852
#76782
#76722
#76652
#76632
#76582
#76472
#76412
#76162
#76102