Finite Group Databasebeta

← #215239

Subgroups — #215239

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#21924C211yesmaximal, characteristic, complemented
3#21924C211yesmaximal, characteristic, complemented
4#9710C211yesmaximal, characteristic, complemented
5#9709C211yesmaximal, characteristic, complemented
6#9704C211yesmaximal, characteristic, complemented
7#9701C211yesmaximal, characteristic, complemented
8#9698C211yesmaximal, characteristic, complemented
9#9697C211yesmaximal, characteristic, complemented
10#9696C211yesmaximal, characteristic, complemented
11#9693C211yesmaximal, characteristic, complemented
12#8344C211yesmaximal, characteristic
13#8344C211yesmaximal, characteristic
14#8340C211yesmaximal, characteristic, complemented
15#8340C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18#2862C2211yescharacteristic
19#2862C2211yescharacteristic
20#2858C2211yescharacteristic
21#2858C2211yescharacteristic
22#2640C412yes—
23#2640C412yes—
24#2563—21no—
25#2563—21no—
26#2547—21no—
27#2547—21no—
28#2504—21no—
29#2504—21no—
30#2436—21no—
31#2436—21no—
32#2311C412yes—
33#2311C412yes—
34C4×#117C2211yescharacteristic
35#1924—21no—
36#1924—21no—
37#1923—21no—
38#1923—21no—
39#1917—21no—
40#1917—21no—
41#1916—21no—
42#1916—21no—
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#332C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69C2×#331C2211yescharacteristic
70C2×#331C2211yescharacteristic
71C22×#41C2×C412yes—
72C23×Q8—21no—
73#532—41no—
74#532—41no—
75#532—41no—
76#532—41no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—41no—
80C2×C4⋊Q8—41no—
81C2×C4⋊Q8—41no—
82C2×C4⋊Q8—41no—
83C2×C4⋊Q8C2×C412yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127C2311yescharacteristic
88C2×#124—21no—
89C2×#124—21no—
90C2×#124—21no—
91C2×#124—21no—
92C2×#124C2×C412yes—
93C2×#121—21no—
94C2×#121—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×#117C2×C411yescharacteristic
98C2×#117C2×C411yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2×C412yes—
101C22×C42C2311yescharacteristic
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#397—22no—
123#397—22no—
124#397—22no—
125#397—22no—
126#393—22no—
127#393—22no—
128#393—22no—
129#393—22no—
130#391—21no—
131#391—21no—
132#391—21no—
133#391—21no—
134#391C2×C412yes—
135#391C2×C412yes—
136#391C2×C412yes—
137#391C2×C412yes—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390C2×C412yes—
143#390C2×C412yes—
144#390C2×C412yes—
145#390C2×C412yes—
146#386—22no—
147#386—22no—
148#386—22no—
149#386—22no—
150#382—21no—
151#382—21no—
152#382—21no—
153#382—21no—
154#380—22no—
155#380—22no—
156#380—22no—
157#380—22no—
158C4×C4⋊C4D411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C4×C4⋊C4D411yescharacteristic
162C4×C22⋊C4D411yescharacteristic
163C4×C22⋊C4D411yescharacteristic
164C4×C22⋊C4D411yescharacteristic
165C4×C22⋊C4D411yescharacteristic
166#376C2×C411yescharacteristic
167#376C2×C411yescharacteristic
168C43C2311yescharacteristic
169C43C2311yescharacteristic
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#333—41no—
183#333—41no—
184#333—41no—
185#333—41no—
186#332—41no—
187#332—41no—
188#332—41no—
189#332—41no—
190#332—41no—
191#332—41no—
192#332—41no—
193#332—41no—
194#331—41no—
195#331—41no—
196#331—41no—
197#331—41no—
198C2×#41—42no—
199C2×#41—42no—
200C2×#41—44no—
201C2×#41—44no—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—81no—
208C22×Q8—81no—
209C22×Q8—81no—
210C22×Q8—81no—
211C22×Q8C22×C412yes—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C22×C412yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243C4⋊Q8—42no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247#127—21no—
248#127—21no—
249#127—21no—
250#127—21no—
251#124—42no—
252#124—42no—
253#124—81no—
254#124—81no—
255#124—81no—
256#124—81no—
257#124—81no—
258#124—81no—
259#124—81no—
260#124—81no—
261#121—42no—
262#121—42no—
263#121—42no—
264#121—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269#117—42no—
270#117—42no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C412yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—81no—
338C4⋊C8—81no—
339C4⋊C8—81no—
340C4⋊C8—81no—
341C22⋊C8—81no—
342C22⋊C8—81no—
343C22⋊C8—81no—
344C22⋊C8—81no—
345#97—81no—
346#97—81no—
347C4×C8—81no—
348C4×C8—81no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359#95C22⋊C411yescharacteristic
360#95C22⋊C411yescharacteristic
361#95C22⋊C411yescharacteristic
362#95C22⋊C411yescharacteristic
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365C24—22no—
366C24C2×C22⋊C411yescharacteristic
367#41—88no—
368#41—88no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8—84no—
389C2×Q8—84no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
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—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42H(5,𝔽2)11yescharacteristic
533C42H(5,𝔽2)11yescharacteristic
534C42H(5,𝔽2)11yescharacteristic
535C42H(5,𝔽2)11yescharacteristic
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23#41312yes—
550C23C22×C22⋊C411yescharacteristic
551Q8—82no—
552Q8—82no—
553Q8—88no—
554Q8—81no—
555Q8—81no—
556Q8—81no—
557Q8—81no—
558Q8—81no—
559Q8—81no—
560Q8—81no—
561Q8—81no—
562D4—82no—
563D4—82no—
564D4—88no—
565D4—81no—
566D4—81no—
567D4—81no—
568D4—81no—
569D4—81no—
570D4—81no—
571D4—81no—
572D4—81no—
573C2×C4—24no—
574C2×C4—24no—
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—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
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—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—88no—
627C2×C4—88no—
628C2×C4#41312yes—
629C2×C4#51811yescharacteristic
630C2×C4#51811yescharacteristic
631C2×C4#53511yescharacteristic
632C2×C4#53511yescharacteristic
633C2×C4#53511yescharacteristic
634C2×C4#53511yescharacteristic
635C2×C4#57611yescharacteristic
636C2×C4#57611yescharacteristic
637C2×C4#57711yescharacteristic
638C2×C4#57711yescharacteristic
639C2×C4#57711yescharacteristic
640C2×C4#57711yescharacteristic
641C2×C4#57811yescharacteristic
642C2×C4#57811yescharacteristic
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C8—81no—
649C8—81no—
650C8—81no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22#230911yescharacteristic
666C22C2×#41311yescharacteristic
667C22C2×#41311yescharacteristic
668C22#289711yescharacteristic
669C22#289711yescharacteristic
670C22#289811yescharacteristic
671C22#289811yescharacteristic
672C4—22no—
673C4—21no—
674C4—21no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#322011yescharacteristic
689C4#322111yescharacteristic
690C4#322211yescharacteristic
691C4#322311yescharacteristic
692C4#322411yescharacteristic
693C4#322411yescharacteristic
694C4#322511yescharacteristic
695C4#322511yescharacteristic
696C2—42no—
697C2—162no—
698C2#1777211yescharacteristic
699C2#1777511yescharacteristic
700C2#1777511yescharacteristic
701C2#1778111yescharacteristic
702C2#2179811yescharacteristic
703C2#2181311yescharacteristic
704C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21924C2semidirect
3. #21924C2semidirect
4. #9710C2semidirect
5. #9709C2semidirect
6. #9704C2semidirect
7. #9701C2semidirect
8. #9698C2semidirect
9. #9697C2semidirect
10. #9696C2semidirect
11. #9693C2semidirect
14. #8340C2semidirect
15. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#219242
#97102
#97092
#97042
#97012
#96982
#96972
#96962
#96932
#83442
#83402