Finite Group Databasebeta

← #165643

Subgroups — #165643

Order 512.

939 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36396C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22787C211yesmaximal, characteristic, complemented
4#7730C211yesmaximal, characteristic, complemented
5#7716C211yesmaximal, characteristic, complemented
6#7704C211yesmaximal, characteristic, complemented
7#7704C211yesmaximal, characteristic, complemented
8#7690C211yesmaximal, characteristic, complemented
9#7676C211yesmaximal, characteristic, complemented
10#7629C211yesmaximal, characteristic, complemented
11#7618C211yesmaximal, characteristic, complemented
12#7612C211yesmaximal, characteristic, complemented
13#7595C211yesmaximal, characteristic, complemented
14#7584C211yesmaximal, characteristic, complemented
15#7578C211yesmaximal, characteristic, complemented
16C2×#553C411yescharacteristic
17C2×#548C411yescharacteristic
18#2861C2211yescharacteristic
19#2856C2211yescharacteristic
20#2850C2211yescharacteristic
21#2565C411yescharacteristic
22#2560—21no—
23#2560—21no—
24#2557—21no—
25#2557—21no—
26#2544C411yescharacteristic
27#2543—21no—
28#2543—21no—
29#2504C411yescharacteristic
30#2499C411yescharacteristic
31#2436C411yescharacteristic
32#2431C411yescharacteristic
33#2364—21no—
34#2364—21no—
35C4×#124C2211yescharacteristic
36#2297C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1924—21no—
41#1924—21no—
42#1924—21no—
43#1917—21no—
44#1869—21no—
45#1839C2211yescharacteristic
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770—21no—
52C4×C4⋊C8C2211yescharacteristic
53#1752—21no—
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1745C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C23×Q8C2×C411yescharacteristic
77C23×C8—21no—
78#553—44no—
79#548—44no—
80C2×#127C2×C411yescharacteristic
81C2×#126—21no—
82C2×#125C2×C411yescharacteristic
83C2×#124—21no—
84C2×#124C2×C411yescharacteristic
85C2×#124C2×C411yescharacteristic
86C2×#124C2×C411yescharacteristic
87C2×#123—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8C2×C411yescharacteristic
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121C2×C411yescharacteristic
95C2×#121C2×C411yescharacteristic
96C2×#121C2×C411yescharacteristic
97C2×C4×Q8C2×C411yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4C2×C411yescharacteristic
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×#117C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123#398—21no—
124#398—21no—
125#397—21no—
126#397—21no—
127#397—21no—
128#397—21no—
129#394—21no—
130#394—21no—
131#393—21no—
132#393—21no—
133#392—21no—
134#392—21no—
135#391—21no—
136#391—21no—
137#391C2×C411yescharacteristic
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#391C2×C411yescharacteristic
141#391C2311yescharacteristic
142#391C2311yescharacteristic
143#390—21no—
144#390—21no—
145#390C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#390C2311yescharacteristic
148#390C2311yescharacteristic
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#386—21no—
172#386—21no—
173#384C2×C411yescharacteristic
174#382C2×C411yescharacteristic
175#382C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C22⋊C4—21no—
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
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C2×#95—21no—
190C2×#95—21no—
191C43C2311yescharacteristic
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C2×#41—22no—
209C2×#41—41no—
210C2×#41—42no—
211C2×#41—81no—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—41no—
215C22×Q8—42no—
216C22×Q8—81no—
217C22×Q8C22×C411yescharacteristic
218C22×Q8C22×C411yescharacteristic
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C22⋊C411yescharacteristic
226C22×D4C22×C411yescharacteristic
227C22×D4C22×C411yescharacteristic
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C22×C411yescharacteristic
233C23×C4C22×C411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—81no—
249#127—81no—
250#126—44no—
251#125—42no—
252#124—22no—
253#124—41no—
254#124—42no—
255#124—44no—
256#124—81no—
257#123—44no—
258C22⋊Q8—44no—
259C22⋊Q8—42no—
260#121—42no—
261#121—42no—
262#121—42no—
263#121—42no—
264#121—42no—
265#121—42no—
266#121—42no—
267#121—44no—
268#121—44no—
269C4×Q8—42no—
270C4×D4—24no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—42no—
276C4×D4—44no—
277C4×D4—44no—
278#117—42no—
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⋊C4C22⋊C411yescharacteristic
288C2×C4⋊C4C22⋊C411yescharacteristic
289C2×C4⋊C4C22⋊C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
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⋊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⋊C4C22⋊C411yescharacteristic
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C42—21no—
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—
340C22⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—81no—
345C22⋊C8—81no—
346C22⋊C8—81no—
347#97—42no—
348#97—81no—
349#97—81no—
350#97—81no—
351C4×C8—42no—
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#95—21no—
366#95—21no—
367#95—21no—
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371#95C22⋊C411yescharacteristic
372#95C22⋊C411yescharacteristic
373#95C22⋊C411yescharacteristic
374C24—21no—
375C24—21no—
376C24C2×C22⋊C411yescharacteristic
377C24C2×C22⋊C411yescharacteristic
378#41—84no—
379#41—84no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—81no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8#13112yes—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4#10014yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C23×C411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—84no—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—22no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—42no—
563C42—42no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×#10012yes—
591C23C2×#10012yes—
592C23C22×C22⋊C411yescharacteristic
593Q8—42no—
594Q8—42no—
595Q8—81no—
596Q8—81no—
597Q8—84no—
598D4—44no—
599D4—81no—
600D4—81no—
601D4—81no—
602D4—81no—
603D4—84no—
604D4—84no—
605D4—84no—
606D4—162no—
607D4—162no—
608D4—162no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—81no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—162no—
677C2×C4—162no—
678C2×C4C2×#10012yes—
679C2×C4C2×#13111yescharacteristic
680C2×C4#56811yescharacteristic
681C2×C4#57511yescharacteristic
682C2×C4#57511yescharacteristic
683C2×C4#57511yescharacteristic
684C2×C4#57511yescharacteristic
685C8—42no—
686C8—42no—
687C8—81no—
688C8—81no—
689C8—81no—
690C8—81no—
691C8—81no—
692C8—81no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22#288011yescharacteristic
722C22#288311yescharacteristic
723C22C22×#10011yescharacteristic
724C22#289611yescharacteristic
725C22#289611yescharacteristic
726C22#290111yescharacteristic
727C22#290111yescharacteristic
728C4—21no—
729C4—21no—
730C4—22no—
731C4—21no—
732C4—21no—
733C4—21no—
734C4—21no—
735C4—41no—
736C4—41no—
737C4—42no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4—81no—
749C4—81no—
750C2—42no—
751C2—42no—
752C2—161no—
753C2—161no—
754C2#1742111yescharacteristic
755C2#1751611yescharacteristic
756C2#1751611yescharacteristic
757C2#1762311yescharacteristic
758C2#1762511yescharacteristic
759C2#1768511yescharacteristic
760C2#1775411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36396C2
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. #22839C2semidirect
3. #22787C2semidirect
4. #7730C2semidirect
5. #7716C2semidirect
6. #7704C2semidirect
7. #7704C2semidirect
8. #7690C2semidirect
9. #7676C2semidirect
10. #7629C2semidirect
11. #7618C2semidirect
12. #7612C2semidirect
13. #7595C2semidirect
14. #7584C2semidirect
15. #7578C2semidirect

Maximal subgroups

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

subgroupindex
#363962
#228392
#227872
#77302
#77162
#77042
#76902
#76762
#76292
#76182
#76122
#75952
#75842
#75782