Finite Group Databasebeta

← #164772

Subgroups — #164772

Order 512.

935 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36375C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic
3#20720C211yesmaximal, characteristic
4#7731C211yesmaximal, characteristic, complemented
5#7715C211yesmaximal, characteristic, complemented
6#7675C211yesmaximal, characteristic, complemented
7#7671C211yesmaximal, characteristic, complemented
8#7661C211yesmaximal, characteristic, complemented
9#7661C211yesmaximal, characteristic, complemented
10#7648C211yesmaximal, characteristic, complemented
11#7641C211yesmaximal, characteristic, complemented
12#7613C211yesmaximal, characteristic, complemented
13#7583C211yesmaximal, characteristic, complemented
14#7573C211yesmaximal, characteristic, complemented
15#7564C211yesmaximal, characteristic, complemented
16C2×#552C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×C8×Q8—21no—
19C2×#433—21no—
20#2855C2211yescharacteristic
21#2855C2211yescharacteristic
22#2851C2211yescharacteristic
23#2584C411yescharacteristic
24#2581C411yescharacteristic
25#2486C411yescharacteristic
26#2485C411yescharacteristic
27#2478—21no—
28#2475—21no—
29#2466—21no—
30#2462—21no—
31#2419—21no—
32#2417C411yescharacteristic
33#2415C411yescharacteristic
34#2406—21no—
35#2359—21no—
36#2352C411yescharacteristic
37#2344—21no—
38C4×C22⋊Q8C2211yescharacteristic
39#2296C2211yescharacteristic
40C2×#389C2211yescharacteristic
41#1954—21no—
42#1954—21no—
43#1954—21no—
44#1817C2211yescharacteristic
45#1817C2211yescharacteristic
46#1798—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1780—21no—
53#1750—21no—
54#1749C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×#41D411yescharacteristic
76#552—44no—
77C2×#126—21no—
78C2×#125—21no—
79C2×#125C2×C411yescharacteristic
80C2×#123—21no—
81C2×#123—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×C22⋊Q8C2×C411yescharacteristic
85C4×#41—44no—
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8D411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4D411yescharacteristic
95C2×#117—21no—
96C2×#117C2×C411yescharacteristic
97C2×#117C2×C411yescharacteristic
98C2×#117D411yescharacteristic
99C2×#117D411yescharacteristic
100C2×#117D411yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C8×Q8—44no—
108#433—44no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×#97—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#400—21no—
132#398—21no—
133#398—21no—
134#398—21no—
135#397—21no—
136#397—21no—
137#397—21no—
138#396—21no—
139#395—21no—
140#395—21no—
141#394—21no—
142#394—21no—
143#391C2×C411yescharacteristic
144#391C2×C411yescharacteristic
145#391C2311yescharacteristic
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389C2×C411yescharacteristic
152#389C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#389C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#389C2311yescharacteristic
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#388—21no—
161#388—21no—
162#388—21no—
163#387—21no—
164#387—21no—
165#387—21no—
166#387—21no—
167#387—21no—
168#387—21no—
169#387—21no—
170#387—21no—
171#387—21no—
172#387—21no—
173#387—21no—
174#386—21no—
175#386—21no—
176#385—21no—
177#385—21no—
178#385—21no—
179#385—21no—
180#385—21no—
181#385—21no—
182#385—21no—
183#382C2×C411yescharacteristic
184#381—21no—
185#381—21no—
186#381—21no—
187#380—21no—
188#380C2×C411yescharacteristic
189#380C2×C411yescharacteristic
190#380C2×C411yescharacteristic
191#380C2×C411yescharacteristic
192#380C2×C411yescharacteristic
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2×C411yescharacteristic
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C2×#95—21no—
200#336—21no—
201#336—21no—
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209#336C2311yescharacteristic
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—42no—
213C2×#41—42no—
214C22×Q8C22⋊C411yescharacteristic
215C22×Q8C22×C411yescharacteristic
216C22×D4C22⋊C411yescharacteristic
217C22×D4C22⋊C411yescharacteristic
218C22×D4C22⋊C411yescharacteristic
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C22×C411yescharacteristic
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238#126—44no—
239#125—42no—
240#125—44no—
241#123—44no—
242#123—44no—
243C22⋊Q8—22no—
244C22⋊Q8—41no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247C4×Q8—24no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×D4—24no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256#117—24no—
257#117—22no—
258#117—42no—
259#117—42no—
260#117—44no—
261#117—41no—
262#117—42no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
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—41no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4C22⋊C411yescharacteristic
286C2×C4⋊C4C22⋊C411yescharacteristic
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×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C22⋊C4C22⋊C411yescharacteristic
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2411yescharacteristic
331C4⋊C8—42no—
332C4⋊C8—42no—
333C4⋊C8—42no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339C22⋊C8—42no—
340C22⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—81no—
346C22⋊C8—81no—
347C22⋊C8—81no—
348#97—42no—
349C4×C8—42no—
350C4×C8—42no—
351C4×C8—42no—
352C4×C8—42no—
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#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376C24C2×C22⋊C411yescharacteristic
377C24C2×C22⋊C411yescharacteristic
378C24C2×C22⋊C411yescharacteristic
379#41—48no—
380#41—48no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C22×C4—21no—
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—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
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—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4#13512yes—
455C22×C4#13512yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C2×C8—22no—
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—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4#9914yes—
518C4⋊C4#13112yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—81no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23#41612yes—
570C23C22×C22⋊C411yescharacteristic
571Q8—28no—
572Q8—81no—
573Q8—81no—
574D4—44no—
575D4—44no—
576D4—44no—
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—22no—
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—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
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—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4C2×#9912yes—
650C2×C4C2×#9912yes—
651C2×C4C2×#9912yes—
652C2×C4#41612yes—
653C2×C4#42112yes—
654C2×C4C2×#13111yescharacteristic
655C2×C4#56811yescharacteristic
656C2×C4C2×#13511yescharacteristic
657C2×C4C2×#13511yescharacteristic
658C2×C4#57711yescharacteristic
659C2×C4#57811yescharacteristic
660C8—42no—
661C8—81no—
662C8—81no—
663C8—81no—
664C8—81no—
665C8—81no—
666C8—81no—
667C8—81no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22#288011yescharacteristic
690C22C22×#9911yescharacteristic
691C22#288911yescharacteristic
692C22C2×#41611yescharacteristic
693C22#289811yescharacteristic
694C22C2×#42111yescharacteristic
695C22#290111yescharacteristic
696C4—21no—
697C4—21no—
698C4—22no—
699C4—22no—
700C4—21no—
701C4—21no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—21no—
706C4—21no—
707C4—41no—
708C4—42no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C2—42no—
720C2—81no—
721C2—81no—
722C2#1741611yescharacteristic
723C2#1750811yescharacteristic
724C2#1752511yescharacteristic
725C2#1757211yescharacteristic
726C2#1760611yescharacteristic
727C2#1768211yescharacteristic
728C2#1772511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7731C2semidirect
5. #7715C2semidirect
6. #7675C2semidirect
7. #7671C2semidirect
8. #7661C2semidirect
9. #7661C2semidirect
10. #7648C2semidirect
11. #7641C2semidirect
12. #7613C2semidirect
13. #7583C2semidirect
14. #7573C2semidirect
15. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#363752
#225302
#207202
#77312
#77152
#76752
#76712
#76612
#76482
#76412
#76132
#75832
#75732
#75642