Finite Group Databasebeta

← #164960

Subgroups — #164960

Order 512.

857 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36271C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic, complemented
3#22736C211yesmaximal, characteristic, complemented
4#7732C211yesmaximal, characteristic, complemented
5#7719C211yesmaximal, characteristic
6#7686C211yesmaximal, characteristic, complemented
7#7679C211yesmaximal, characteristic
8#7672C211yesmaximal, characteristic, complemented
9#7666C211yesmaximal, characteristic, complemented
10#7642C211yesmaximal, characteristic, complemented
11#7625C211yesmaximal, characteristic, complemented
12#7614C211yesmaximal, characteristic, complemented
13#7588C211yesmaximal, characteristic, complemented
14#7574C211yesmaximal, characteristic, complemented
15#7560C211yesmaximal, characteristic, complemented
16C2×#556—21no—
17C2×#446—21no—
18#2860C2211yescharacteristic
19#2860C2211yescharacteristic
20#2858C2211yescharacteristic
21#2601—21no—
22#2601—21no—
23#2592—21no—
24#2560—21no—
25#2560—21no—
26#2548C411yescharacteristic
27#2546C411yescharacteristic
28#2545—21no—
29#2512C411yescharacteristic
30#2512C411yescharacteristic
31#2452C411yescharacteristic
32#2452C411yescharacteristic
33#2367—21no—
34#2366C411yescharacteristic
35#2365C411yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37#2293C2211yescharacteristic
38#2293C2211yescharacteristic
39C42×Q8C2211yescharacteristic
40#1984—21no—
41#1940—21no—
42#1940—21no—
43#1939—21no—
44#1916—21no—
45#1916—21no—
46#1839C2211yescharacteristic
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1831C2211yescharacteristic
52C8×C4⋊C4—21no—
53#1767C2211yescharacteristic
54#1767C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57#1744C2211yescharacteristic
58#1744C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75#556—44no—
76#556—44no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#127C2×C411yescharacteristic
81C2×#126—21no—
82C2×#126—21no—
83C2×#126—21no—
84C2×#126—21no—
85C2×#124—21no—
86C2×#124—21no—
87C2×#124—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8D411yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2×C411yescharacteristic
98C2×#117—21no—
99C2×#117—21no—
100#446—44no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#398—21no—
116#398—21no—
117#397—21no—
118#397—21no—
119#396—21no—
120#396—21no—
121#396—21no—
122#396—21no—
123#394—21no—
124#394—21no—
125#393—21no—
126#393—21no—
127#391—21no—
128#391—21no—
129#391—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#390—21no—
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#390C2311yescharacteristic
143#390C2311yescharacteristic
144#389—21no—
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#389C2311yescharacteristic
152#389C2311yescharacteristic
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#388—21no—
158#388—21no—
159#388—21no—
160#388—21no—
161#388—21no—
162#388—21no—
163#388—21no—
164#388—21no—
165#387—21no—
166#387—21no—
167#386—21no—
168#386—21no—
169#385—21no—
170#385—21no—
171#384C2×C411yescharacteristic
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C4⋊C4D411yescharacteristic
185C4×C4⋊C4D411yescharacteristic
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#336—21no—
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#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8C22⋊C411yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C22×C411yescharacteristic
215C22×D4C22×C411yescharacteristic
216C23×C4—21no—
217C23×C4—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C4⋊Q8—22no—
227C4⋊Q8—41no—
228C4⋊Q8—42no—
229C4⋊Q8—44no—
230#127—42no—
231#126—44no—
232#126—44no—
233#126—42no—
234#126—42no—
235#126—42no—
236#126—42no—
237#124—42no—
238#124—42no—
239#124—44no—
240#124—42no—
241#124—42no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C4×Q8—41no—
245C4×Q8—41no—
246C4×Q8—42no—
247C4×Q8—44no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×Q8D814yes—
251C4×Q8SD1614yes—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—44no—
257#117—42no—
258#117—42no—
259#117—42no—
260#117—42no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22⋊C411yescharacteristic
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
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⋊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×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2411yescharacteristic
333C4⋊C8—42no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341#97—42no—
342#97—42no—
343C4×C8—42no—
344C4×C8—42no—
345C4×C8—42no—
346C4×C8—42no—
347#95—21no—
348#95—21no—
349#95—21no—
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#95C22⋊C411yescharacteristic
360#95C22⋊C411yescharacteristic
361#95C22⋊C411yescharacteristic
362#95C22⋊C411yescharacteristic
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365C24—21no—
366C24—21no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8#10214yes—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—81no—
389C2×D4—81no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C23×C411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4#10214yes—
469C4⋊C4#10214yes—
470C4⋊C4#10214yes—
471C4⋊C4#13112yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42C2×D812yes—
525C42C2×D812yes—
526C42C2×D812yes—
527C42C2×SD1612yes—
528C42C2×SD1612yes—
529C42C2×SD1612yes—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C22×C22⋊C411yescharacteristic
545Q8—44no—
546Q8—84no—
547Q8—84no—
548D4—81no—
549D4—81no—
550D4—81no—
551D4—81no—
552D4—84no—
553D4—162no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
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—22no—
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—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4#41312yes—
610C2×C4C2×#10212yes—
611C2×C4C2×#10212yes—
612C2×C4C2×#10212yes—
613C2×C4C2×#10212yes—
614C2×C4C2×#10212yes—
615C2×C4C2×#10212yes—
616C2×C4C2×#13111yescharacteristic
617C2×C4#56811yescharacteristic
618C2×C4C22×D811yescharacteristic
619C2×C4C22×SD1611yescharacteristic
620C2×C4#57511yescharacteristic
621C2×C4#57511yescharacteristic
622C8—42no—
623C8—42no—
624C8—81no—
625C8—81no—
626C8—81no—
627C8—81no—
628C8—81no—
629C8—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22#288011yescharacteristic
645C22#288411yescharacteristic
646C22C2×#41311yescharacteristic
647C22C22×#10211yescharacteristic
648C22#289611yescharacteristic
649C22#290111yescharacteristic
650C22#290111yescharacteristic
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—41no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—81no—
675C2—161no—
676C2—161no—
677C2#1742811yescharacteristic
678C2#1750711yescharacteristic
679C2#1751611yescharacteristic
680C2#1761811yescharacteristic
681C2#1762411yescharacteristic
682C2#1768611yescharacteristic
683C2#1770411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22828C2semidirect
3. #22736C2semidirect
4. #7732C2semidirect
6. #7686C2semidirect
8. #7672C2semidirect
9. #7666C2semidirect
10. #7642C2semidirect
11. #7625C2semidirect
12. #7614C2semidirect
13. #7588C2semidirect
14. #7574C2semidirect
15. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#362712
#228282
#227362
#77322
#77192
#76862
#76792
#76722
#76662
#76422
#76252
#76142
#75882
#75742
#75602