Finite Group Databasebeta

← #164762

Subgroups — #164762

Order 512.

747 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#22589C211yesmaximal, characteristic
3#20813C211yesmaximal, characteristic
4#7733C211yesmaximal, characteristic, complemented
5#7727C211yesmaximal, characteristic
6#7689C211yesmaximal, characteristic, complemented
7#7673C211yesmaximal, characteristic, complemented
8#7669C211yesmaximal, characteristic, complemented
9#7667C211yesmaximal, characteristic
10#7653C211yesmaximal, characteristic, complemented
11#7648C211yesmaximal, characteristic
12#7624C211yesmaximal, characteristic, complemented
13#7590C211yesmaximal, characteristic, complemented
14#7589C211yesmaximal, characteristic
15#7571C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2863C2211yescharacteristic
18#2863C2211yescharacteristic
19#2857C2211yescharacteristic
20#2606—21no—
21#2606—21no—
22#2552C411yescharacteristic
23#2546C411yescharacteristic
24#2522—21no—
25#2522—21no—
26#2507—21no—
27#2441—21no—
28#2429C411yescharacteristic
29#2429C411yescharacteristic
30#2423C411yescharacteristic
31#2423C411yescharacteristic
32#2370—21no—
33#2365C411yescharacteristic
34Q8×C4⋊C4C411yescharacteristic
35#2347—21no—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C22⋊Q8C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1916—21no—
41#1916—21no—
42#1850—21no—
43#1843—21no—
44#1839C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1798—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1768—21no—
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56C8×C22⋊C4—21no—
57#1744C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1272C2211yescharacteristic
68#1272C2211yescharacteristic
69#1272C2211yescharacteristic
70#1272C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C23×C8—21no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#125—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×#117D411yescharacteristic
89C2×#117D411yescharacteristic
90C22×C42C2311yescharacteristic
91#433—44no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#399—21no—
108#399—21no—
109#398—21no—
110#398—21no—
111#398—21no—
112#398—21no—
113#398—21no—
114#398—21no—
115#398—21no—
116#398—21no—
117#396—21no—
118#396—21no—
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2311yescharacteristic
124#391C2311yescharacteristic
125#391C2311yescharacteristic
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2311yescharacteristic
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#385—21no—
161#384—21no—
162#384C2×C411yescharacteristic
163#384C2×C411yescharacteristic
164#384C2×C411yescharacteristic
165#383—21no—
166#383—21no—
167#380—21no—
168#380—21no—
169#380C2×C411yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187#376D411yescharacteristic
188#376D411yescharacteristic
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#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8C22×C411yescharacteristic
208C22×Q8C22×C411yescharacteristic
209C22×Q8C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—81no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—41no—
227C4⋊Q8—41no—
228C4⋊Q8—42no—
229#125—44no—
230C22⋊Q8—22no—
231C22⋊Q8—41no—
232C22⋊Q8—81no—
233C22⋊Q8—81no—
234C4×Q8—42no—
235C4×Q8—81no—
236C4×Q8—81no—
237C4×Q8—81no—
238C4×Q8—81no—
239#117—42no—
240#117SD1614yes—
241#117Q1614yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—42no—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2411yescharacteristic
307C4⋊C8—42no—
308C4⋊C8—42no—
309C4⋊C8—42no—
310C4⋊C8—42no—
311C4⋊C8—42no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C4⋊C8—81no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318#97—42no—
319C4×C8—42no—
320C4×C8—42no—
321C4×C8—81no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351C24C2×C22⋊C411yescharacteristic
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8#13112yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×SD1612yes—
394C22×C4C2×Q1612yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—84no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4#10314yes—
458C4⋊C4#10314yes—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4#10314yes—
466C22⋊C4#10314yes—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—42no—
487C42—81no—
488C42—81no—
489C42—81no—
490C42—81no—
491C42C2×SD1612yes—
492C42C2×SD1612yes—
493C42C2×Q1612yes—
494C42C2×Q1612yes—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C2×#10312yes—
502C23C22×C22⋊C411yescharacteristic
503Q8—42no—
504Q8—42no—
505Q8—81no—
506Q8—81no—
507Q8—81no—
508Q8—81no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4#41012yes—
572C2×C4C2×#10312yes—
573C2×C4C2×#10312yes—
574C2×C4C2×#10312yes—
575C2×C4C2×#10312yes—
576C2×C4C2×#10312yes—
577C2×C4#42112yes—
578C2×C4C2×#13111yescharacteristic
579C2×C4#56811yescharacteristic
580C2×C4C22×SD1611yescharacteristic
581C2×C4C22×Q1611yescharacteristic
582C2×C4#57711yescharacteristic
583C2×C4#57811yescharacteristic
584C8—42no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C8—81no—
590C8—81no—
591C8—81no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#288011yescharacteristic
600C22C2×#41011yescharacteristic
601C22#288811yescharacteristic
602C22C22×#10311yescharacteristic
603C22#289811yescharacteristic
604C22C2×#42111yescharacteristic
605C22#290111yescharacteristic
606C4—21no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—21no—
614C4—21no—
615C4—22no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C2—42no—
632C2#1741511yescharacteristic
633C2#1750911yescharacteristic
634C2#1752411yescharacteristic
635C2#1757511yescharacteristic
636C2#1760911yescharacteristic
637C2#1768411yescharacteristic
638C2#1772211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7733C2semidirect
6. #7689C2semidirect
7. #7673C2semidirect
8. #7669C2semidirect
10. #7653C2semidirect
12. #7624C2semidirect
13. #7590C2semidirect
15. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#364812
#225892
#208132
#77332
#77272
#76892
#76732
#76692
#76672
#76532
#76482
#76242
#75902
#75892
#75712