Finite Group Databasebeta

← #164667

Subgroups — #164667

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36372C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20781C211yesmaximal, characteristic, complemented
4#7731C211yesmaximal, characteristic, complemented
5#7717C211yesmaximal, characteristic, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7671C211yesmaximal, characteristic, complemented
8#7660C211yesmaximal, characteristic, complemented
9#7657C211yesmaximal, characteristic, complemented
10#7647C211yesmaximal, characteristic
11#7642C211yesmaximal, characteristic, complemented
12#7614C211yesmaximal, characteristic, complemented
13#7582C211yesmaximal, characteristic
14#7580C211yesmaximal, characteristic, complemented
15#7560C211yesmaximal, characteristic, complemented
16#2864C2211yescharacteristic
17#2859C2211yescharacteristic
18#2851C2211yescharacteristic
19#2609—21no—
20#2573—21no—
21#2570C411yescharacteristic
22#2566C411yescharacteristic
23#2530—21no—
24#2497C411yescharacteristic
25#2495C411yescharacteristic
26#2475—21no—
27#2470—21no—
28#2460—21no—
29#2422C411yescharacteristic
30#2420C411yescharacteristic
31#2370C411yescharacteristic
32#2359—21no—
33#2349C411yescharacteristic
34#2343—21no—
35C4×#121C2211yescharacteristic
36#2293C2211yescharacteristic
37#2293C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1939—21no—
40#1923—21no—
41#1923—21no—
42#1839C2211yescharacteristic
43#1839C2211yescharacteristic
44#1832—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1798—21no—
50#1797C2211yescharacteristic
51C8×C4⋊C4—21no—
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1750—21no—
55C4×C22⋊C8C2211yescharacteristic
56#1746—21no—
57#1744C2211yescharacteristic
58#1744C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C2×#125—21no—
76C2×#125C2×C411yescharacteristic
77C2×#123—21no—
78C2×#121C2×C411yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102#400—21no—
103#400—21no—
104#400—21no—
105#398—21no—
106#397—21no—
107#397—21no—
108#397—21no—
109#396—21no—
110#396—21no—
111#396—21no—
112#394—21no—
113#394—21no—
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2311yescharacteristic
117#389—21no—
118#389—21no—
119#389—21no—
120#389C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2311yescharacteristic
127#389C2311yescharacteristic
128#389C2311yescharacteristic
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#386—21no—
148#386—21no—
149#385—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#384C2×C411yescharacteristic
157#384C2×C411yescharacteristic
158#383—21no—
159#382—21no—
160#382C2×C411yescharacteristic
161#382C2×C411yescharacteristic
162#382C2×C411yescharacteristic
163#381—21no—
164#380—21no—
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168C4×C4⋊C4—21no—
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4C2×C411yescharacteristic
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183#376D411yescharacteristic
184#376D411yescharacteristic
185C2×#95—21no—
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×D4—21no—
207C22×D4C22×C411yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222#125—42no—
223#125—81no—
224#125—81no—
225#123—44no—
226#121—22no—
227#121—41no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—42no—
233C4×D4—81no—
234C4×D4—81no—
235#117—22no—
236#117—22no—
237#117—42no—
238#117—42no—
239#117—42no—
240#117—81no—
241#117—81no—
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⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—42no—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
305C4⋊C8—42no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308C4⋊C8—81no—
309C4⋊C8—81no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—81no—
314#97—42no—
315C4×C8—42no—
316C4×C8—42no—
317C4×C8—42no—
318C4×C8—42no—
319#95—21no—
320#95—21no—
321#95—21no—
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#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348C24—21no—
349C24C2×C22⋊C411yescharacteristic
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—81no—
356C2×D4#13112yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
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×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C2×C8—22no—
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—41no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—41no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—42no—
495C42—81no—
496C42—81no—
497C42#13512yes—
498C42#13512yes—
499C23—22no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C22×C22⋊C411yescharacteristic
514D4—42no—
515D4—42no—
516D4—84no—
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—21no—
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—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—42no—
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—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—162no—
582C2×C4#41012yes—
583C2×C4#41612yes—
584C2×C4#41612yes—
585C2×C4#42112yes—
586C2×C4C2×#13111yescharacteristic
587C2×C4#56811yescharacteristic
588C2×C4C2×#13511yescharacteristic
589C2×C4C2×#13511yescharacteristic
590C2×C4#57611yescharacteristic
591C2×C4#57711yescharacteristic
592C8—42no—
593C8—81no—
594C8—81no—
595C8—81no—
596C8—81no—
597C8—81no—
598C8—81no—
599C8—81no—
600C22—41no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22#288011yescharacteristic
615C22C2×#41011yescharacteristic
616C22#288911yescharacteristic
617C22C2×#41611yescharacteristic
618C22#289711yescharacteristic
619C22C2×#42111yescharacteristic
620C22#290111yescharacteristic
621C4—21no—
622C4—21no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—21no—
627C4—21no—
628C4—22no—
629C4—21no—
630C4—21no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C2—81no—
646C2—82no—
647C2#1741311yescharacteristic
648C2#1750811yescharacteristic
649C2#1752211yescharacteristic
650C2#1757611yescharacteristic
651C2#1761011yescharacteristic
652C2#1768011yescharacteristic
653C2#1772711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×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
2. #22632C2semidirect
3. #20781C2semidirect
4. #7731C2semidirect
5. #7717C2semidirect
6. #7683C2semidirect
7. #7671C2semidirect
8. #7660C2semidirect
9. #7657C2semidirect
11. #7642C2semidirect
12. #7614C2semidirect
14. #7580C2semidirect
15. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#363722
#226322
#207812
#77312
#77172
#76832
#76712
#76602
#76572
#76472
#76422
#76142
#75822
#75802
#75602