Finite Group Databasebeta

← #162519

Subgroups — #162519

Order 512.

758 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35039C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic
3#22014C211yesmaximal, characteristic
4#7728C211yesmaximal, characteristic, complemented
5#7715C211yesmaximal, characteristic, complemented
6#7709C211yesmaximal, characteristic, complemented
7#7697C211yesmaximal, characteristic, complemented
8#7689C211yesmaximal, characteristic, complemented
9#7677C211yesmaximal, characteristic, complemented
10#7668C211yesmaximal, characteristic, complemented
11#7655C211yesmaximal, characteristic, complemented
12#7600C211yesmaximal, characteristic, complemented
13#7598C211yesmaximal, characteristic, complemented
14#7589C211yesmaximal, characteristic, complemented
15#7579C211yesmaximal, characteristic, complemented
16C2×C8×D4—21no—
17#2864C2211yescharacteristic
18#2863C2211yescharacteristic
19#2857C2211yescharacteristic
20#2855C2211yescharacteristic
21#2652—21no—
22#2638—21no—
23#2588—21no—
24#2577—21no—
25#2523—21no—
26#2506C411yescharacteristic
27#2499C411yescharacteristic
28#2495C411yescharacteristic
29#2485C411yescharacteristic
30#2460—21no—
31#2451C411yescharacteristic
32#2439C411yescharacteristic
33#2421C411yescharacteristic
34#2415C411yescharacteristic
35#2336—21no—
36#2332—21no—
37C4×C22⋊Q8C2211yescharacteristic
38#2296C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1955—21no—
41#1940—21no—
42#1939—21no—
43#1924—21no—
44#1924—21no—
45#1850—21no—
46#1850—21no—
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54#1767C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56#1745C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
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
75C23×C8—21no—
76C23×C8—21no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#125—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117C2×C411yescharacteristic
87C2×#117C2×C411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C8×D4—44no—
91C2×C4⋊C8—21no—
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×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107#400—21no—
108#400—21no—
109#400—21no—
110#398—21no—
111#398—21no—
112#397—21no—
113#397—21no—
114#397—21no—
115#397—21no—
116#397—21no—
117#397—21no—
118#396—21no—
119#395—21no—
120#394—21no—
121#394—21no—
122#393—21no—
123#391—21no—
124#391—21no—
125#391—21no—
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#391C2311yescharacteristic
130#389—21no—
131#389—21no—
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2311yescharacteristic
140#389C2311yescharacteristic
141#389C2311yescharacteristic
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#387—21no—
147#387—21no—
148#387—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#387—21no—
161#387—21no—
162#387—21no—
163#385—21no—
164#385—21no—
165#385—21no—
166#385—21no—
167#384C2×C411yescharacteristic
168#383—21no—
169#383—21no—
170#382C2×C411yescharacteristic
171#382C2×C411yescharacteristic
172#381—21no—
173#380—21no—
174#380C2×C411yescharacteristic
175#380C2×C411yescharacteristic
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4C2×C411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C4⋊C4D411yescharacteristic
185C4×C22⋊C4—21no—
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4C2×C411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C4×C22⋊C4D411yescharacteristic
192C43C2311yescharacteristic
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×Q8—21no—
208C22×Q8C22×C411yescharacteristic
209C22×D4C22⋊C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—81no—
229C22×C8—81no—
230C4⋊Q8—81no—
231#125—44no—
232C22⋊Q8—24no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—81no—
236C4×Q8—81no—
237C4×D4—24no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—42no—
241#117—24no—
242#117—81no—
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⋊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×C4⋊C4C22×C411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—81no—
288C2×C42—81no—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2411yescharacteristic
302C4⋊C8—42no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305C4⋊C8—81no—
306C4⋊C8—81no—
307C4⋊C8—81no—
308C4⋊C8—81no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—81no—
312C22⋊C8—81no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315#97—42no—
316#97—81no—
317C4×C8—42no—
318#95—21no—
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#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351C24C2×C22⋊C411yescharacteristic
352C24C2×C22⋊C411yescharacteristic
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—42no—
357C2×Q8—81no—
358C2×Q8—81no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
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×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4#13114yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4#13712yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C2×C8—22no—
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—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—84no—
430C2×C8—84no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—42no—
490C42—42no—
491C42—81no—
492C42—81no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#40812yes—
506C23#41712yes—
507C23C22×C22⋊C411yescharacteristic
508Q8—82no—
509Q8—162no—
510D4—28no—
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—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—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
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—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4#40812yes—
579C2×C4#41712yes—
580C2×C4#42112yes—
581C2×C4C2×#13112yes—
582C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
583C2×C4C2×#13711yescharacteristic
584C2×C4#57711yescharacteristic
585C2×C4#57811yescharacteristic
586C8—42no—
587C8—81no—
588C8—81no—
589C8—81no—
590C8—81no—
591C8—81no—
592C8—81no—
593C8—81no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22C2×#40811yescharacteristic
609C22#288311yescharacteristic
610C22#288911yescharacteristic
611C22C2×#41711yescharacteristic
612C22#289811yescharacteristic
613C22C2×#42111yescharacteristic
614C22#290111yescharacteristic
615C4—22no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C2—42no—
639C2—42no—
640C2#1738711yescharacteristic
641C2#1746611yescharacteristic
642C2#1748411yescharacteristic
643C2#1760011yescharacteristic
644C2#1762211yescharacteristic
645C2#1768311yescharacteristic
646C2#1776311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
4. #7728C2semidirect
5. #7715C2semidirect
6. #7709C2semidirect
7. #7697C2semidirect
8. #7689C2semidirect
9. #7677C2semidirect
10. #7668C2semidirect
11. #7655C2semidirect
12. #7600C2semidirect
13. #7598C2semidirect
14. #7589C2semidirect
15. #7579C2semidirect

Maximal subgroups

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

subgroupindex
#350392
#226962
#220142
#77282
#77152
#77092
#76972
#76892
#76772
#76682
#76552
#76002
#75982
#75892
#75792