Finite Group Databasebeta

← #164158

Subgroups — #164158

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36306C211yesmaximal, characteristic
2#22589C211yesmaximal, characteristic, complemented
3#20368C211yesmaximal, characteristic, complemented
4#7693C211yesmaximal, characteristic, complemented
5#7678C211yesmaximal, characteristic
6#7676C211yesmaximal, characteristic, complemented
7#7669C211yesmaximal, characteristic, complemented
8#7656C211yesmaximal, characteristic, complemented
9#7654C211yesmaximal, characteristic
10#7618C211yesmaximal, characteristic, complemented
11#7611C211yesmaximal, characteristic, complemented
12#7609C211yesmaximal, characteristic, complemented
13#7567C211yesmaximal, characteristic, complemented
14#7562C211yesmaximal, characteristic, complemented
15C2×#1272C211yesmaximal, characteristic, complemented
16C2×#405—21no—
17#2863C2211yescharacteristic
18#2861C2211yescharacteristic
19#2854C2211yescharacteristic
20#2589—21no—
21#2532—21no—
22#2532—21no—
23#2527—21no—
24#2514C411yescharacteristic
25#2510—21no—
26#2498—21no—
27#2453C411yescharacteristic
28#2448C411yescharacteristic
29#2447C411yescharacteristic
30#2442C411yescharacteristic
31#2436C411yescharacteristic
32#2361C411yescharacteristic
33#2359—21no—
34#2353—21no—
35Q8×C22⋊C4C411yescharacteristic
36C4×#124C2211yescharacteristic
37#2295C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1916—21no—
41#1916—21no—
42#1850—21no—
43#1839C2211yescharacteristic
44#1831C2211yescharacteristic
45#1817C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C8×C22⋊C4—21no—
48#1746—21no—
49#1746—21no—
50#1745C2211yescharacteristic
51#1745C2211yescharacteristic
52#1744C2211yescharacteristic
53#1744C2211yescharacteristic
54#1744C2211yescharacteristic
55#1729—21no—
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×Q8C2×C411yescharacteristic
76C22×M16—21no—
77C23×C8—21no—
78C23×C8—21no—
79C2×#125C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×#123—21no—
83C2×#121C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4C2×C411yescharacteristic
88C2×#117D411yescharacteristic
89C2×#117D411yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C42C2311yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C22⋊C8—21no—
95#405—28no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102#401—21no—
103#400—21no—
104#400—21no—
105#400—21no—
106#399—21no—
107#398—21no—
108#398—21no—
109#397—21no—
110#397—21no—
111#397—21no—
112#393—21no—
113#393—21no—
114#391—21no—
115#391—21no—
116#391—21no—
117#391—21no—
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2311yescharacteristic
125#391C2311yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2311yescharacteristic
129#389—21no—
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2311yescharacteristic
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
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#386—21no—
154#386—21no—
155#386—21no—
156#386—21no—
157#385—21no—
158#385—21no—
159#385—21no—
160#383—21no—
161#383—21no—
162#382C2×C411yescharacteristic
163#381—21no—
164#380—21no—
165#380—21no—
166#380C2×C411yescharacteristic
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4C2×C411yescharacteristic
177C4×C22⋊C4C2×C411yescharacteristic
178C4×C22⋊C4D411yescharacteristic
179C4×C22⋊C4D411yescharacteristic
180#376—21no—
181#376D411yescharacteristic
182#376D411yescharacteristic
183C2×#95—21no—
184C2×#95—21no—
185C43C2311yescharacteristic
186C43C2311yescharacteristic
187#336—21no—
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#336—21no—
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—41no—
210C22×Q8—42no—
211C22×Q8—81no—
212C22×Q8C22×C411yescharacteristic
213C22×Q8C22×C411yescharacteristic
214C22×D4—21no—
215C22×D4C22×C411yescharacteristic
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C2×M16—24no—
223C2×M16—42no—
224C2×M16—42no—
225C22×C8—24no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—42no—
238C22×C8—81no—
239#125—81no—
240#124—22no—
241#124—41no—
242#124—81no—
243#123—42no—
244#123—42no—
245#121—42no—
246C4×Q8—22no—
247C4×Q8—41no—
248C4×Q8—81no—
249C4×D4—44no—
250C4×D4—81no—
251#117—22no—
252#117—22no—
253#117—24no—
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—81no—
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
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×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⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—42no—
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2411yescharacteristic
311C4⋊C8—42no—
312C4⋊C8—81no—
313C22⋊C8—42no—
314#97—24no—
315#97—24no—
316C4×C8—24no—
317C4×C8—24no—
318C4×C8—42no—
319C4×C8—81no—
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
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353C24—21no—
354C24C2×C22⋊C411yescharacteristic
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—81no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8#13112yes—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4C2×C22⋊C411yescharacteristic
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×C4#13512yes—
413C22×C4#13512yes—
414C22×C4C23×C411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417M16—48no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—48no—
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—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—84no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4#10414yes—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—42no—
507C42—42no—
508C42—81no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23#41012yes—
521C23#41612yes—
522C23C2×#10412yes—
523C23C22×C22⋊C411yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—81no—
527Q8—81no—
528Q8—84no—
529D4—81no—
530D4—81no—
531D4—84no—
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—24no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
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—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—162no—
597C2×C4—162no—
598C2×C4#41612yes—
599C2×C4C2×#10412yes—
600C2×C4C2×#10412yes—
601C2×C4C2×#13111yescharacteristic
602C2×C4#56811yescharacteristic
603C2×C4C2×#13511yescharacteristic
604C2×C4C2×#13511yescharacteristic
605C2×C4#57711yescharacteristic
606C2×C4#57811yescharacteristic
607C8—42no—
608C8—42no—
609C8—42no—
610C8—42no—
611C8—42no—
612C8—81no—
613C8—81no—
614C8—81no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#150014yes—
629C22#288011yescharacteristic
630C22C2×#41011yescharacteristic
631C22#288811yescharacteristic
632C22C2×#41611yescharacteristic
633C22#289811yescharacteristic
634C22C22×#10411yescharacteristic
635C22#290111yescharacteristic
636C4—21no—
637C4—21no—
638C4—22no—
639C4—21no—
640C4—21no—
641C4—22no—
642C4—21no—
643C4—21no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—81no—
660C2—24no—
661C2—161no—
662C2#1741211yescharacteristic
663C2#1751011yescharacteristic
664C2#1751911yescharacteristic
665C2C2×#150011yescharacteristic
666C2#1760911yescharacteristic
667C2#1765211yescharacteristic
668C2#1772611yescharacteristic
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. #22589C2semidirect
3. #20368C2semidirect
4. #7693C2semidirect
6. #7676C2semidirect
7. #7669C2semidirect
8. #7656C2semidirect
10. #7618C2semidirect
11. #7611C2semidirect
12. #7609C2semidirect
13. #7567C2semidirect
14. #7562C2semidirect
15. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
#363062
#225892
#203682
#76932
#76782
#76762
#76692
#76562
#76542
#76182
#76112
#76092
#75672
#75622
C2×#12722