Finite Group Databasebeta

← #214132

Subgroups — #214132

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic, complemented
3#21373C211yesmaximal, characteristic, complemented
4#9074C212yesmaximal, complemented
5#9072C212yesmaximal, complemented
6#8940C212yesmaximal, complemented
7#8939C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8310C212yesmaximal, complemented
10C2×#535C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic, complemented
14C2×#431C2211yescharacteristic, complemented
15C22×C4⋊C8C2211yescharacteristic, complemented
16C22×C4×C8C2211yescharacteristic, complemented
17#2328—21no—
18#2326—21no—
19#2324C412yes—
20#2324C412yes—
21#2321—21no—
22#2321—21no—
23#2317C411yescharacteristic
24#2311C411yescharacteristic
25#2309C411yescharacteristic
26C4×#127—21no—
27C4×#124—21no—
28C4×#121C2212yes—
29C4×#121C2212yes—
30C4×#120—21no—
31C4×#120—21no—
32C4×#117C2211yescharacteristic
33#1984—21no—
34#1984—21no—
35#1954—21no—
36#1954—21no—
37#1869—21no—
38#1869—21no—
39#1869—21no—
40#1869—21no—
41#1797C2212yes—
42#1797C2212yescomplemented
43C4×C22⋊C8C2212yes—
44C4×C22⋊C8C2212yescomplemented
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yescomplemented
50#1279C2212yescomplemented
51#1279C2212yescomplemented
52#1279C2212yescomplemented
53C22×#41C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C24×C4—21no—
56C23×C8—21no—
57C23×C8—21no—
58#535—24no—
59#535C818yescomplemented
60C2×#127—21no—
61C2×#124—21no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#120—21no—
65C2×#120—21no—
66C4×#41—41no—
67C4×#41—41no—
68C4×#41—41no—
69C4×#41—41no—
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—41no—
77C2×C4×D4—41no—
78C2×C4×D4—41no—
79C2×C4×D4—41no—
80C2×C4×D4C2×C412yes—
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×C4×D4C2312yes—
88C2×#117C2×C411yescharacteristic
89C2×#117C2×C411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C22⋊C4—21no—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#431—41no—
99#431—41no—
100#431—41no—
101#431—41no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8C2312yes—
119C2×C4×C8C2312yes—
120#391—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—22no—
125#390—22no—
126#390C2×C412yes—
127#390C2×C412yes—
128#389C2×C412yes—
129#389C2×C412yes—
130#388—22no—
131#388—22no—
132#388—22no—
133#388—22no—
134#386—21no—
135#386—21no—
136#386—22no—
137#386—22no—
138#386—22no—
139#386—22no—
140#385—22no—
141#385—22no—
142#385—21no—
143#385—21no—
144#384—21no—
145#382—21no—
146#380C2×C412yes—
147#380C2×C412yes—
148#379—21no—
149#379—21no—
150C4×C4⋊C4C2×C412yes—
151C4×C4⋊C4C2312yes—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156#376—21no—
157C43—21no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162D4⋊C8—42no—
163D4⋊C8—42no—
164D4⋊C8—42no—
165D4⋊C8—42no—
166C25—21no—
167C2×#41—41no—
168C2×#41—41no—
169C2×#41—41no—
170C2×#41—41no—
171C2×#41C2×C814yes—
172C2×#41M1614yes—
173C22×Q8C22⋊C411yescharacteristic
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—22no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C2×C814yescomplemented
186C22×D4C22×C412yes—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—82no—
193C23×C4—82no—
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—44no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—82no—
211#127—28no—
212#124—28no—
213#121—24no—
214#121—24no—
215#121C2×C818yescomplemented
216#121C2×C818yescomplemented
217#120—28no—
218#120—28no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—82no—
246C4×D4—82no—
247C4×D4—82no—
248C4×D4—82no—
249#117—22no—
250#117—41no—
251#117—41no—
252#117—41no—
253#117—41no—
254#117—41no—
255#117—41no—
256#117—41no—
257#117—41no—
258#117C2×C814yescomplemented
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C412yes—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2411yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—42no—
324C4⋊C8—42no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329C22⋊C8—82no—
330C22⋊C8—82no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—22no—
334C4×C8—22no—
335C4×C8—42no—
336C4×C8—42no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343C24—22no—
344C24—24no—
345C24—42no—
346C24—42no—
347C24—82no—
348C24—82no—
349C24C2×C22⋊C411yescharacteristic
350C24C2×C22⋊C411yescharacteristic
351C24C2×C22⋊C411yescharacteristic
352#41—44no—
353#41—44no—
354#41—44no—
355#41—44no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8C22⋊C814yes—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4C22⋊C814yes—
384C2×D4C22⋊C814yes—
385C2×D4C22⋊C814yes—
386C2×D4C22×C814yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—88no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C22×C812yes—
447C22×C4C22×C812yes—
448C22×C4C22×C812yes—
449C22×C4C2×M1612yes—
450C22×C4C2×M1612yes—
451C22×C4C2×M1612yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C2×C8—24no—
456C2×C8—42no—
457C2×C8—48no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—84no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C22×C814yes—
484C22⋊C4—24no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—82no—
526C42H(5,𝔽2)11yescharacteristic
527C42H(5,𝔽2)11yescharacteristic
528C42H(5,𝔽2)11yescharacteristic
529C42H(5,𝔽2)11yescharacteristic
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—48no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—88no—
551C23—84no—
552C23—84no—
553C23—88no—
554C23C2×C22⋊C812yes—
555C23C2×C22⋊C812yes—
556C23C2×C22⋊C812yes—
557C23C22×C22⋊C411yescharacteristic
558Q8—44no—
559D4—24no—
560D4—24no—
561D4—44no—
562D4—44no—
563D4—44no—
564D4—48no—
565D4—42no—
566D4—42no—
567D4—88no—
568D4—88no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4C2×C22⋊C812yes—
631C2×C4C2×C22⋊C812yes—
632C2×C4C2×C22⋊C812yes—
633C2×C4#51811yescharacteristic
634C2×C4#51811yescharacteristic
635C2×C4#53411yescharacteristic
636C2×C4#53411yescharacteristic
637C2×C4#53411yescharacteristic
638C2×C4#53411yescharacteristic
639C2×C4C23×C811yescharacteristic
640C2×C4C22×M1611yescharacteristic
641C2×C4#56811yescharacteristic
642C2×C4#56811yescharacteristic
643C2×C4#57612yes—
644C2×C4#57712yes—
645C8—44no—
646C8—82no—
647C8—82no—
648C8—82no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#230811yescharacteristic
673C22C22×C22⋊C811yescharacteristic
674C22#287911yescharacteristic
675C22#289712yes—
676C22#290112yes—
677C4—21no—
678C4—21no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4#297211yescharacteristic
692C4#297211yescharacteristic
693C4#297311yescharacteristic
694C4#297311yescharacteristic
695C4#319812yes—
696C4#319911yescharacteristic
697C4#320011yescharacteristic
698C2—42no—
699C2—42no—
700C2—42no—
701C2—84no—
702C2#1743812yes—
703C2#1750512yes—
704C2#2173811yescharacteristic
705C2#2177011yescharacteristic
706C2#2177511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#535C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21847C2semidirect
3. #21373C2semidirect
4. #9074C2semidirect
5. #9072C2semidirect
6. #8940C2semidirect
7. #8939C2semidirect
8. #8708C2semidirect
9. #8310C2semidirect
13. C2×#431C22semidirect
14. C2×#431C22semidirect
15. C22×C4⋊C8C22semidirect
16. C22×C4×C8C22semidirect
42. #1797C22semidirect
44. C4×C22⋊C8C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
59. #535C8semidirect
185. C22×D4C2×C8semidirect
215. #121C2×C8semidirect
216. #121C2×C8semidirect
258. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#218472
#213732
#90742
#90722
#89402
#89392
#87082
#83102