Finite Group Databasebeta

← #363237

Subgroups — #363237

Order 512.

2994 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#555C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic, complemented
8#10969C214yesmaximal, complemented
9#10731C214yesmaximal, complemented
10C2×#555—24no—
11C2×#555—24no—
12C2×#555—24no—
13C2×#555—24no—
14C2×#555—24no—
15C2×#555C418yescomplemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124—22no—
18C22×#121—21no—
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117—21no—
23C22×#117C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×C22⋊C8C2214yes—
28C22×C4×C8C2211yescharacteristic
29C2×#390—21no—
30C2×#384C2211yescharacteristic
31C2×#380C2214yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990—24no—
34#1987—24no—
35#1954—24no—
36#1802—24no—
37C4×#102—24no—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1439—22no—
43C4⋊#95—21no—
44C23×Q8—21no—
45C23×D4—22no—
46C23×D4D411yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52#555—232no—
53#555—232no—
54#555—232no—
55#555—232no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D414yes—
62C2×#124—216no—
63C2×#124—216no—
64C2×#124—216no—
65C2×#121—28no—
66C2×#121—28no—
67C2×#121—28no—
68C2×#121—24no—
69C2×#121—24no—
70C2×#121—24no—
71C2×#121—24no—
72C2×#121—24no—
73C2×#121C2×C418yes—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4C2×C414yes—
82C2×C4×D4C2×C414yes—
83C2×C4×D4D414yes—
84C2×C4×D4D414yes—
85C2×C4×D4D414yes—
86C2×#117—28no—
87C2×#117—28no—
88C2×#117—24no—
89C2×#117—28no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117C2×C418yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4D412yes—
108C22×C42—22no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102—24no—
118C2×#102—44no—
119C2×C22⋊C8—28no—
120C2×C22⋊C8—28no—
121C2×C22⋊C8—44no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125#390—44no—
126#384—22no—
127#384—22no—
128#380—28no—
129#380—44no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C2×#95—24no—
133C2×#95—21no—
134#340—24no—
135#340—28no—
136#340—28no—
137#340—24no—
138C25—21no—
139C25C2×D412yes—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—82no—
144C22×Q8—82no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—84no—
162C22×D4—84no—
163C22×D4C22⋊C414yes—
164C22×D4C22×C412yes—
165C22×D4C2×D412yes—
166C22×D4C2×D412yes—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C22×C414yes—
187C23×C4C2×D412yes—
188C23×C4C2×D414yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—24no—
202C22×C8—24no—
203C22×C8#4114yes—
204C4⋊Q8—28no—
205C4⋊Q8—28no—
206C4⋊Q8—28no—
207C4⋊Q8C22⋊C4116yescomplemented
208#124—2128no—
209#121—216no—
210#121—216no—
211#121—264no—
212#121—216no—
213#121—216no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4C22⋊C4116yescomplemented
222#117—216no—
223#117—216no—
224#117—264no—
225#117—216no—
226#117—216no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C22⋊C414yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—84no—
274C2×C22⋊C4C22⋊C418yes—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C22×C412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C4⋊C8—44no—
288#102—216no—
289#102—48no—
290C22⋊C8—416no—
291C4×C8—44no—
292#95—48no—
293#95—42no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—24no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—42no—
308C24—42no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—42no—
314C24—42no—
315C24C2×C22⋊C414yes—
316C24C22×D411yescharacteristic
317C2×Q8—216no—
318C2×Q8—88no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—88no—
322C2×D4—232no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—816no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—816no—
351C2×D4C2×C22⋊C414yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—24no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4C2×C22⋊C414yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4#12012yes—
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4#12012yes—
414C22×C4#13512yes—
415C22×C4#13512yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4#13712yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+412yes—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—44no—
439C2×C8—44no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4C2×C22⋊C414yes—
458C4⋊C4C2×C22⋊C418yes—
459C22⋊C4—232no—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—816no—
464C42—232no—
465C42—84no—
466C42C2×C22⋊C414yes—
467C42C2×C22⋊C414yes—
468C42#12014yes—
469C42#12014yes—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23C22×C22⋊C411yescharacteristic
517C23C2×#12011yescharacteristic
518C23C2×#12011yescharacteristic
519C23#54612yes—
520C23#54712yes—
521C23C2×#13511yescharacteristic
522C23C2×#13511yescharacteristic
523C23C2×AGL(1,ℤ/8)11yescharacteristic
524C23C2×#13711yescharacteristic
525C23#57612yes—
526C23#57712yes—
527Q8—816no—
528D4—24no—
529D4—24no—
530D4—44no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—416no—
535D4—832no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—432no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4#37914yes—
577C2×C4#37914yes—
578C2×C4#41614yes—
579C2×C4#41714yes—
580C2×C4#44914yes—
581C2×C4#44914yes—
582C2×C4#45214yes—
583C2×C4#45214yes—
584C2×C4C22×C22⋊C411yescharacteristic
585C2×C4C22×C22⋊C411yescharacteristic
586C2×C4C4×#4112yes—
587C2×C4#52012yes—
588C2×C4C2×#12011yescharacteristic
589C2×C4C2×#12011yescharacteristic
590C2×C4#52512yes—
591C2×C4#53211yescharacteristic
592C2×C4#53211yescharacteristic
593C2×C4#53511yescharacteristic
594C2×C4#53611yescharacteristic
595C2×C4#53712yes—
596C2×C4#54612yes—
597C2×C4#54712yes—
598C2×C4#54812yes—
599C2×C4#54812yes—
600C2×C4#55012yes—
601C2×C4#55012yes—
602C8—44no—
603C8—82no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—48no—
611C22—48no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22C2×#37911yescharacteristic
627C22#233912yes—
628C22#240712yes—
629C22#243111yescharacteristic
630C22#243211yescharacteristic
631C22C2×#41611yescharacteristic
632C22C2×#41711yescharacteristic
633C22#289712yes—
634C22C2×#44911yescharacteristic
635C22C2×#44911yescharacteristic
636C22C2×#45211yescharacteristic
637C22C2×#45211yescharacteristic
638C22#300811yescharacteristic
639C22#300911yescharacteristic
640C22#301212yes—
641C22#330312yes—
642C22#330512yes—
643C22#332512yes—
644C22#332912yes—
645C22#334012yes—
646C22#334112yes—
647C22#335412yes—
648C22#335712yes—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—48no—
653C4—84no—
654C4—162no—
655C4#178818yes—
656C4#229711yescharacteristic
657C4#231111yescharacteristic
658C4#234112yes—
659C4#240712yes—
660C4#243111yescharacteristic
661C4#243211yescharacteristic
662C2—84no—
663C2—84no—
664C2#1393711yescharacteristic
665C2C2×#178811yescharacteristic
666C2#2060211yescharacteristic
667C2#2248912yes—
668C2#2250312yes—
669C2#2411712yes—
670C2#2412112yes—
671C2#2556811yescharacteristic
672C2#2557211yescharacteristic
673C2#2570011yescharacteristic
674C2#2570411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#555C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1987C2semidirect
4. C2×#1954C2semidirect
5. C2×#1802C2semidirect
6. C2×C4×#102C2semidirect
7. #13906C2semidirect
8. #10969C2semidirect
9. #10731C2semidirect
15. C2×#555C4semidirect
207. C4⋊Q8C22⋊C4semidirect
221. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5552
C2×#19902
C2×#19872
C2×#19542
C2×#18022
C2×C4×#1022
#139062
#109692
#107312