Finite Group Databasebeta

← #360631

Subgroups — #360631

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2C2×#1826C211yesmaximal, characteristic
3C2×#1810C211yesmaximal, characteristic
4#20628C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6#11329C212yesmaximal, complemented
7#11291C212yesmaximal, complemented
8#10910C212yesmaximal, complemented
9#10887C212yesmaximal, complemented
10#10733C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15#2924—24no—
16#2924—24no—
17C2×#433—22no—
18C2×#432—21no—
19C2×#431—21no—
20C22×#108C2212yes—
21C2×#424C2212yes—
22C2×#424C2212yes—
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C2×#405C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1826—24no—
29#1810—24no—
30#1798—22no—
31#1797—21no—
32#1797—21no—
33C2×#341C2212yes—
34#1735C411yescharacteristic, complemented
35#1735C411yescharacteristic, complemented
36#1735C411yescharacteristic, complemented
37#1735C411yescharacteristic, complemented
38#1735C2211yescharacteristic
39#1735C2211yescharacteristic
40#1735C2211yescharacteristic
41#1735C2211yescharacteristic
42C2×#340C411yescharacteristic, complemented
43C2×#340C411yescharacteristic, complemented
44C2×#340C411yescharacteristic, complemented
45C2×#340C411yescharacteristic, complemented
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50#1729C2212yes—
51C2×#336C2212yes—
52C2×#336C2212yes—
53C24×C4C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2312yes—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—24no—
59C23×C8—42no—
60C23×C8C2×C411yescharacteristic
61C23×C8C2×C411yescharacteristic
62C23×C8C2×C411yescharacteristic
63C23×C8C2×C411yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C2×#117—21no—
68C2×#117—22no—
69C2×#117—21no—
70C2×#117—22no—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—24no—
74C2×#117—42no—
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117D414yes—
80C2×#117Q814yes—
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2312yes—
95#433—216no—
96#432—28no—
97#431—28no—
98C2×#108—24no—
99C2×#108—24no—
100C2×#108—42no—
101#424—24no—
102#424—24no—
103#424—24no—
104#424—42no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109#407—22no—
110#407—22no—
111#407—22no—
112#407D414yes—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D412yes—
122#405—24no—
123#405—24no—
124C2×#97—22no—
125C2×C4×C8—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129#341—24no—
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#339—24no—
147#339—24no—
148#339—24no—
149#339—42no—
150#336—22no—
151#336—22no—
152#336—24no—
153#336—42no—
154C25#4111yescharacteristic
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q812yes—
171C23×C4#4111yescharacteristic
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16C2×D414yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—41no—
210C22×C8—41no—
211C22×C8—41no—
212C22×C8—41no—
213C22×C8—48no—
214C22×C8—48no—
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8#4114yes—
222#117—22no—
223#117—22no—
224#117—24no—
225#117—24no—
226#117—24no—
227#117—24no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—44no—
238#117—44no—
239#117—41no—
240#117—41no—
241#117C4⋊C414yes—
242#117C4⋊C414yes—
243#117C2×D412yes—
244#117C2×D412yes—
245#117C2×Q812yes—
246#117C2×Q812yes—
247#117#4114yes—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4C4⋊C414yes—
259C2×C4⋊C4C4⋊C414yes—
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—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—44no—
299#108—48no—
300C4⋊C8—28no—
301C4⋊C8—28no—
302C4⋊C8—44no—
303C4⋊C8—44no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311#97—28no—
312C4×C8—28no—
313#95—42no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24C4×D411yescharacteristic
321C24C4×D411yescharacteristic
322C24C22⋊Q812yes—
323C24#12312yes—
324C24C2×#4111yescharacteristic
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4C2×C4⋊C412yes—
375C22×C4C2×C4⋊C412yes—
376C22×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4#12012yes—
381C22×C4#12012yes—
382C22×C4C22⋊Q812yes—
383C22×C4C22⋊Q812yes—
384C22×C4C22⋊Q812yes—
385C22×C4#12312yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×Q811yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C42−1+412yes—
396M16—24no—
397M16—24no—
398M16—42no—
399M16—42no—
400M16—42no—
401M16—42no—
402M16—42no—
403M16—42no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8#11712yes—
439C2×C8#11712yes—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4C2×C4⋊C412yes—
455C4⋊C4C2×C4⋊C412yes—
456C4⋊C4C2×C4⋊C412yes—
457C4⋊C4C2×C4⋊C412yes—
458C4⋊C4#11712yes—
459C4⋊C4#11712yes—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—48no—
470C22⋊C4—42no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—24no—
477C42—24no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—24no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23#38512yes—
510C23#38512yes—
511C23#44312yes—
512C23#44412yes—
513C23C2×C4×D411yescharacteristic
514C23C2×C22⋊Q811yescharacteristic
515C23C2×#12311yescharacteristic
516C23#55112yes—
517C23#55212yes—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4#38512yes—
572C2×C4#38512yes—
573C2×C4C22×C4⋊C411yescharacteristic
574C2×C4C2×#11711yescharacteristic
575C2×C4C2×#11711yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4#51811yescharacteristic
578C2×C4#51911yescharacteristic
579C2×C4C2×#12011yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4#53212yes—
582C2×C4#53312yes—
583C2×C4#53712yes—
584C2×C4#54412yes—
585C2×C4#54612yes—
586C2×C4#54612yes—
587C2×C4#54712yes—
588C2×C4#54712yes—
589C2×C4#55112yes—
590C2×C4#55212yes—
591C8—22no—
592C8—22no—
593C8—41no—
594C8—41no—
595C8—42no—
596C8—42no—
597C8—44no—
598C8—82no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—48no—
617C22—48no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22#186614yes—
623C22C2×#38511yescharacteristic
624C22#234211yescharacteristic
625C22#234411yescharacteristic
626C22#243812yes—
627C22#249812yes—
628C22C2×#44311yescharacteristic
629C22C2×#44411yescharacteristic
630C22#295711yescharacteristic
631C22#295711yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—84no—
641C4—81no—
642C4—81no—
643C4#228511yescharacteristic
644C4#231611yescharacteristic
645C4#234211yescharacteristic
646C4#234411yescharacteristic
647C4#243012yes—
648C4#243812yes—
649C2—22no—
650C2—24no—
651C2—22no—
652C2—42no—
653C2#1387811yescharacteristic
654C2C2×#186611yescharacteristic
655C2#2088611yescharacteristic
656C2#2258611yescharacteristic
657C2#2258711yescharacteristic
658C2#2260611yescharacteristic
659C2#2260711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11329C2semidirect
7. #11291C2semidirect
8. #10910C2semidirect
9. #10887C2semidirect
34. #1735C4semidirect
35. #1735C4semidirect
36. #1735C4semidirect
37. #1735C4semidirect
42. C2×#340C4semidirect
43. C2×#340C4semidirect
44. C2×#340C4semidirect
45. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C2×#18262
C2×#18102
#206282
C2×#17352
#113292
#112912
#109102
#108872
#107332
#107142