Finite Group Databasebeta

← #108455

Subgroups — #108455

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#380C211yesmaximal, characteristic
2#10712C211yesmaximal, characteristic
3C2×#1294C212yesmaximal
4C2×#1294C212yesmaximal
5#7554C211yesmaximal, characteristic
6C23×C42C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2211yescharacteristic
9C2×#380C2211yescharacteristic
10C2×C4×C4⋊C4C411yescharacteristic, complemented
11C2×C4×C4⋊C4C411yescharacteristic, complemented
12C2×C4×C22⋊C4—21no—
13C22×#95—21no—
14C22×#95—21no—
15C22×#95C2212yes—
16C2×#336—21no—
17C2×#336—21no—
18C4⋊#95C411yescharacteristic, complemented
19C4⋊#95C411yescharacteristic, complemented
20#1434—21no—
21C22⋊#95—21no—
22C4×#95—21no—
23#1294—24no—
24#1294C418yescomplemented
25C24×C4—22no—
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4C2311yescharacteristic
29C23×C8—22no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C42—22no—
37C22×C42—22no—
38C22×C42—21no—
39C22×C42—21no—
40C22×C42—42no—
41C22×C42—42no—
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8C2×C414yes—
52#380—22no—
53#380—21no—
54#380—21no—
55#380C2×C414yes—
56#380D412yes—
57#380Q812yes—
58C4×C4⋊C4—42no—
59C4×C4⋊C4—42no—
60C4×C22⋊C4—44no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—24no—
68C2×#95—24no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—42no—
77C2×#95—42no—
78C2×#95—42no—
79C2×#95—42no—
80C2×#95C2×C414yes—
81#336—42no—
82#336—42no—
83C25C2×D411yescharacteristic
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4SD1612yes—
126C23×C4Q1612yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q811yescharacteristic
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4#4112yes—
137C22×C8—24no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C4212yes—
157C2×C4⋊C4C4212yes—
158C2×C4⋊C4C4⋊C412yes—
159C2×C4⋊C4C4⋊C412yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—48no—
179C2×C42—48no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194#95—24no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—42no—
200#95—48no—
201#95—48no—
202#95—42no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—42no—
208#95—42no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24C2×C22⋊C411yescharacteristic
218C24C4×D412yes—
219C24#12111yescharacteristic
220C24C22⋊Q811yescharacteristic
221C24#12312yes—
222C24C2×SD1611yescharacteristic
223C24C2×Q1611yescharacteristic
224C24#13512yes—
225C24AGL(1,ℤ/8)11yescharacteristic
226C24AGL(1,ℤ/8)12yes—
227C24#13711yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4#9912yes—
358C22×C4#9912yes—
359C22×C4#10312yes—
360C22×C4#10312yes—
361C22×C4#10412yes—
362C22×C4#10412yes—
363C22×C4C2×C4211yescharacteristic
364C22×C4C2×C4⋊C411yescharacteristic
365C22×C4#11711yescharacteristic
366C22×C4#11711yescharacteristic
367C22×C4#11711yescharacteristic
368C22×C4#11711yescharacteristic
369C22×C4#11712yes—
370C22×C4#11712yes—
371C22×C4C4×D412yes—
372C22×C4C4×D412yes—
373C22×C4C4×Q812yes—
374C22×C4#12111yescharacteristic
375C22×C4C22⋊Q811yescharacteristic
376C22×C4#12312yes—
377C22×C4#12412yes—
378C22×C4#12512yes—
379C22×C4#12612yes—
380C22×C4#12612yes—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—82no—
385C2×C8—82no—
386C2×C8—82no—
387C2×C8—82no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C42—24no—
401C42—24no—
402C42—44no—
403C42—44no—
404C42—42no—
405C42—42no—
406C42—84no—
407C42—84no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23C4×C22⋊C411yescharacteristic
449C23#38011yescharacteristic
450C23#38111yescharacteristic
451C23#38712yes—
452C23#38812yes—
453C23C2×#9911yescharacteristic
454C23C2×#9911yescharacteristic
455C23#41011yescharacteristic
456C23#41011yescharacteristic
457C23C2×#10311yescharacteristic
458C23#41611yescharacteristic
459C23#41711yescharacteristic
460C23#41811yescharacteristic
461C23C2×#10411yescharacteristic
462C23C2×#10411yescharacteristic
463C23#42111yescharacteristic
464C23#42111yescharacteristic
465C23C4×SD1612yes—
466C23C4×Q1612yes—
467C23#44012yes—
468C23#44212yes—
469C23#46011yescharacteristic
470C23#46211yescharacteristic
471C23#46312yes—
472C23#47511yescharacteristic
473C23#47711yescharacteristic
474C23#47812yes—
475C23#48112yes—
476C23#48212yes—
477C23#48412yes—
478C23#48512yes—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4#32814yes—
582C2×C4#37611yescharacteristic
583C2×C4C4×C4⋊C411yescharacteristic
584C2×C4#38211yescharacteristic
585C2×C4#38311yescharacteristic
586C2×C4#38712yes—
587C2×C4#38812yes—
588C8—84no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22#143411yescharacteristic
622C22C2×#32811yescharacteristic
623C22#150011yescharacteristic
624C22#150111yescharacteristic
625C22#150311yescharacteristic
626C22#150411yescharacteristic
627C22#150511yescharacteristic
628C22#150711yescharacteristic
629C22#150911yescharacteristic
630C22#151011yescharacteristic
631C22#151211yescharacteristic
632C22#151311yescharacteristic
633C22#151411yescharacteristic
634C22#151611yescharacteristic
635C22#151811yescharacteristic
636C22#152011yescharacteristic
637C22#152111yescharacteristic
638C22C4×#9911yescharacteristic
639C22C4×#9911yescharacteristic
640C22C4×#10411yescharacteristic
641C22#175911yescharacteristic
642C22C4×#10311yescharacteristic
643C22#176211yescharacteristic
644C22#179911yescharacteristic
645C22#179911yescharacteristic
646C22#180311yescharacteristic
647C22#180411yescharacteristic
648C22#180611yescharacteristic
649C22#180611yescharacteristic
650C22#181911yescharacteristic
651C22#182011yescharacteristic
652C22#182411yescharacteristic
653C22#182511yescharacteristic
654C22#182611yescharacteristic
655C22#182611yescharacteristic
656C22#191912yes—
657C22#192112yes—
658C22#192612yes—
659C22#192812yes—
660C4—24no—
661C4—22no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—88no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C2—22no—
676C2—22no—
677C2—22no—
678C2—22no—
679C2C4×#32811yescharacteristic
680C2#821611yescharacteristic
681C2#821811yescharacteristic
682C2#822111yescharacteristic
683C2#857211yescharacteristic
684C2#857311yescharacteristic
685C2#857511yescharacteristic
686C2#857811yescharacteristic
687C2#871311yescharacteristic
688C2#871411yescharacteristic
689C2#871511yescharacteristic
690C2#871711yescharacteristic
691C2#1080111yescharacteristic
692C2#1080711yescharacteristic
693C2#1080811yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×C4×C4⋊C4C4semidirect
11. C2×C4×C4⋊C4C4semidirect
18. C4⋊#95C4semidirect
19. C4⋊#95C4semidirect
24. #1294C4semidirect

Maximal subgroups

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

subgroupindex
C4×#3802
#107122
C2×#12942
#75542