Finite Group Databasebeta

← #361563

Subgroups — #361563

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1841C211yesmaximal, characteristic, complemented
2C4×#407C211yesmaximal, characteristic
3C22×#339C211yesmaximal, characteristic
4#11300C211yesmaximal, characteristic, complemented
5#11300C211yesmaximal, characteristic, complemented
6#11299C211yesmaximal, characteristic, complemented
7#11299C211yesmaximal, characteristic, complemented
8#10825C211yesmaximal, characteristic, complemented
9#10813C211yesmaximal, characteristic, complemented
10#10805C211yesmaximal, characteristic, complemented
11#10738C211yesmaximal, characteristic
12#10737C211yesmaximal, characteristic
13#10719C211yesmaximal, characteristic
14C22⋊#338C211yesmaximal, characteristic
15#10716C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2280C2211yescharacteristic
25#1841—24no—
26#1749C414yescomplemented
27C4×C22⋊C8C414yescomplemented
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#339—24no—
37C2×#339C2211yescharacteristic
38C2×#339C2211yescharacteristic
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#338C2211yescharacteristic
46C2×#338C2211yescharacteristic
47C2×#338C2211yescharacteristic
48C2×#338C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53#1728C2211yescharacteristic
54#1728C2211yescharacteristic
55C24×C4—21no—
56C24×C4C2311yescharacteristic
57C22×M16—21no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—41no—
63C22×M16—41no—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8—21no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—41no—
79C2×#117—41no—
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42—41no—
92C22×C42—41no—
93C22×C42C2×C412yes—
94C22×C42C2×C412yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#407—21no—
98#407—21no—
99#407—22no—
100#407—21no—
101#407—21no—
102#407—22no—
103#407—21no—
104#407—21no—
105#407—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8C2×C414yes—
108C2×C22⋊C8C2×C414yes—
109C4×M16—22no—
110C4×M16—22no—
111C4×M16—41no—
112C4×M16—41no—
113C2×#97C2×C412yescomplemented
114C2×#97C2×C412yescomplemented
115C2×C4×C8C2×C412yescomplemented
116C2×C4×C8C2×C412yescomplemented
117#381—21no—
118#380—22no—
119#379—21no—
120C4×C22⋊C4—22no—
121#344—22no—
122#344—22no—
123#344—22no—
124#344—22no—
125#344—41no—
126#344—41no—
127#344—41no—
128#344—41no—
129#339—22no—
130#339—22no—
131#339—22no—
132#339—22no—
133#339—22no—
134#339—41no—
135#339—41no—
136#339D414yes—
137#338—22no—
138#338—22no—
139#338—41no—
140#338—41no—
141H(3,ℤ/4)—22no—
142H(3,ℤ/4)—22no—
143H(3,ℤ/4)—41no—
144H(3,ℤ/4)—41no—
145#336C2×C414yescomplemented
146#336C2×C414yescomplemented
147C25#4111yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—48no—
163C23×C4—41no—
164C23×C4C22×C412yes—
165C23×C4#4111yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4#4111yescharacteristic
169C23×C4#4111yescharacteristic
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—28no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—42no—
222C2×M16—42no—
223C2×M16—44no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16—42no—
227C2×M16C2×D412yes—
228C2×M16#4112yes—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C22×C8C22×C412yes—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—24no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—42no—
256#117—42no—
257#117—42no—
258#117—42no—
259#117—42no—
260#117C2×D412yes—
261#117#4112yes—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—41no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—42no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—41no—
302C2×C42C22×C412yes—
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C2×D412yes—
306C2×C42#4112yes—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313#97—24no—
314#97—42no—
315C4×C8—24no—
316C4×C8—42no—
317#95—42no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—44no—
323C24#11712yes—
324C24#12611yescharacteristic
325C24#12612yes—
326C24#12611yescharacteristic
327C24C2×#4111yescharacteristic
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—44no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—84no—
387C22×C4#11712yes—
388C22×C4#11712yes—
389C22×C4#11712yes—
390C22×C4#11712yes—
391C22×C4#11712yes—
392C22×C4#12612yes—
393C22×C4#12612yes—
394C22×C4#12612yes—
395C22×C4#12612yes—
396C22×C4#12612yes—
397C22×C4#12611yescharacteristic
398C22×C4#12611yescharacteristic
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—48no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—48no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—44no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—84no—
490C42—21no—
491C42—21no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—44no—
499C42—44no—
500C42—44no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42C4×D412yes—
510C42C4×D412yes—
511C23—24no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—21no—
519C23—22no—
520C23—21no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23#38312yes—
531C23#38312yes—
532C23C2×#11711yescharacteristic
533C23C2×#12611yescharacteristic
534C23C2×#12611yescharacteristic
535C23#54611yescharacteristic
536C23#55111yescharacteristic
537C23#55111yescharacteristic
538C23#55211yescharacteristic
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—88no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4#38312yes—
586C2×C4#38312yes—
587C2×C4C2×#11711yescharacteristic
588C2×C4C2×#11711yescharacteristic
589C2×C4C2×C4×D411yescharacteristic
590C2×C4C4×#4111yescharacteristic
591C2×C4#51811yescharacteristic
592C2×C4#52011yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4C2×#12611yescharacteristic
596C2×C4C2×#12611yescharacteristic
597C2×C4#52911yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#53811yescharacteristic
600C2×C4#53811yescharacteristic
601C2×C4#53911yescharacteristic
602C2×C4#53911yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54611yescharacteristic
605C2×C4#54811yescharacteristic
606C2×C4#55111yescharacteristic
607C2×C4#55111yescharacteristic
608C2×C4#55111yescharacteristic
609C2×C4#55111yescharacteristic
610C2×C4#55211yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55311yescharacteristic
613C2×C4#55311yescharacteristic
614C2×C4#55311yescharacteristic
615C8—42no—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C8—42no—
621C8—81no—
622C8—81no—
623C22—24no—
624C22—24no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22#184114yescomplemented
644C22C2×#38311yescharacteristic
645C22#234211yescharacteristic
646C22#234311yescharacteristic
647C22#248011yescharacteristic
648C22#248111yescharacteristic
649C22#249311yescharacteristic
650C22#251011yescharacteristic
651C4—24no—
652C4—22no—
653C4—22no—
654C4—44no—
655C4—42no—
656C4—82no—
657C4—82no—
658C4—161no—
659C4—161no—
660C4#229011yescharacteristic
661C4#231811yescharacteristic
662C4#235211yescharacteristic
663C4#235311yescharacteristic
664C4#248011yescharacteristic
665C4#248111yescharacteristic
666C4#249311yescharacteristic
667C4#251011yescharacteristic
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2—24no—
673C2—21no—
674C2—21no—
675C2—42no—
676C2#1390111yescharacteristic
677C2C2×#184112yes—
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#383
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#407 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1841C2semidirect
4. #11300C2semidirect
5. #11300C2semidirect
6. #11299C2semidirect
7. #11299C2semidirect
8. #10825C2semidirect
9. #10813C2semidirect
10. #10805C2semidirect
26. #1749C4semidirect
27. C4×C22⋊C8C4semidirect
113. C2×#97C2×C4semidirect
114. C2×#97C2×C4semidirect
115. C2×C4×C8C2×C4semidirect
116. C2×C4×C8C2×C4semidirect
145. #336C2×C4semidirect
146. #336C2×C4semidirect
643. C22#1841semidirect

Maximal subgroups

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

subgroupindex
C2×#18412
C4×#4072
C22×#3392
#113002
#112992
#108252
#108132
#108052
#107382
#107372
#107192
C22⋊#3382
#107162