Finite Group Databasebeta

← #361564

Subgroups — #361564

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1841C211yesmaximal, characteristic, complemented
2#20377C211yesmaximal, characteristic
3C2×#1733C211yesmaximal, 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#10720C211yesmaximal, characteristic
14C22⋊#338C211yesmaximal, characteristic
15#10716C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2280C2211yescharacteristic
25#1841—24no—
26#1750C414yescomplemented
27#1750C414yescomplemented
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1733—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×C4C2311yescharacteristic
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—41no—
62C22×M16—41no—
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8—21no—
72C2×#117—22no—
73C2×#117—21no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—41no—
81C2×#117—41no—
82C2×#117—41no—
83C2×#117—41no—
84C2×#117C2×C412yes—
85C2×#117C2×C412yes—
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C42—21no—
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—21no—
157C23×C4—21no—
158C23×C4—41no—
159C23×C4C22×C412yes—
160C23×C4#4111yescharacteristic
161C23×C4#4111yescharacteristic
162C23×C4#4111yescharacteristic
163C23×C4#4111yescharacteristic
164C23×C4#4112yes—
165C23×C4#4111yescharacteristic
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
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—28no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
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—42no—
217C2×M16—42no—
218C2×M16—44no—
219C2×M16—44no—
220C2×M16—44no—
221C2×M16—42no—
222C2×M16C2×D412yes—
223C2×M16#4112yes—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—42no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235#117—21no—
236#117—21no—
237#117—24no—
238#117—24no—
239#117—21no—
240#117—21no—
241#117—24no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—42no—
253#117—42no—
254#117—44no—
255#117—44no—
256#117—42no—
257#117—42no—
258#117—42no—
259#117—42no—
260#117—42no—
261#117C2×D412yes—
262#117#4112yes—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—41no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—42no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—41no—
301C2×C42C22×C412yes—
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42#4112yes—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312#97—24no—
313#97—42no—
314C4×C8—24no—
315C4×C8—42no—
316#95—42no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—82no—
322C24#11712yes—
323C24#12611yescharacteristic
324C24#12612yes—
325C24#12611yescharacteristic
326C24C2×#4111yescharacteristic
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—84no—
378C22×C4#11712yes—
379C22×C4#11712yes—
380C22×C4#11712yes—
381C22×C4#11712yes—
382C22×C4#11712yes—
383C22×C4#12612yes—
384C22×C4#12612yes—
385C22×C4#12612yes—
386C22×C4#12612yes—
387C22×C4#12612yes—
388C22×C4#12611yescharacteristic
389C22×C4#12611yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C42−1+411yescharacteristic
404M16—22no—
405M16—22no—
406M16—22no—
407M16—22no—
408M16—48no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
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—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—48no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—84no—
485C42—21no—
486C42—21no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—44no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42C4×D412yes—
503C42C4×D412yes—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23#38312yes—
524C23#38312yes—
525C23C2×#11711yescharacteristic
526C23C2×#12611yescharacteristic
527C23C2×#12611yescharacteristic
528C23#54611yescharacteristic
529C23#55111yescharacteristic
530C23#55111yescharacteristic
531C23#55211yescharacteristic
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
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—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—88no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4#38312yes—
576C2×C4#38312yes—
577C2×C4C2×#11711yescharacteristic
578C2×C4C2×#11711yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4#51811yescharacteristic
582C2×C4#52011yescharacteristic
583C2×C4#52511yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4C2×#12611yescharacteristic
586C2×C4C2×#12611yescharacteristic
587C2×C4#52911yescharacteristic
588C2×C4#52911yescharacteristic
589C2×C4#53811yescharacteristic
590C2×C4#53811yescharacteristic
591C2×C4#53911yescharacteristic
592C2×C4#53911yescharacteristic
593C2×C4#54611yescharacteristic
594C2×C4#54611yescharacteristic
595C2×C4#54811yescharacteristic
596C2×C4#55111yescharacteristic
597C2×C4#55111yescharacteristic
598C2×C4#55111yescharacteristic
599C2×C4#55111yescharacteristic
600C2×C4#55211yescharacteristic
601C2×C4#55211yescharacteristic
602C2×C4#55311yescharacteristic
603C2×C4#55311yescharacteristic
604C2×C4#55311yescharacteristic
605C8—42no—
606C8—42no—
607C8—42no—
608C8—42no—
609C8—42no—
610C8—42no—
611C8—81no—
612C8—81no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#38311yescharacteristic
635C22#234211yescharacteristic
636C22#234311yescharacteristic
637C22#248011yescharacteristic
638C22#248111yescharacteristic
639C22#249311yescharacteristic
640C22#251011yescharacteristic
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—161no—
649C4—161no—
650C4#229011yescharacteristic
651C4#231811yescharacteristic
652C4#235211yescharacteristic
653C4#235311yescharacteristic
654C4#248011yescharacteristic
655C4#248111yescharacteristic
656C4#249311yescharacteristic
657C4#251011yescharacteristic
658C2—21no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—42no—
665C2—42no—
666C2#1390111yescharacteristic
667C2#2077612yes—
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. #1750C4semidirect
27. #1750C4semidirect
113. C2×#97C2×C4semidirect
114. C2×#97C2×C4semidirect
115. C2×C4×C8C2×C4semidirect
116. C2×C4×C8C2×C4semidirect
145. #336C2×C4semidirect
146. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18412
#203772
C2×#17332
#113002
#112992
#108252
#108132
#108052
#107382
#107372
#107202
C22⋊#3382
#107162