Finite Group Databasebeta

← #356590

Subgroups — #356590

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×D4C211yesmaximal, characteristic
2C2×#1869C212yesmaximal
3C2×#1752C212yesmaximal, complemented
4C2×#1749C211yesmaximal, characteristic, complemented
5C22×#336C211yesmaximal, characteristic, complemented
6#10922C212yesmaximal, complemented
7#10875C212yesmaximal, complemented
8#10795C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic, complemented
10#10712C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C24×C8C2212yes—
13C22×C4×D4C412yes—
14C22×C4×D4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2212yes—
21C22×#97C2212yes—
22C42×D4—28no—
23C2×C4×C4⋊C4C412yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C412yes—
26C2×C4×C22⋊C4C412yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1869C4116yescomplemented
31#1752—28no—
32#1749—24no—
33C2×#336—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8C2×C418yescomplemented
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4×D4—28no—
61C2×C4×D4—28no—
62C2×C4×D4—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—28no—
66C2×C4×D4C2×C418yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C412yes—
73C22×C42—24no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—24no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—42no—
80C22×C42—42no—
81C22×C42C2×C412yes—
82C22×C42C2×C412yes—
83C22×C42C2×C412yes—
84C22×C42C2×C412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8C2×C418yescomplemented
97C2×C22⋊C8C2×C418yescomplemented
98C2×C22⋊C8C2×C418yescomplemented
99C2×#97—24no—
100C2×#97—24no—
101C2×#97—24no—
102C2×#97—24no—
103C2×#97—42no—
104C2×#97—42no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—28no—
107C4×C22⋊C4—28no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—28no—
110C4×C22⋊C4D418yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C43—24no—
115#336—24no—
116#336—22no—
117#336—22no—
118#336—24no—
119#336—24no—
120#336—22no—
121#336—42no—
122#336C2×C418yescomplemented
123#336C2×C418yescomplemented
124#336D414yescomplemented
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4C4214yes—
131C22×D4M1618yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4M1618yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C414yes—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C414yes—
170C23×C4C22×C412yes—
171C23×C4C2×D414yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—216no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—216no—
199C22×C8—48no—
200C22×C8—48no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8—48no—
204C22×C8C22×C414yes—
205C22×C8C22×C414yes—
206C22×C8C22×C414yes—
207C22×C8C22×C414yes—
208C22×C8C2×D414yes—
209C22×C8#4114yes—
210C4×D4—264no—
211C4×D4—216no—
212C4×D4—216no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4C4214yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4C4214yes—
229C2×C22⋊C4C4214yes—
230C2×C22⋊C4C22⋊C418yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—48no—
256C2×C42—48no—
257C2×C42—48no—
258C2×C42—48no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C22×C414yes—
263C2×C42C2×D414yes—
264C2×C42C2×D414yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C22⋊C8—216no—
268C22⋊C8—216no—
269C22⋊C8—48no—
270#97—48no—
271#97—48no—
272#95—24no—
273#95—24no—
274#95C22⋊C418yes—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24C2×C4212yes—
291C24C2×C4212yes—
292C24C2×C22⋊C412yes—
293C24#11712yes—
294C24C2×M1614yes—
295C24C2×M1614yes—
296C24C2×M1614yes—
297C24C23×C411yescharacteristic
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—216no—
301C2×D4—216no—
302C2×D4#97116yescomplemented
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—216no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—216no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—216no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4C2×C4212yes—
387C22×C4C2×C4212yes—
388C22×C4C2×C4212yes—
389C22×C4C2×C4212yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4#11712yes—
394C22×C4C4×D418yes—
395C22×C4C4×D418yes—
396C22×C4C4×D414yes—
397C22×C4C2×M1614yes—
398C22×C4C2×M1614yes—
399C22×C4C2×M1614yes—
400C22×C4#13114yes—
401C22×C4C23×C412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+411yescharacteristic
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—432no—
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—42no—
439C2×C8C4×D418yescomplemented
440C4⋊C4—216no—
441C4⋊C4—28no—
442C22⋊C4—216no—
443C22⋊C4—28no—
444C22⋊C4—216no—
445C22⋊C4—28no—
446C42—216no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—48no—
456C42—48no—
457C42—48no—
458C42—42no—
459C42—42no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C4×C22⋊C414yes—
501C23C2×#9714yes—
502C23C2×#9714yes—
503C23C4×M1614yes—
504C23#40718yes—
505C23#43218yes—
506C23C22×C4211yescharacteristic
507C23C22×C22⋊C411yescharacteristic
508C23C2×#11711yescharacteristic
509C23C2×C4×D411yescharacteristic
510C23C4×#4111yescharacteristic
511C23#51811yescharacteristic
512C23#52011yescharacteristic
513C23C22×M1612yes—
514C23C22×M1612yes—
515C23C2×#13112yes—
516C23#56812yes—
517D4—232no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—432no—
554C2×C4—432no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4C4×C22⋊C414yes—
578C2×C4C2×#9714yes—
579C2×C4#43518yes—
580C2×C4C2×C4×D412yes—
581C2×C4C2×C4×D412yes—
582C2×C4C2×C4×D412yes—
583C2×C4C4×#4112yes—
584C2×C4#51812yes—
585C2×C4#52012yes—
586C2×C4D4212yes—
587C2×C4#54611yescharacteristic
588C2×C4#54612yes—
589C2×C4#54612yes—
590C2×C4#54712yes—
591C2×C4#54811yescharacteristic
592C2×C4#55112yes—
593C2×C4#55211yescharacteristic
594C2×C4#55312yes—
595C2×C4#55311yescharacteristic
596C8—28no—
597C8—44no—
598C8—44no—
599C8—44no—
600C22—28no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—28no—
606C22—24no—
607C22—28no—
608C22—28no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22#174918yescomplemented
622C22C2×C4×C22⋊C411yescharacteristic
623C22C42×D411yescharacteristic
624C22#227511yescharacteristic
625C22D4×C22⋊C411yescharacteristic
626C22#233911yescharacteristic
627C22#235211yescharacteristic
628C22#235311yescharacteristic
629C22C22×#9711yescharacteristic
630C22C2×C4×M1611yescharacteristic
631C22C4×#13111yescharacteristic
632C22#287511yescharacteristic
633C22C2×#40712yes—
634C22#288012yes—
635C22C2×#43212yes—
636C22#292512yes—
637C22C2×#43512yes—
638C22D4×M1612yes—
639C22#296512yes—
640C22#297012yes—
641C22#297312yes—
642C22#297412yes—
643C22#298912yes—
644C22#299012yes—
645C4—28no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4#235812yes—
654C4#235912yes—
655C4#236112yes—
656C4#236412yes—
657C2—28no—
658C2—44no—
659C2#1381311yescharacteristic
660C2C2×#174911yescharacteristic
661C2#2037811yescharacteristic
662C2C4×#43511yescharacteristic
663C2D4×#9711yescharacteristic
664C2#2055611yescharacteristic
665C2#2055811yescharacteristic
666C2#2239912yes—
667C2#2247112yes—
668C2#2275012yes—
669C2#2278712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1752C2semidirect
4. C2×#1749C2semidirect
5. C22×#336C2semidirect
6. #10922C2semidirect
7. #10875C2semidirect
8. #10795C2semidirect
9. #10712C2semidirect
10. #10712C2semidirect
11. #10693C2semidirect
30. #1869C4semidirect
55. C23×C8C2×C4semidirect
96. C2×C22⋊C8C2×C4semidirect
97. C2×C22⋊C8C2×C4semidirect
98. C2×C22⋊C8C2×C4semidirect
122. #336C2×C4semidirect
123. #336C2×C4semidirect
124. #336D4semidirect
302. C2×D4#97semidirect
439. C2×C8C4×D4semidirect
621. C22#1749semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D42
C2×#18692
C2×#17522
C2×#17492
C22×#3362
#109222
#108752
#107952
#107122
#106932