Finite Group Databasebeta

← #381433

Subgroups — #381433

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3C2×#1878C212yesmaximal
4#14703C211yesmaximal, characteristic
5#11267C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11050C211yesmaximal, characteristic
8#11024C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10795C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C4×C4⋊Q8—28no—
20C2×#391C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—21no—
30C2×C43C2211yescharacteristic
31#2016C418yescomplemented
32#1878—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45#1434—21no—
46C23×Q8C2311yescharacteristic
47C23×Q8C2312yes—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—42no—
51C22×Q16—42no—
52C22×Q16—42no—
53C22×Q16—42no—
54C22×Q16C2×C418yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C2×C4×Q8D418yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C42—22no—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103C2×C418yes—
97#391—24no—
98#391—24no—
99#391—22no—
100#391—24no—
101#391—42no—
102#391C2×C414yes—
103#389—28no—
104#389—28no—
105#389—42no—
106#384—24no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—28no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C43—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—24no—
127#336—22no—
128#336—42no—
129#336C2×C414yes—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8C22×C412yes—
151C22×Q8C22×C414yes—
152C22×Q8C2×D414yes—
153C22×Q8#4114yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×Q16—216no—
168C2×Q16—48no—
169C2×Q16—48no—
170C2×Q16—48no—
171C2×Q16—48no—
172C2×Q16—48no—
173C2×Q16—48no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C22×C414yes—
188C4⋊Q8—216no—
189C4⋊Q8—216no—
190C4×Q8—48no—
191C4×Q8—48no—
192C4×Q8—48no—
193C4×Q8—48no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—28no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42Q1614yescomplemented
254C2×C42C22×C412yes—
255C2×C42C2×D414yes—
256C2×C42#4112yes—
257#103—48no—
258#103—48no—
259#103—48no—
260#103—48no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274C24C22×D411yescharacteristic
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8C4×D418yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4C4×D412yes—
392C22×C4C4×D412yes—
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4C2×Q1612yes—
396C22×C4C2×Q1612yes—
397C22×C4C2×Q1612yes—
398C22×C4#13512yes—
399C22×C4#13712yes—
400C22×C4#13712yes—
401C22×C4#13712yes—
402C22×C4#13712yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4111yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413Q16—432no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—416no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—84no—
444C42—28no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—28no—
450C42—44no—
451C42—44no—
452C42—416no—
453C42—42no—
454C42—42no—
455C23C2×C4×D411yescharacteristic
456C23C2×#12011yescharacteristic
457C23#52511yescharacteristic
458C23#53511yescharacteristic
459C23#53511yescharacteristic
460C23#53511yescharacteristic
461C23#53711yescharacteristic
462C23C22×Q1611yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×#13711yescharacteristic
465C23C2×#13711yescharacteristic
466C23C2×#13711yescharacteristic
467C23C2×#13711yescharacteristic
468C23#57511yescharacteristic
469C23#57511yescharacteristic
470Q8—28no—
471Q8—28no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—82no—
481Q8—82no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
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—28no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4C4×Q1614yes—
564C2×C4#44114yes—
565C2×C4#45114yes—
566C2×C4C2×C4×D412yes—
567C2×C4#51811yescharacteristic
568C2×C4#51812yes—
569C2×C4#52011yescharacteristic
570C2×C4#52512yes—
571C2×C4#53412yes—
572C2×C4#53412yes—
573C2×C4#53512yes—
574C2×C4#53712yes—
575C2×C4#53812yes—
576C2×C4#53812yes—
577C2×C4#53812yes—
578C2×C4#53912yes—
579C2×C4#54012yes—
580C2×C4#54011yescharacteristic
581C2×C4#54211yescharacteristic
582C8—44no—
583C8—82no—
584C22C4×#12011yescharacteristic
585C22#232411yescharacteristic
586C22#232811yescharacteristic
587C22#261311yescharacteristic
588C22#261511yescharacteristic
589C22#263111yescharacteristic
590C22#263611yescharacteristic
591C22C2×C4×Q1611yescharacteristic
592C22C2×#44111yescharacteristic
593C22C4×#13711yescharacteristic
594C22C4×#13711yescharacteristic
595C22C2×#45111yescharacteristic
596C22#300211yescharacteristic
597C22#300411yescharacteristic
598C22#300711yescharacteristic
599C22#310411yescharacteristic
600C22#310811yescharacteristic
601C22#311311yescharacteristic
602C22#311711yescharacteristic
603C22#320911yescharacteristic
604C22#320911yescharacteristic
605C22#320911yescharacteristic
606C22#321011yescharacteristic
607C22#321011yescharacteristic
608C22#321211yescharacteristic
609C22#321511yescharacteristic
610C22#321611yescharacteristic
611C22#321611yescharacteristic
612C22#321811yescharacteristic
613C22#321811yescharacteristic
614C22#321911yescharacteristic
615C22#324811yescharacteristic
616C22#324911yescharacteristic
617C22#325211yescharacteristic
618C22#325511yescharacteristic
619C4—24no—
620C4—24no—
621C4—24no—
622C4—42no—
623C4—48no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—84no—
628C4—82no—
629C4—161no—
630C4—161no—
631C4#232112yes—
632C4#232512yes—
633C4#263012yes—
634C4#263512yes—
635C2#1429911yescharacteristic
636C2C4×#45111yescharacteristic
637C2#2145511yescharacteristic
638C2#2214811yescharacteristic
639C2#2215411yescharacteristic
640C2#2227011yescharacteristic
641C2#2227311yescharacteristic
642C2#2962711yescharacteristic
643C2#2962911yescharacteristic
644C2#2966411yescharacteristic
645C2#2966811yescharacteristic
646C2#3004211yescharacteristic
647C2#3004311yescharacteristic
648C2#3013111yescharacteristic
649C2#3013211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #2016C4semidirect
253. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20162
C2×#18782
#147032
#112672
#110672
#110502
#110242
#110102
#107952
#107892