Finite Group Databasebeta

← C4×#2001 #358998

Subgroups — C4×#2001 #358998

Order 512.

2268 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C212yesmaximal
3C2×#2001C211yesmaximal, characteristic
4C2×C4×#103C212yesmaximal, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic
6#11328C211yesmaximal, characteristic, complemented
7#11096C211yesmaximal, characteristic
8#11067C212yesmaximal, complemented
9#11024C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11C4×#340C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×Q8C2212yes—
15C23×C42C2211yescharacteristic
16C2×C4×Q16—24no—
17C2×C4×Q16—28no—
18C22×#107C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C4×C22⋊Q8—28no—
27C4×C22⋊Q8—28no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#385—22no—
32C2×#380C2212yes—
33C2×C4×C4⋊C4—22no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4—22no—
36#2001C418yesdirect product factor
37C4×#103—28no—
38C4×C22⋊C8—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C23×Q8C2312yes—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16C2×C414yescomplemented
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8C2×C418yes—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8D418yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42—22no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C4×Q16—232no—
90C2×#107—22no—
91C2×#107—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103C2×C418yescomplemented
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8C2×C414yes—
107C2×C4×C8—24no—
108C2×C4×C8—22no—
109C2×C4×C8—24no—
110C2×C4×C8D414yes—
111#391—24no—
112#391—24no—
113#389—24no—
114#389—24no—
115#387—216no—
116#385—216no—
117#380—24no—
118#380—24no—
119C4×C4⋊C4—216no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4D418yes—
122C4×C22⋊C4—216no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128#340—22no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340C2×C414yescomplemented
133#336—22no—
134#336—22no—
135C25C2×D411yescharacteristic
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C22×C414yes—
148C22×Q8C2×D414yes—
149C22×Q8#4114yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4Q1614yescomplemented
172C23×C4C22×C412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180C2×Q16—28no—
181C2×Q16—216no—
182C2×Q16—28no—
183C2×Q16—216no—
184C2×Q16—28no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C22×C412yes—
200C22×C8C22×C412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8#4112yes—
204C22×C8#4112yes—
205C22⋊Q8—216no—
206C22⋊Q8—216no—
207C22⋊Q8—216no—
208C22⋊Q8—216no—
209C4×Q8—216no—
210C4×Q8—216no—
211C4×Q8—48no—
212C4×Q8—48no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4#4114yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—48no—
262C2×C42—44no—
263C2×C42C2×D414yes—
264C2×C42C2×D412yes—
265C2×C42C2×D414yes—
266C2×C42#4112yes—
267#107—28no—
268#103—216no—
269#103—216no—
270#103—216no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273C4×C8—28no—
274C4×C8—44no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280C24—22no—
281C24—21no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24C4×D412yes—
289C24C2×Q1612yes—
290C24#13512yes—
291C24C22×D411yescharacteristic
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
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×Q8—42no—
319C2×Q8C4×D418yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—216no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C4×D412yes—
392C22×C4#12014yes—
393C22×C4#12014yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C2×Q1612yes—
397C22×C4C2×Q1612yes—
398C22×C4#13512yes—
399C22×C4#13512yes—
400C22×C4#13714yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411Q16—232no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8C4×D414yes—
432C4⋊C4—232no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—416no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—416no—
446C4⋊C4C4×D418yes—
447C22⋊C4—232no—
448C22⋊C4—416no—
449C42—41no—
450C42—42no—
451C42—41no—
452C42—44no—
453C42—44no—
454C42—416no—
455C42—416no—
456C42#12014yes—
457C42#12118yes—
458C42#12714yes—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23C4×Q1614yescomplemented
482C23C2×C4×D411yescharacteristic
483C23C2×#12012yes—
484C23C2×#12111yescharacteristic
485C23#52512yes—
486C23#53211yescharacteristic
487C23C22×Q1611yescharacteristic
488C23C2×#13511yescharacteristic
489C23C2×#13511yescharacteristic
490C23C2×#13511yescharacteristic
491C23C2×#13712yes—
492C23#57512yes—
493Q8—28no—
494Q8—28no—
495Q8—44no—
496Q8—44no—
497Q8—44no—
498Q8—44no—
499Q8—44no—
500Q8—44no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—416no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—416no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#39214yes—
557C2×C4#39214yes—
558C2×C4#45118yes—
559C2×C4#45218yes—
560C2×C4#46714yes—
561C2×C4#47014yes—
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×C4×D412yes—
564C2×C4C2×C4×D412yes—
565C2×C4C4×#4111yescharacteristic
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×#12112yes—
568C2×C4#52512yes—
569C2×C4#52511yescharacteristic
570C2×C4#52512yes—
571C2×C4#52512yes—
572C2×C4#52511yescharacteristic
573C2×C4#52512yes—
574C2×C4#52911yescharacteristic
575C2×C4#52912yes—
576C2×C4C2×#12711yescharacteristic
577C2×C4#53211yescharacteristic
578C2×C4#53212yes—
579C2×C4#53212yes—
580C8—24no—
581C8—42no—
582C8—42no—
583C8—42no—
584C22—24no—
585C22—28no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22C4×#12012yes—
596C22C4×#12111yescharacteristic
597C22C2×#39211yescharacteristic
598C22#238812yes—
599C22#239211yescharacteristic
600C22C2×C4×Q1611yescharacteristic
601C22C4×#13511yescharacteristic
602C22C4×#13712yes—
603C22C2×#45112yes—
604C22C2×#45212yes—
605C22#300212yes—
606C22#300212yes—
607C22C2×#46711yescharacteristic
608C22#305011yescharacteristic
609C22C2×#47011yescharacteristic
610C22#305411yescharacteristic
611C22#310212yes—
612C22#310412yes—
613C22#310712yes—
614C22#310812yes—
615C22#316811yescharacteristic
616C22#317011yescharacteristic
617C22#317212yes—
618C4—24no—
619C4—42no—
620C4—48no—
621C4—42no—
622C4—42no—
623C4—48no—
624C4—84no—
625C4—84no—
626C4#200118yesdirect product factor
627C4C4×#12011yescharacteristic
628C4C4×#12112yes—
629C4C4×#12711yescharacteristic
630C4#238811yescharacteristic
631C4#239212yes—
632C4#239711yescharacteristic
633C2—44no—
634C2C4×#39211yescharacteristic
635C2C4×#45112yes—
636C2C4×#45212yes—
637C2C4×#46711yescharacteristic
638C2C4×#47011yescharacteristic
639C2C2×#200111yescharacteristic
640C2#2338611yescharacteristic
641C2#2357112yes—
642C2#2357212yes—
643C2#2369811yescharacteristic
644C2#2369911yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×Q16C2semidirect
4. C2×C4×#103C2semidirect
6. #11328C2semidirect
8. #11067C2semidirect
9. #11024C2semidirect
11. C4×#340C2semidirect
36. #2001C4direct (N × H)
54. C22×Q16C2×C4semidirect
100. C2×#103C2×C4semidirect
132. #340C2×C4semidirect
171. C23×C4Q16semidirect
481. C23C4×Q16semidirect
626. C4#2001direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×C4×C22⋊Q82
C2×#20012
C2×C4×#1032
C2×C4×C22⋊C82
#113282
#110962
#110672
#110242
#109702
C4×#3402