Finite Group Databasebeta

← C4×#1905 #354249

Subgroups — C4×#1905 #354249

Order 512.

1835 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#137C211yesmaximal, characteristic
2C4×#519C211yesmaximal, characteristic, complemented
3C2×#1905C212yesmaximal
4C4×#419C211yesmaximal, characteristic, complemented
5C4×#421C211yesmaximal, characteristic
6C4×#410C211yesmaximal, characteristic
7#10954C212yesmaximal
8#10903C212yesmaximal, complemented
9#10786C212yesmaximal, complemented
10C4×#344C211yesmaximal, characteristic, complemented
11C4×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
12C22×#137C2211yescharacteristic
13C2×#519C2212yes—
14C2×C4×#41C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C4×#137—22no—
17C4×#137—22no—
18C4×#137—22no—
19C4×#137—22no—
20C2×#441C412yescomplemented
21C2×#440C412yescomplemented
22C2×C4×Q16C412yescomplemented
23C2×C4×SD16C412yescomplemented
24C2×#428C2211yescharacteristic
25C2×#421C2212yes—
26C2×#417C2211yescharacteristic
27C2×#419C2212yes—
28C2×#419C2211yescharacteristic
29C2×#410C2212yes—
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2276—22no—
33#2276—22no—
34#2274—22no—
35#2274C2212yes—
36C42×Q8—22no—
37C42×Q8—22no—
38C4×#117—22no—
39C4×#117C2211yescharacteristic
40C4×#117C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#1905C4116yesdirect product factor
43#1763—22no—
44C4×#103—22no—
45#1759—22no—
46C4×#104—22no—
47C4×#99—22no—
48C4×#99—22no—
49C2×#344C2212yes—
50C2×#344C2212yes—
51#1740C2212yes—
52#1736C2212yes—
53#1733C2212yes—
54C2×H(3,ℤ/4)C2212yes—
55C2×H(3,ℤ/4)C2212yes—
56C22×#41C2311yescharacteristic
57C23×Q8C2311yescharacteristic
58C2×#137—22no—
59C2×#137—22no—
60C2×#137—24no—
61C2×#137C2×C418yes—
62C22×Q16C2×C412yes—
63C22×SD16C2×C412yes—
64C22×M16C2312yes—
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67#519—24no—
68#519—24no—
69#519—24no—
70#519C2×C418yescomplemented
71C4×#41—21no—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41—21no—
78C4×#41—21no—
79C2×C4×Q8—24no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—24no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C412yes—
90C2×C4×Q8C2×C412yes—
91C2×C4×Q8D412yes—
92C2×C4×Q8D412yes—
93C2×C4×Q8D412yes—
94C2×C4×Q8D412yes—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C411yescharacteristic
99C2×#117—24no—
100C2×#117—21no—
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C42—21no—
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110#441—24no—
111#441—24no—
112#440—24no—
113#440—24no—
114C4×Q16—24no—
115C4×Q16—24no—
116C4×SD16—24no—
117C4×SD16—24no—
118#428—22no—
119#428—22no—
120C2×#107C2×C412yes—
121C2×#106C2×C412yes—
122#421—24no—
123#421C2×C418yes—
124C2×#104—24no—
125#419—22no—
126#419—22no—
127#419—24no—
128#419C2×C418yescomplemented
129#417—22no—
130#417—22no—
131C2×#103—24no—
132C2×#103C2×C412yes—
133C2×#103C2×C412yes—
134C2×#103C2×C412yes—
135C2×#102C2×C412yes—
136#410—24no—
137#410C2×C418yes—
138C2×#99—24no—
139C4×M16—21no—
140C4×M16—21no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16—21no—
146C4×M16—21no—
147C2×#97—21no—
148C2×#97C2×C412yes—
149C2×C4×C8—21no—
150C2×C4×C8C2×C412yes—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4D412yes—
169C4×C22⋊C4D412yes—
170#376—24no—
171#376—21no—
172#376—21no—
173#376D412yes—
174C43—24no—
175C43—21no—
176C43—21no—
177C43D412yes—
178#344—24no—
179#344C2×C418yescomplemented
180#342—24no—
181#340—24no—
182#339—24no—
183H(3,ℤ/4)—24no—
184H(3,ℤ/4)C2×C418yescomplemented
185C2×#41—22no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41C22×C414yes—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8C22×C411yescharacteristic
199C22×Q8C22×C412yes—
200C22×Q8C22×C414yes—
201C22×Q8C2×D412yes—
202C22×Q8C2×D412yes—
203C22×Q8#4114yes—
204C22×D4—21no—
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2411yescharacteristic
214#137—216no—
215#137—216no—
216C2×Q16—24no—
217C2×Q16—24no—
218C2×Q16C4218yescomplemented
219C2×SD16—24no—
220C2×SD16—24no—
221C2×SD16C4218yescomplemented
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—22no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—22no—
228C2×M16—22no—
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—22no—
233C2×M16—22no—
234C2×M16C22×C414yes—
235C2×M16C22×C414yes—
236C22×C8—22no—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C4×Q8—216no—
240C4×Q8—216no—
241C4×Q8—216no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—216no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C22⋊C414yes—
264C4×Q8C22⋊C414yes—
265C4×Q8C22⋊C414yes—
266C4×Q8C22⋊C414yes—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—41no—
283#117—216no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—216no—
289#117—22no—
290#117—22no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—22no—
296#117—41no—
297#117—41no—
298#117—41no—
299#117—41no—
300#117C22×C414yes—
301#117C22×C414yes—
302#117C22×C414yes—
303#117#4114yes—
304#117#4114yes—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C22×C411yescharacteristic
321C2×C4⋊C4C22×C411yescharacteristic
322C2×C4⋊C4C22×C412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4C2×D414yes—
331C2×C42—28no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—28no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—41no—
349C2×C42—41no—
350C2×C42—41no—
351C2×C42—41no—
352C2×C42—41no—
353C2×C42—41no—
354C2×C42C22×C411yescharacteristic
355C2×C42C22×C411yescharacteristic
356C2×C42C2×D414yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42#4111yescharacteristic
364C2×C42#4111yescharacteristic
365C2×C42#4111yescharacteristic
366C2×C42#4111yescharacteristic
367#107—24no—
368#106—24no—
369#104—216no—
370#103—24no—
371#103—24no—
372#103—216no—
373#103—24no—
374#102—24no—
375#99—216no—
376#97—22no—
377#97—22no—
378#97—41no—
379#97—41no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—41no—
383C4×C8—41no—
384#95—28no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—41no—
390#95—41no—
391C24—21no—
392C24C22×D411yescharacteristic
393#41—24no—
394#41—24no—
395#41—24no—
396#41—24no—
397#41—42no—
398#41—42no—
399#41—42no—
400#41—42no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8C2×C4212yes—
420C2×Q8C2×C4214yes—
421C2×Q8C2×C22⋊C414yes—
422C2×Q8#11714yes—
423C2×Q8C4×D418yes—
424C2×Q8C4×D418yes—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—42no—
434C2×D4C2×C4212yes—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—82no—
459C22×C4—81no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4C4×D411yescharacteristic
463C22×C4C4×D411yescharacteristic
464C22×C4C4×D411yescharacteristic
465C22×C4C4×D411yescharacteristic
466C22×C4#12011yescharacteristic
467C22×C4#12011yescharacteristic
468C22×C4#12011yescharacteristic
469C22×C4#12011yescharacteristic
470C22×C4#12111yescharacteristic
471C22×C4#12111yescharacteristic
472C22×C4#12111yescharacteristic
473C22×C4#12111yescharacteristic
474C22×C4C23×C411yescharacteristic
475C22×C4C23×C412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2×#4111yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4111yescharacteristic
483C22×C4C2×#4112yes—
484Q16—216no—
485SD16—216no—
486M16—24no—
487M16—24no—
488M16—24no—
489M16—24no—
490M16—42no—
491M16—42no—
492M16—42no—
493M16—42no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8C2×C4214yes—
503C4⋊C4—232no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4C2×C22⋊C414yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4C4×D418yes—
538C42—232no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—24no—
547C42—44no—
548C42—44no—
549C42—41no—
550C42—41no—
551C42—42no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—82no—
557C42—82no—
558C42—82no—
559C42—88no—
560C42—82no—
561C42C2×C22⋊C411yescharacteristic
562C42C2×C22⋊C411yescharacteristic
563C42C2×C22⋊C411yescharacteristic
564C42C2×C22⋊C411yescharacteristic
565C42C4×D418yes—
566C42C4×D411yescharacteristic
567C42C4×D411yescharacteristic
568C42C4×D411yescharacteristic
569C42C4×D411yescharacteristic
570C42#12111yescharacteristic
571C42#12111yescharacteristic
572C42#12111yescharacteristic
573C42#12111yescharacteristic
574C42#12311yescharacteristic
575C42#12311yescharacteristic
576C42#12311yescharacteristic
577C42#12311yescharacteristic
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—81no—
584C23—81no—
585C23C2×C4×D411yescharacteristic
586C23C2×C4×D412yes—
587C23C2×#12011yescharacteristic
588C23C2×#12111yescharacteristic
589C23#52512yes—
590Q8—24no—
591Q8—24no—
592Q8—24no—
593Q8—28no—
594Q8—28no—
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599Q8C4×C22⋊C418yes—
600D4—22no—
601D4—22no—
602D4—44no—
603D4—44no—
604D4—41no—
605D4—41no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4#38611yescharacteristic
650C2×C4#38611yescharacteristic
651C2×C4#38611yescharacteristic
652C2×C4#38611yescharacteristic
653C2×C4#38611yescharacteristic
654C2×C4#38611yescharacteristic
655C2×C4#38611yescharacteristic
656C2×C4#38611yescharacteristic
657C2×C4C22×C4211yescharacteristic
658C2×C4C22×C22⋊C411yescharacteristic
659C2×C4C2×#11711yescharacteristic
660C2×C4C2×C4×D412yes—
661C2×C4C2×C4×D412yes—
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C2×C4×D412yes—
664C2×C4C4×#4111yescharacteristic
665C2×C4C4×#4112yes—
666C2×C4C4×#4111yescharacteristic
667C2×C4C4×#4112yes—
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12311yescharacteristic
670C2×C4#52512yes—
671C2×C4#52512yes—
672C2×C4#52912yes—
673C2×C4#52912yes—
674C2×C4#53212yes—
675C8—24no—
676C8—42no—
677C8—44no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—82no—
688C22—82no—
689C22—81no—
690C22—81no—
691C22C42×D411yescharacteristic
692C22C2×#38611yescharacteristic
693C22#229011yescharacteristic
694C22C4×#12012yes—
695C22C4×#12112yes—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—28no—
704C4—48no—
705C4—48no—
706C4—44no—
707C4—42no—
708C4—84no—
709C4—81no—
710C4—81no—
711C4#190512yesdirect product factor
712C4C2×C4×C22⋊C411yescharacteristic
713C4C42×D411yescharacteristic
714C4#229011yescharacteristic
715C4#229711yescharacteristic
716C4C4×#12112yes—
717C4C4×#12312yes—
718C2—22no—
719C2—22no—
720C2—21no—
721C2—21no—
722C2—82no—
723C2C4×#38611yescharacteristic
724C2C2×#190511yescharacteristic
725C2#2099811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#519C2semidirect
4. C4×#419C2semidirect
8. #10903C2semidirect
9. #10786C2semidirect
10. C4×#344C2semidirect
11. C4×H(3,ℤ/4)C2semidirect
20. C2×#441C4semidirect
21. C2×#440C4semidirect
22. C2×C4×Q16C4semidirect
23. C2×C4×SD16C4semidirect
42. #1905C4direct (N × H)
70. #519C2×C4semidirect
128. #419C2×C4semidirect
179. #344C2×C4semidirect
184. H(3,ℤ/4)C2×C4semidirect
218. C2×Q16C42semidirect
221. C2×SD16C42semidirect
711. C4#1905direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1372
C4×#5192
C2×#19052
C4×#4192
C4×#4212
C4×#4102
#109542
#109032
#107862
C4×#3442
C4×H(3,ℤ/4)2