Finite Group Databasebeta

← C2×#16514 #611063

Subgroups — C2×#16514 #611063

Order 512.

5806 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#396C212yesmaximal, complemented
2C22×#394C211yesmaximal, characteristic
3C22×#394C212yesmaximal
4C22×#388C212yesmaximal, complemented
5C22×#384C211yesmaximal, characteristic, complemented
6#16514C2116yesmaximal, direct product factor
7C2×#1441C212yesmaximal, complemented
8C2×#1441C211yesmaximal, characteristic, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1439C212yesmaximal, complemented
11C24×D4C2211yescharacteristic
12C23×C4⋊C4C2212yes—
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C42C2211yescharacteristic
25C2×#396—216no—
26C2×#396C22116yescomplemented
27C2×#394—28no—
28C2×#394—216no—
29C2×#394C2218yes—
30C2×#394C22116yes—
31C2×#388—216no—
32C2×#388C22116yescomplemented
33C2×#384—28no—
34C2×#384C2218yescomplemented
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2211yescharacteristic
42#1441C22116yescomplemented
43#1441C2218yescomplemented
44#1439C22116yescomplemented
45#1439C22116yescomplemented
46C26C2312yes—
47C23×D4—28no—
48C23×D4—24no—
49C23×D4—44no—
50C23×D4—42no—
51C23×D4—42no—
52C23×D4C2314yes—
53C24×C4C2312yes—
54C24×C4C2312yes—
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C22×C4⋊C4—28no—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—28no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—28no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4D418yes—
78C22×C4⋊C4C2318yes—
79C22×C4⋊C4C2318yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2318yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4D4116yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2318yes—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42D418yescomplemented
111C22×C42C2314yes—
112#396—264no—
113#394—232no—
114#394—264no—
115#388—264no—
116#384—232no—
117C2×#95—28no—
118C2×#95—24no—
119C2×#95—28no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95—42no—
123C2×#95—44no—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2314yes—
131C25—24no—
132C25—24no—
133C25—24no—
134C25—24no—
135C25—24no—
136C25—24no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—24no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—24no—
145C25C2×D418yes—
146C25C2414yes—
147C25C2411yescharacteristic
148C22×D4—216no—
149C22×D4—28no—
150C22×D4—48no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—432no—
154C22×D4—416no—
155C22×D4—416no—
156C22×D4—432no—
157C22×D4—416no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4C2×D414yes—
202C23×C4C2×D418yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D418yes—
206C23×C4C2×D414yes—
207C23×C4C2×D418yes—
208C23×C4#4118yes—
209C23×C4#4114yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C2×C4⋊C4—216no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—416no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—432no—
236C2×C4⋊C4—432no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—432no—
240C2×C4⋊C4C2×D4116yes—
241C2×C22⋊C4—264no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—432no—
260C2×C22⋊C4—432no—
261C2×C22⋊C4—432no—
262C2×C22⋊C4—432no—
263C2×C22⋊C4—432no—
264C2×C22⋊C4C2×D4132yes—
265C2×C42—232no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42C2×D4116yescomplemented
269#95—216no—
270#95—216no—
271#95—216no—
272#95—28no—
273#95—216no—
274#95—44no—
275#95—48no—
276C24—216no—
277C24—216no—
278C24—216no—
279C24—216no—
280C24—216no—
281C24—216no—
282C24—216no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24—48no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—48no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24C22×D418yes—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C22×D414yes—
329C24C2×#4112yes—
330C24C2×#4114yes—
331C24H(5,𝔽2)12yes—
332C24H(5,𝔽2)12yes—
333C24H(5,𝔽2)14yes—
334C242−1+412yes—
335C242−1+412yes—
336C242−1+414yes—
337C24C2511yescharacteristic
338C2×D4—432no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—432no—
342C2×D4—4128no—
343C2×D4—416no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—216no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—216no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4#121116yes—
438C22×C4#121132yes—
439C22×C4C22×D418yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D418yes—
444C22×C4C22×D414yes—
445C22×C4C22×D418yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4118yes—
448C4⋊C4—416no—
449C4⋊C4—432no—
450C4⋊C4—432no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—432no—
454C22⋊C4—264no—
455C22⋊C4—432no—
456C22⋊C4—432no—
457C22⋊C4—432no—
458C22⋊C4—432no—
459C22⋊C4—432no—
460C42—232no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—432no—
469C23—44no—
470C23—44no—
471C23—432no—
472C23—432no—
473C23—432no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—432no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—432no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—432no—
503C23—816no—
504C23—816no—
505C23—816no—
506C23—816no—
507C23—816no—
508C23—816no—
509C23—816no—
510C23—816no—
511C23C2×#12114yes—
512C23C2×#12118yes—
513C23#53414yes—
514C23#53518yes—
515C23#53614yes—
516C23#53618yes—
517C23#53714yes—
518C23#53718yes—
519C23#53714yes—
520C23#53914yes—
521C23#53914yes—
522C23#54218yes—
523C23#54218yes—
524C23D4218yes—
525C23D4214yes—
526C23#54718yes—
527C23#54718yes—
528C23#54818yes—
529C23#55318yes—
530C23#55514yes—
531C23#56014yes—
532C23#56318yes—
533C23#56314yes—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C23×D412yes—
537C23C22×#4111yescharacteristic
538C23C22×#4112yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+411yescharacteristic
544C23C2×2−1+412yes—
545D4—464no—
546C2×C4—232no—
547C2×C4—28no—
548C2×C4—264no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—432no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4—816no—
604C2×C4C2×#12118yes—
605C2×C4C2×#121116yes—
606C22—48no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—864no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22#2416116yes—
623C22#253718yes—
624C22#261818yes—
625C22#2620116yes—
626C22#266918yes—
627C22#2674116yes—
628C22#268018yes—
629C22#2718116yes—
630C22#273118yes—
631C22#279518yes—
632C22#282318yes—
633C22C22×#12112yes—
634C22C22×#12111yescharacteristic
635C22C2×#53411yescharacteristic
636C22C2×#53512yes—
637C22C2×#53612yes—
638C22C2×#53611yescharacteristic
639C22C2×#53711yescharacteristic
640C22C2×#53712yes—
641C22C2×#53711yescharacteristic
642C22C2×#53911yescharacteristic
643C22C2×#53911yescharacteristic
644C22C2×#54212yes—
645C22C2×#54212yes—
646C22C2×D4211yescharacteristic
647C22C2×D4212yes—
648C22C2×#54712yes—
649C22C2×#54712yes—
650C22C2×#54812yes—
651C22C2×#55312yes—
652C22C2×#55511yescharacteristic
653C22C2×#56011yescharacteristic
654C22C2×#56312yes—
655C22C2×#56311yescharacteristic
656C4—216no—
657C4—28no—
658C4—48no—
659C4—44no—
660C4—44no—
661C4—84no—
662C4—84no—
663C4—84no—
664C2—88no—
665C2#16514116yesdirect product factor
666C2C2×#261811yescharacteristic
667C2C2×#262012yes—
668C2C2×#241612yes—
669C2C2×#253711yescharacteristic
670C2C2×#282311yescharacteristic
671C2C2×#266911yescharacteristic
672C2C2×#267412yes—
673C2C2×#268011yescharacteristic
674C2C2×#271812yes—
675C2C2×#273111yescharacteristic
676C2C2×#279511yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#396C2semidirect
4. C22×#388C2semidirect
5. C22×#384C2semidirect
6. #16514C2direct (N × H)
7. C2×#1441C2semidirect
8. C2×#1441C2semidirect
9. C2×#1439C2semidirect
10. C2×#1439C2semidirect
26. C2×#396C22semidirect
32. C2×#388C22semidirect
34. C2×#384C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
110. C22×C42D4semidirect
268. C2×C42C2×D4semidirect
665. C2#16514direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3962
C22×#3942
C22×#3882
C22×#3842
#165142
C2×#14412
C2×#14392