Finite Group Databasebeta

← C2×#14401 #520705

Subgroups — C2×#14401 #520705

Order 512.

7298 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#390C211yesmaximal, characteristic
2C22×#386C211yesmaximal, characteristic
3C22×#386C211yesmaximal, characteristic
4C22×#385C212yesmaximal, complemented
5C22×#380C212yesmaximal, complemented
6#14401C2116yesmaximal, direct product factor
7C2×#1444C212yesmaximal, complemented
8C2×#1439C212yesmaximal, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1436C212yesmaximal, complemented
11C24×D4C2211yescharacteristic
12C25×C4C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2212yes—
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#390—28no—
26C2×#390C2218yes—
27C2×#386—28no—
28C2×#386C4116yescomplemented
29C2×#386C2218yes—
30C2×#386C2218yes—
31C2×#385—216no—
32C2×#385C22116yescomplemented
33C2×#380—216no—
34C2×#380C22116yescomplemented
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42#1444C22116yescomplemented
43#1439C22116yescomplemented
44#1439C22116yescomplemented
45#1436C22116yescomplemented
46C26C2311yescharacteristic
47C26C2311yescharacteristic
48C23×D4—42no—
49C23×D4—42no—
50C23×D4—42no—
51C23×D4—42no—
52C23×D4C2×C418yes—
53C23×D4D418yes—
54C23×D4Q818yes—
55C23×D4C2314yes—
56C24×C4—28no—
57C24×C4—28no—
58C24×C4—24no—
59C24×C4—28no—
60C24×C4—42no—
61C24×C4—42no—
62C24×C4—44no—
63C24×C4—42no—
64C24×C4—42no—
65C24×C4C2×C418yescomplemented
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2314yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2318yes—
88C22×C4⋊C4C2318yes—
89C22×C4⋊C4C2318yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—42no—
101C22×C22⋊C4—42no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—42no—
106C22×C22⋊C4—42no—
107C22×C22⋊C4C2×C4116yes—
108C22×C22⋊C4C2×C418yescomplemented
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42—24no—
116C22×C42—42no—
117C22×C42—42no—
118C22×C42C2314yes—
119#390—232no—
120#386—232no—
121#386C2×C4164yescomplemented
122#385—264no—
123#380—264no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—44no—
128C2×#95—44no—
129C2×#95C2×C4116yescomplemented
130C2×#95D4116yescomplemented
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C25—24no—
139C25—24no—
140C25—24no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—22no—
145C25—22no—
146C25—22no—
147C25—22no—
148C25—22no—
149C25—22no—
150C25—22no—
151C25—22no—
152C25—24no—
153C25—24no—
154C25C22×C414yes—
155C25C22×C414yes—
156C25C2×D414yes—
157C25C2×D414yes—
158C25C2×Q814yes—
159C25C2×Q814yes—
160C25C2412yes—
161C25C2412yes—
162C25C2411yescharacteristic
163C22×D4—416no—
164C22×D4—416no—
165C22×D4—416no—
166C22×D4—416no—
167C22×D4—416no—
168C22×D4—416no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4C4⋊C4164yescomplemented
174C22×D4C22×C4116yes—
175C22×D4C2×D4116yes—
176C22×D4C2×Q8116yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
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×C4—24no—
202C23×C4—24no—
203C23×C4—264no—
204C23×C4—216no—
205C23×C4—216no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—216no—
222C23×C4—416no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—48no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—416no—
229C23×C4—432no—
230C23×C4—416no—
231C23×C4—416no—
232C23×C4—432no—
233C23×C4—432no—
234C23×C4—416no—
235C23×C4—416no—
236C23×C4C22×C418yes—
237C23×C4C22×C414yes—
238C23×C4C22×C418yes—
239C23×C4C22×C4116yescomplemented
240C23×C4C2×D418yes—
241C23×C4C2×D418yes—
242C23×C4C2×D414yes—
243C23×C4C2×D418yes—
244C23×C4C2×Q814yes—
245C23×C4#4118yes—
246C23×C4C2414yes—
247C23×C4C2414yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2414yes—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—432no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—432no—
269C2×C22⋊C4—264no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—416no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—432no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—432no—
290C2×C22⋊C4—416no—
291C2×C22⋊C4C22×C4132yes—
292C2×C22⋊C4C22×C4116yescomplemented
293C2×C42—28no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—416no—
297#95—216no—
298#95—216no—
299#95—216no—
300#95—48no—
301#95—48no—
302#95C22×C4132yescomplemented
303#95C2×D4132yescomplemented
304C24—216no—
305C24—216no—
306C24—28no—
307C24—28no—
308C24—28no—
309C24—28no—
310C24—216no—
311C24—216no—
312C24—28no—
313C24—28no—
314C24—28no—
315C24—28no—
316C24—28no—
317C24—28no—
318C24—28no—
319C24—28no—
320C24—216no—
321C24—216no—
322C24—216no—
323C24—216no—
324C24—216no—
325C24—216no—
326C24—216no—
327C24—216no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—48no—
352C24—44no—
353C24—44no—
354C24—48no—
355C24—48no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24—44no—
360C24—48no—
361C24—48no—
362C24—44no—
363C24—44no—
364C24C2×C4⋊C4116yes—
365C24C2×C4⋊C4116yes—
366C24C23×C414yes—
367C24C23×C414yes—
368C24C23×C412yes—
369C24C22×D414yes—
370C24C22×D414yes—
371C24C22×D412yes—
372C24C22×D414yes—
373C24C22×Q814yes—
374C24C22×Q814yes—
375C24C22×Q812yes—
376C24C2×#4114yes—
377C24H(5,𝔽2)12yes—
378C24H(5,𝔽2)12yes—
379C24H(5,𝔽2)12yes—
380C24H(5,𝔽2)12yes—
381C24H(5,𝔽2)14yes—
382C242−1+414yes—
383C24C2511yescharacteristic
384C2×D4—4128no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4C2×C4⋊C4164yescomplemented
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—264no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—416no—
455C22×C4—4128no—
456C22×C4—416no—
457C22×C4—432no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—416no—
463C22×C4—4128no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4C2×C4⋊C4116yes—
492C22×C4C4×D4132yes—
493C22×C4C23×C418yes—
494C22×C4C23×C414yes—
495C22×C4C23×C418yes—
496C22×C4C22×D418yes—
497C22×C4C22×D418yes—
498C22×C4C22×D414yes—
499C22×C4C22×D418yes—
500C22×C4C22×Q814yes—
501C22×C4C2×#4118yes—
502C4⋊C4—432no—
503C4⋊C4—432no—
504C4⋊C4—416no—
505C4⋊C4—432no—
506C22⋊C4—264no—
507C22⋊C4—416no—
508C22⋊C4—432no—
509C22⋊C4—432no—
510C22⋊C4—432no—
511C22⋊C4—416no—
512C42—416no—
513C23—232no—
514C23—232no—
515C23—264no—
516C23—264no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—432no—
542C23—432no—
543C23—416no—
544C23—416no—
545C23—416no—
546C23—416no—
547C23—416no—
548C23—416no—
549C23—432no—
550C23—432no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—416no—
556C23—416no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—432no—
562C23—432no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—416no—
570C23—416no—
571C23—416no—
572C23—416no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23C22×C4⋊C418yes—
580C23C22×C4⋊C418yes—
581C23C22×C4⋊C414yes—
582C23C2×C4×D418yes—
583C23#51814yes—
584C23#51814yes—
585C23#52018yes—
586C23#53414yes—
587C23#53414yes—
588C23#53518yes—
589C23#53718yes—
590C23#54018yes—
591C23#54218yes—
592C23#54314yes—
593C23#54314yes—
594C23D4218yes—
595C23D4214yes—
596C23#54618yes—
597C23#54718yes—
598C23D4×Q818yes—
599C23#55114yes—
600C23#55418yes—
601C23#56014yes—
602C23#56218yes—
603C23#56314yes—
604C23C24×C411yescharacteristic
605C23C23×D411yescharacteristic
606C23C23×D412yes—
607C23C23×Q811yescharacteristic
608C23C22×#4112yes—
609C23C2×H(5,𝔽2)11yescharacteristic
610C23C2×H(5,𝔽2)11yescharacteristic
611C23C2×H(5,𝔽2)11yescharacteristic
612C23C2×H(5,𝔽2)11yescharacteristic
613C23C2×H(5,𝔽2)12yes—
614C23C2×2−1+412yes—
615D4—464no—
616C2×C4—232no—
617C2×C4—264no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—464no—
640C2×C4—464no—
641C2×C4—432no—
642C2×C4—432no—
643C2×C4—432no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—816no—
671C2×C4—816no—
672C2×C4C22×C4⋊C418yes—
673C2×C4C2×C4×D4116yes—
674C22—28no—
675C22—28no—
676C22—216no—
677C22—216no—
678C22—48no—
679C22—48no—
680C22—44no—
681C22—44no—
682C22—464no—
683C22—464no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—48no—
689C22—48no—
690C22—44no—
691C22—44no—
692C22—48no—
693C22—48no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22#231518yes—
699C22#2324116yes—
700C22D4×C4⋊C4116yes—
701C22#235818yes—
702C22#237618yes—
703C22#266918yes—
704C22#2671116yes—
705C22#268318yes—
706C22#2692116yes—
707C22#271718yes—
708C22#274818yes—
709C22C23×C4⋊C411yescharacteristic
710C22C22×C4×D412yes—
711C22C2×#51811yescharacteristic
712C22C2×#51811yescharacteristic
713C22C2×#52012yes—
714C22C2×#53411yescharacteristic
715C22C2×#53411yescharacteristic
716C22C2×#53512yes—
717C22C2×#53712yes—
718C22C2×#54012yes—
719C22C2×#54212yes—
720C22C2×#54311yescharacteristic
721C22C2×#54311yescharacteristic
722C22C2×D4212yes—
723C22C2×D4211yescharacteristic
724C22C2×#54612yes—
725C22C2×#54712yes—
726C22C2×D4×Q812yes—
727C22C2×#55111yescharacteristic
728C22C2×#55412yes—
729C22C2×#56011yescharacteristic
730C22C2×#56212yes—
731C22C2×#56311yescharacteristic
732C4—28no—
733C4—216no—
734C4—48no—
735C4—48no—
736C4—48no—
737C4—84no—
738C4—84no—
739C2—48no—
740C2—48no—
741C2#14401116yesdirect product factor
742C2C2×#231511yescharacteristic
743C2C2×#232412yes—
744C2C2×D4×C4⋊C412yes—
745C2C2×#235811yescharacteristic
746C2C2×#237611yescharacteristic
747C2C2×#266911yescharacteristic
748C2C2×#267112yes—
749C2C2×#268311yescharacteristic
750C2C2×#269212yes—
751C2C2×#271711yescharacteristic
752C2C2×#274811yescharacteristic
subgroupG/H
center, socleC25C24
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#385C2semidirect
5. C22×#380C2semidirect
6. #14401C2direct (N × H)
7. C2×#1444C2semidirect
8. C2×#1439C2semidirect
9. C2×#1439C2semidirect
10. C2×#1436C2semidirect
28. C2×#386C4semidirect
32. C2×#385C22semidirect
34. C2×#380C22semidirect
42. #1444C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1436C22semidirect
65. C24×C4C2×C4semidirect
108. C22×C22⋊C4C2×C4semidirect
121. #386C2×C4semidirect
129. C2×#95C2×C4semidirect
130. C2×#95D4semidirect
173. C22×D4C4⋊C4semidirect
239. C23×C4C22×C4semidirect
292. C2×C22⋊C4C22×C4semidirect
302. #95C22×C4semidirect
303. #95C2×D4semidirect
391. C2×D4C2×C4⋊C4semidirect
741. C2#14401direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3902
C22×#3862
C22×#3852
C22×#3802
#144012
C2×#14442
C2×#14392
C2×#14362