Finite Group Databasebeta

← C2×#14840 #607799

Subgroups — C2×#14840 #607799

Order 512.

5246 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic
4C22×C4×C22⋊C4C211yesmaximal, characteristic
5#14840C2116yesmaximal, direct product factor
6C2×#1445C211yesmaximal, characteristic, complemented
7C2×#1444C211yesmaximal, characteristic, complemented
8C2×#1441C211yesmaximal, characteristic, complemented
9C2×#1439C212yesmaximal
10C2×#1439C212yesmaximal
11C2×#1436C212yesmaximal, complemented
12C2×C4⋊#95C212yesmaximal, complemented
13C25×C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2212yes—
25C2×#398—28no—
26C2×#398C2218yescomplemented
27C2×#385—28no—
28C2×#385C2218yes—
29C2×#380—28no—
30C2×#380C2218yes—
31C2×C4×C22⋊C4—28no—
32C2×C4×C22⋊C4C2218yes—
33C22×#95C2212yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44#1445C2218yescomplemented
45#1444C2218yescomplemented
46#1441C2218yescomplemented
47#1439C22116yes—
48#1439C22116yes—
49#1436C22116yescomplemented
50C4⋊#95C22116yescomplemented
51C26C2311yescharacteristic
52C24×C4—24no—
53C24×C4—24no—
54C24×C4—28no—
55C24×C4—44no—
56C24×C4D418yes—
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2314yes—
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C22×C4⋊C4—28no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4—44no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2318yes—
88C22×C4⋊C4C2318yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4Q8116yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2314yes—
105C22×C42—28no—
106C22×C42—28no—
107C22×C42—44no—
108C22×C42C2318yes—
109#398—232no—
110#385—232no—
111#380—232no—
112C4×C22⋊C4—232no—
113C2×#95—28no—
114C2×#95—24no—
115C2×#95—28no—
116C2×#95—24no—
117C2×#95—24no—
118C2×#95—28no—
119C2×#95—28no—
120C2×#95—44no—
121C2×#95—44no—
122C2×#95—44no—
123C2×#95D418yes—
124C2×#95C2318yes—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C25—22no—
136C25—24no—
137C25—22no—
138C25—22no—
139C25—22no—
140C25—24no—
141C25—22no—
142C25—22no—
143C25—24no—
144C25C2×D414yes—
145C25C2×Q818yes—
146C25C2412yes—
147C25C2411yescharacteristic
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—232no—
174C23×C4—232no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—216no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
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—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—416no—
203C23×C4—48no—
204C23×C4—432no—
205C23×C4—432no—
206C23×C4C2×D4116yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D418yes—
210C23×C4C2×D414yes—
211C23×C4C2×Q818yes—
212C23×C4C2×Q818yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4#4118yes—
217C23×C4#4118yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—432no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—432no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C22⋊C4—232no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—264no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—416no—
263C2×C22⋊C4—432no—
264C2×C22⋊C4C2×Q8132yes—
265C2×C42—216no—
266C2×C42—216no—
267C2×C42—48no—
268C2×C42—432no—
269#95—28no—
270#95—216no—
271#95—28no—
272#95—28no—
273#95—216no—
274#95—216no—
275#95—216no—
276#95—48no—
277#95—48no—
278#95—48no—
279#95C2×D4116yes—
280C24—28no—
281C24—216no—
282C24—28no—
283C24—28no—
284C24—28no—
285C24—216no—
286C24—28no—
287C24—28no—
288C24—216no—
289C24—216no—
290C24—216no—
291C24—216no—
292C24—22no—
293C24—24no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—48no—
305C24—48no—
306C24—44no—
307C24—44no—
308C24—48no—
309C24—44no—
310C24—44no—
311C24C4⋊Q8116yes—
312C24C22×D414yes—
313C24C22×D412yes—
314C24C22×D412yes—
315C24C22×Q818yes—
316C24C22×Q814yes—
317C24C2×#4112yes—
318C24C2×#4114yes—
319C24C2×#4114yes—
320C24C2×#4112yes—
321C24C2×#4112yes—
322C24H(5,𝔽2)14yes—
323C24H(5,𝔽2)12yes—
324C24H(5,𝔽2)12yes—
325C24C2511yescharacteristic
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—216no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—216no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—216no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—232no—
374C22×C4—232no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
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—22no—
399C22×C4—416no—
400C22×C4—4128no—
401C22×C4—432no—
402C22×C4—432no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
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—48no—
438C22×C4#121116yes—
439C22×C4C22⋊Q8132yes—
440C22×C4#123116yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D418yes—
444C22×C4C22×D414yes—
445C22×C4C22×Q818yes—
446C22×C4C22×Q818yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4118yes—
452C4⋊C4—432no—
453C4⋊C4—416no—
454C4⋊C4—432no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C22⋊C4—232no—
459C22⋊C4—264no—
460C22⋊C4—416no—
461C22⋊C4—432no—
462C42—432no—
463C23—232no—
464C23—232no—
465C23—24no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—232no—
478C23—432no—
479C23—416no—
480C23—416no—
481C23—416no—
482C23—416no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—432no—
494C23—432no—
495C23—432no—
496C23C2×#12114yes—
497C23C2×C22⋊Q818yes—
498C23C2×#12314yes—
499C23#52514yes—
500C23#52918yes—
501C23#52918yes—
502C23C2×C4⋊Q818yes—
503C23C2×C4⋊Q814yes—
504C23#53214yes—
505C23#53318yes—
506C23#53414yes—
507C23#53514yes—
508C23#53818yes—
509C23#54018yes—
510C23#54618yes—
511C23#54614yes—
512C23#55014yes—
513C23#55118yes—
514C23#55314yes—
515C23#55418yes—
516C23#55418yes—
517C23#55418yes—
518C23#55418yes—
519C23#55514yes—
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×Q812yes—
523C23C22×#4111yescharacteristic
524C23C22×#4112yes—
525C23C22×#4112yes—
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—232no—
534C2×C4—232no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—264no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—464no—
557C2×C4—416no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—416no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—432no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
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—88no—
596C2×C4—816no—
597C2×C4—88no—
598C2×C4—816no—
599C2×C4C2×#12118yes—
600C2×C4C2×C22⋊Q8116yes—
601C2×C4C2×#12318yes—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—44no—
606C22—48no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—48no—
612C22—44no—
613C22—464no—
614C22#239518yes—
615C22#241218yes—
616C22#241418yes—
617C22#2450116yes—
618C22#2454116yes—
619C22#250218yes—
620C22#2522116yes—
621C22#2574116yes—
622C22#258318yes—
623C22#266118yes—
624C22#285718yes—
625C22C22×#12111yescharacteristic
626C22C22×C22⋊Q812yes—
627C22C22×#12311yescharacteristic
628C22C2×#52511yescharacteristic
629C22C2×#52912yes—
630C22C2×#52912yes—
631C22C22×C4⋊Q811yescharacteristic
632C22C2×#53211yescharacteristic
633C22C2×#53312yes—
634C22C2×#53411yescharacteristic
635C22C2×#53511yescharacteristic
636C22C2×#53812yes—
637C22C2×#54012yes—
638C22C2×#54612yes—
639C22C2×#54611yescharacteristic
640C22C2×#55011yescharacteristic
641C22C2×#55112yes—
642C22C2×#55311yescharacteristic
643C22C2×#55412yes—
644C22C2×#55412yes—
645C22C2×#55412yes—
646C22C2×#55412yes—
647C22C2×#55511yescharacteristic
648C4—28no—
649C4—28no—
650C4—216no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—84no—
658C2—48no—
659C2#14840116yesdirect product factor
660C2C2×#239511yescharacteristic
661C2C2×#266111yescharacteristic
662C2C2×#241211yescharacteristic
663C2C2×#245012yes—
664C2C2×#245412yes—
665C2C2×#241411yescharacteristic
666C2C2×#250211yescharacteristic
667C2C2×#252212yes—
668C2C2×#258311yescharacteristic
669C2C2×#257412yes—
670C2C2×#285711yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
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×#398C2semidirect
5. #14840C2direct (N × H)
6. C2×#1445C2semidirect
7. C2×#1444C2semidirect
8. C2×#1441C2semidirect
11. C2×#1436C2semidirect
12. C2×C4⋊#95C2semidirect
26. C2×#398C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1441C22semidirect
49. #1436C22semidirect
50. C4⋊#95C22semidirect
659. C2#14840direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3852
C22×#3802
C22×C4×C22⋊C42
#148402
C2×#14452
C2×#14442
C2×#14412
C2×#14392
C2×#14362
C2×C4⋊#952