Finite Group Databasebeta

← C2×#15065 #608154

Subgroups — C2×#15065 #608154

Order 512.

5098 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×#400C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic, complemented
4C22×#393C211yesmaximal, characteristic
5C22×#391C211yesmaximal, characteristic, complemented
6C22×#388C211yesmaximal, characteristic
7#15065C2116yesmaximal, direct product factor
8C2×#1445C212yesmaximal, complemented
9C2×#1439C212yesmaximal
10C2×#1437C212yesmaximal
11C2×C4⋊#95C212yesmaximal, complemented
12C2×C4×#95C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C42C2212yes—
24C23×C42C2211yescharacteristic
25C2×#400—28no—
26C2×#400C2218yescomplemented
27C2×#399—28no—
28C2×#399C2218yes—
29C2×#395—28no—
30C2×#395C2218yescomplemented
31C2×#393—28no—
32C2×#393C2218yes—
33C2×#391—28no—
34C2×#391C2218yescomplemented
35C2×#388—28no—
36C2×#388C2218yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C22116yescomplemented
49#1439C22116yes—
50#1437C22116yes—
51C4⋊#95C22116yescomplemented
52C4×#95C2218yescomplemented
53C26C2311yescharacteristic
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—28no—
58C23×Q8—44no—
59C23×Q8C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2312yes—
63C24×C4C2312yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2318yes—
85C22×C4⋊C4C2318yes—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4Q818yescomplemented
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42—28no—
106C22×C42—28no—
107C22×C42—44no—
108C22×C42—44no—
109C22×C42D418yes—
110C22×C42C2318yes—
111C22×C42C2314yes—
112#400—232no—
113#399—232no—
114#395—232no—
115#393—232no—
116#391—232no—
117#388—232no—
118C2×#95—28no—
119C2×#95—28no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95D418yes—
128C2×#95D418yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C25—22no—
141C25—22no—
142C25—22no—
143C25—22no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—22no—
149C25—22no—
150C25C2×Q814yes—
151C25C2412yes—
152C25C2411yescharacteristic
153C22×Q8—232no—
154C22×Q8—28no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—216no—
158C22×Q8—48no—
159C22×Q8—432no—
160C22×Q8—432no—
161C22×Q8—416no—
162C22×Q8—416no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
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—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D418yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×Q814yes—
218C23×C4C2×Q814yes—
219C23×C4#4118yes—
220C23×C4#4114yes—
221C23×C4#4118yes—
222C23×C4#4118yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—432no—
246C2×C4⋊C4—432no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—416no—
249C2×C22⋊C4—232no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—432no—
264C2×C22⋊C4—416no—
265C2×C22⋊C4—432no—
266C2×C22⋊C4—416no—
267C2×C22⋊C4C2×Q8116yescomplemented
268C2×C42—216no—
269C2×C42—216no—
270C2×C42—416no—
271C2×C42—48no—
272C2×C42—48no—
273C2×C42—432no—
274C2×C42C2×D4116yes—
275#95—28no—
276#95—28no—
277#95—216no—
278#95—216no—
279#95—216no—
280#95—216no—
281#95—216no—
282#95—216no—
283#95—216no—
284#95C2×D4116yes—
285#95C2×D4116yes—
286C24—28no—
287C24—216no—
288C24—28no—
289C24—28no—
290C24—216no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—44no—
302C24—48no—
303C24—48no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—48no—
308C24—44no—
309C24—44no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—44no—
315C24—44no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24C22×D412yes—
320C24C22×D412yes—
321C24C22×D412yes—
322C24C22×Q814yes—
323C24C22×Q812yes—
324C24C2×#4112yes—
325C24C2×#4114yes—
326C24C2×#4114yes—
327C24C2×#4114yes—
328C24H(5,𝔽2)14yes—
329C242−1+412yes—
330C242−1+412yes—
331C24C2511yescharacteristic
332C2×Q8—232no—
333C2×Q8—432no—
334C2×Q8—432no—
335C2×Q8—416no—
336C2×Q8—4128no—
337C2×Q8—416no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4C22⋊Q8116yes—
443C22×C4#124132yes—
444C22×C4#127116yes—
445C22×C4C22×D418yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D418yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C2×#4118yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4118yes—
458C4⋊C4—432no—
459C4⋊C4—432no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C22⋊C4—232no—
463C22⋊C4—432no—
464C22⋊C4—416no—
465C22⋊C4—432no—
466C22⋊C4—416no—
467C42—416no—
468C42—432no—
469C23—24no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—416no—
475C23—416no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—432no—
495C23—416no—
496C23—432no—
497C23—88no—
498C23—88no—
499C23—88no—
500C23—816no—
501C23—816no—
502C23—88no—
503C23C2×C22⋊Q814yes—
504C23#52514yes—
505C23C2×#12418yes—
506C23#52918yes—
507C23C2×#12714yes—
508C23#53214yes—
509C23#53214yes—
510C23#53218yes—
511C23#53318yes—
512C23#53414yes—
513C23#53614yes—
514C23#53918yes—
515C23#54218yes—
516C23#54618yes—
517C23#54618yes—
518C23#54714yes—
519C23#54714yes—
520C23D4×Q814yes—
521C23D4×Q814yes—
522C23#55118yes—
523C23#55218yes—
524C23#55418yes—
525C23#55618yes—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C23×D411yescharacteristic
529C23C23×Q811yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4112yes—
532C23C22×#4112yes—
533C23C22×#4112yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+411yescharacteristic
537Q8—464no—
538C2×C4—232no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—264no—
554C2×C4—232no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—416no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—432no—
583C2×C4—432no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4C2×C22⋊Q818yes—
606C2×C4C2×#124116yes—
607C2×C4C2×#12718yes—
608C22—48no—
609C22—44no—
610C22—44no—
611C22—48no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—84no—
618C22—84no—
619C22—832no—
620C22#239618yes—
621C22#2488116yes—
622C22#2492116yes—
623C22#2521116yes—
624C22#253918yes—
625C22#254218yes—
626C22#2556116yes—
627C22#256418yes—
628C22#260418yes—
629C22#266318yes—
630C22#285818yes—
631C22C22×C22⋊Q811yescharacteristic
632C22C2×#52511yescharacteristic
633C22C22×#12412yes—
634C22C2×#52912yes—
635C22C22×#12711yescharacteristic
636C22C2×#53212yes—
637C22C2×#53211yescharacteristic
638C22C2×#53211yescharacteristic
639C22C2×#53312yes—
640C22C2×#53411yescharacteristic
641C22C2×#53611yescharacteristic
642C22C2×#53912yes—
643C22C2×#54212yes—
644C22C2×#54612yes—
645C22C2×#54612yes—
646C22C2×#54711yescharacteristic
647C22C2×#54711yescharacteristic
648C22C2×D4×Q811yescharacteristic
649C22C2×D4×Q811yescharacteristic
650C22C2×#55112yes—
651C22C2×#55212yes—
652C22C2×#55412yes—
653C22C2×#55612yes—
654C4—28no—
655C4—28no—
656C4—216no—
657C4—48no—
658C4—44no—
659C4—44no—
660C4—48no—
661C4—82no—
662C4—84no—
663C4—82no—
664C2—84no—
665C2#15065116yesdirect product factor
666C2C2×#239611yescharacteristic
667C2C2×#266311yescharacteristic
668C2C2×#248812yes—
669C2C2×#252112yes—
670C2C2×#249212yes—
671C2C2×#253911yescharacteristic
672C2C2×#255612yes—
673C2C2×#256411yescharacteristic
674C2C2×#260411yescharacteristic
675C2C2×#254211yescharacteristic
676C2C2×#285811yescharacteristic
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×#400C2semidirect
3. C22×#395C2semidirect
5. C22×#391C2semidirect
7. #15065C2direct (N × H)
8. C2×#1445C2semidirect
11. C2×C4⋊#95C2semidirect
12. C2×C4×#95C2semidirect
26. C2×#400C22semidirect
30. C2×#395C22semidirect
34. C2×#391C22semidirect
48. #1445C22semidirect
51. C4⋊#95C22semidirect
52. C4×#95C22semidirect
99. C22×C22⋊C4Q8semidirect
267. C2×C22⋊C4C2×Q8semidirect
665. C2#15065direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4002
C22×#3992
C22×#3952
C22×#3932
C22×#3912
C22×#3882
#150652
C2×#14452
C2×#14392
C2×#14372
C2×C4⋊#952
C2×C4×#952