Finite Group Databasebeta

← C2×#15265 #608790

Subgroups — C2×#15265 #608790

Order 512.

7818 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C23×C22⋊Q8C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic, complemented
4C22×#393C211yesmaximal, characteristic
5C22×#385C212yesmaximal
6#15265C2116yesmaximal, direct product factor
7C2×#1446C211yesmaximal, characteristic, complemented
8C2×#1444C212yesmaximal, complemented
9C2×#1441C212yesmaximal, complemented
10C2×#1439C212yesmaximal
11C2×#1437C211yesmaximal, characteristic
12C2×#1436C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C24×Q8C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C22×C4⋊Q8—28no—
17C22×C4⋊Q8—216no—
18C22×C4⋊Q8C2218yescomplemented
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8—28no—
21C22×C22⋊Q8—28no—
22C22×C22⋊Q8C2218yes—
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30C23×C22⋊C4C2212yes—
31C23×C22⋊C4C2211yescharacteristic
32C23×C22⋊C4C2212yes—
33C23×C42C2211yescharacteristic
34C2×#395—28no—
35C2×#395C2218yescomplemented
36C2×#393—28no—
37C2×#393C2218yes—
38C2×#385—216no—
39C2×#385C22116yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2212yes—
49#1446C2218yescomplemented
50#1444C22116yescomplemented
51#1441C22116yescomplemented
52#1439C22116yes—
53#1437C2218yes—
54#1436C2218yescomplemented
55C26C2311yescharacteristic
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—28no—
59C23×Q8—28no—
60C23×Q8—24no—
61C23×Q8—24no—
62C23×Q8—24no—
63C23×Q8—44no—
64C23×Q8—44no—
65C23×Q8D418yes—
66C23×Q8C2314yes—
67C23×Q8C2314yes—
68C24×C4—24no—
69C24×C4—24no—
70C24×C4—24no—
71C24×C4—28no—
72C24×C4—44no—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C4⋊Q8—2128no—
85C2×C4⋊Q8—232no—
86C2×C4⋊Q8—264no—
87C2×C22⋊Q8—2128no—
88C2×C22⋊Q8—232no—
89C2×C22⋊Q8—232no—
90C2×C22⋊Q8—232no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4D4116yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2318yes—
112C22×C4⋊C4C2318yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2318yes—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—28no—
121C22×C22⋊C4—28no—
122C22×C22⋊C4—28no—
123C22×C22⋊C4—28no—
124C22×C22⋊C4—44no—
125C22×C22⋊C4D418yes—
126C22×C22⋊C4Q8116yescomplemented
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2318yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2318yes—
131C22×C42—24no—
132C22×C42—28no—
133C22×C42C2314yes—
134#395—232no—
135#393—232no—
136#385—264no—
137C2×#95—28no—
138C2×#95—24no—
139C2×#95—28no—
140C2×#95—24no—
141C2×#95—28no—
142C2×#95—28no—
143C2×#95—28no—
144C2×#95—28no—
145C2×#95—44no—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C25—22no—
156C25—22no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—24no—
162C25C2×D414yes—
163C25C2×Q818yes—
164C25C2412yes—
165C25C2411yescharacteristic
166C22×Q8—232no—
167C22×Q8—232no—
168C22×Q8—232no—
169C22×Q8—28no—
170C22×Q8—28no—
171C22×Q8—216no—
172C22×Q8—216no—
173C22×Q8—28no—
174C22×Q8—28no—
175C22×Q8—28no—
176C22×Q8—416no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8—432no—
180C22×Q8—432no—
181C22×Q8—432no—
182C22×Q8—432no—
183C22×Q8—416no—
184C22×Q8—416no—
185C22×Q8C2×D4116yes—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—232no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
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—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—216no—
235C23×C4—48no—
236C23×C4—432no—
237C23×C4—432no—
238C23×C4—416no—
239C23×C4—416no—
240C23×C4C2×D414yes—
241C23×C4C2×D418yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D418yes—
250C23×C4C2×D418yes—
251C23×C4C2×Q818yes—
252C23×C4C2×Q818yes—
253C23×C4#4118yes—
254C23×C4C2414yes—
255C23×C4C2414yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2414yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2414yes—
264C4⋊Q8—2256no—
265C22⋊Q8—2256no—
266C2×C4⋊C4—232no—
267C2×C4⋊C4—232no—
268C2×C4⋊C4—264no—
269C2×C4⋊C4—232no—
270C2×C4⋊C4—264no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—216no—
282C2×C4⋊C4—216no—
283C2×C4⋊C4—216no—
284C2×C4⋊C4—216no—
285C2×C4⋊C4—216no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—432no—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D4132yes—
290C2×C22⋊C4—232no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—232no—
293C2×C22⋊C4—264no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—432no—
304C2×C22⋊C4C2×D4116yes—
305C2×C22⋊C4C2×Q8132yescomplemented
306C2×C42—232no—
307C2×C42—28no—
308C2×C42—216no—
309#95—216no—
310#95—216no—
311#95—216no—
312#95—28no—
313#95—28no—
314#95—216no—
315#95—216no—
316#95—216no—
317#95—48no—
318C24—28no—
319C24—216no—
320C24—28no—
321C24—216no—
322C24—216no—
323C24—216no—
324C24—216no—
325C24—216no—
326C24—216no—
327C24—216no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24—48no—
339C24—44no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24C4⋊Q8116yescomplemented
346C24C22×D414yes—
347C24C22×D412yes—
348C24C22×D414yes—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C22×Q818yes—
352C24C22×Q814yes—
353C24C2×#4114yes—
354C24H(5,𝔽2)14yes—
355C242−1+414yes—
356C242−1+414yes—
357C24C2511yescharacteristic
358C2×Q8—232no—
359C2×Q8—232no—
360C2×Q8—232no—
361C2×Q8—416no—
362C2×Q8—4128no—
363C2×Q8—432no—
364C2×Q8—432no—
365C2×Q8—432no—
366C2×Q8—432no—
367C2×Q8—4128no—
368C2×Q8—416no—
369C2×Q8—416no—
370C22×C4—216no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—232no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
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—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—432no—
446C22×C4—432no—
447C22×C4—4128no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4#120116yes—
476C22×C4#120116yes—
477C22×C4C22⋊Q8132yescomplemented
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D418yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D418yes—
489C22×C4C22×Q818yes—
490C22×C4C22×Q818yes—
491C22×C4C2×#4118yes—
492C4⋊C4—232no—
493C4⋊C4—232no—
494C4⋊C4—264no—
495C4⋊C4—232no—
496C4⋊C4—264no—
497C4⋊C4—432no—
498C22⋊C4—232no—
499C22⋊C4—232no—
500C22⋊C4—264no—
501C22⋊C4—432no—
502C42—232no—
503C23—24no—
504C23—28no—
505C23—24no—
506C23—28no—
507C23—232no—
508C23—28no—
509C23—264no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—416no—
516C23—42no—
517C23—432no—
518C23—432no—
519C23—42no—
520C23—432no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—416no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—432no—
530C23—432no—
531C23—44no—
532C23C2×#12014yes—
533C23C2×#12014yes—
534C23C2×C22⋊Q818yes—
535C23C2×C4⋊Q814yes—
536C23C2×C4⋊Q818yescomplemented
537C23#53214yes—
538C23#53414yes—
539C23#53618yes—
540C23#53614yes—
541C23#53614yes—
542C23#54218yes—
543C23#54418yes—
544C23D4218yes—
545C23D4218yes—
546C23#54718yes—
547C23#54818yes—
548C23#54818yes—
549C23D4×Q818yes—
550C23D4×Q818yes—
551C23D4×Q818yes—
552C23#55418yes—
553C23#56318yes—
554C23C23×D411yescharacteristic
555C23C23×D412yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C23×Q812yes—
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+412yes—
562C23C2×2−1+412yes—
563Q8—464no—
564Q8—464no—
565C2×C4—232no—
566C2×C4—232no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—264no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—464no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—432no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—432no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—816no—
631C2×C4C2×#12018yes—
632C2×C4C2×#12018yes—
633C2×C4C2×C22⋊Q8116yescomplemented
634C22—216no—
635C22—28no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—464no—
643C22—48no—
644C22#240318yes—
645C22#2409116yes—
646C22#243218yes—
647C22#243418yes—
648C22#2440116yes—
649C22#2452116yes—
650C22#2674116yes—
651C22#2677116yes—
652C22#2729116yes—
653C22C22×#12011yescharacteristic
654C22C22×#12011yescharacteristic
655C22C22×C22⋊Q812yes—
656C22C22×C4⋊Q811yescharacteristic
657C22C2×#53211yescharacteristic
658C22C2×#53411yescharacteristic
659C22C2×#53612yes—
660C22C2×#53611yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#54212yes—
663C22C2×#54412yes—
664C22C2×D4212yes—
665C22C2×D4212yes—
666C22C2×#54712yes—
667C22C2×#54812yes—
668C22C2×#54812yes—
669C22C2×D4×Q812yes—
670C22C2×D4×Q812yes—
671C22C2×D4×Q812yes—
672C22C2×#55412yes—
673C22C2×#56312yes—
674C4—28no—
675C4—28no—
676C4—216no—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—48no—
681C4—44no—
682C4—44no—
683C4—84no—
684C2—48no—
685C2#15265116yesdirect product factor
686C2C2×#240311yescharacteristic
687C2C2×#244012yes—
688C2C2×#245212yes—
689C2C2×#240912yes—
690C2C2×#243411yescharacteristic
691C2C2×#243211yescharacteristic
692C2C2×#267412yes—
693C2C2×#267712yes—
694C2C2×#272912yes—
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. C23×C4⋊Q8C2semidirect
3. C22×#395C2semidirect
6. #15265C2direct (N × H)
7. C2×#1446C2semidirect
8. C2×#1444C2semidirect
9. C2×#1441C2semidirect
12. C2×#1436C2semidirect
18. C22×C4⋊Q8C22semidirect
35. C2×#395C22semidirect
49. #1446C22semidirect
50. #1444C22semidirect
51. #1441C22semidirect
54. #1436C22semidirect
126. C22×C22⋊C4Q8semidirect
305. C2×C22⋊C4C2×Q8semidirect
345. C24C4⋊Q8semidirect
477. C22×C4C22⋊Q8semidirect
536. C23C2×C4⋊Q8semidirect
633. C2×C4C2×C22⋊Q8semidirect
685. C2#15265direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C23×C22⋊Q82
C22×#3952
C22×#3932
C22×#3852
#152652
C2×#14462
C2×#14442
C2×#14412
C2×#14392
C2×#14372
C2×#14362