Finite Group Databasebeta

← C2×#15951 #609883

Subgroups — C2×#15951 #609883

Order 512.

8978 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×C22⋊Q8C212yesmaximal, complemented
2C22×#386C212yesmaximal
3C22×#385C212yesmaximal, complemented
4#15951C2116yesmaximal, direct product factor
5C2×#1446C211yesmaximal, characteristic, complemented
6C2×#1444C211yesmaximal, characteristic, complemented
7C2×#1441C212yesmaximal, complemented
8C2×#1439C212yesmaximal, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1438C211yesmaximal, characteristic
11C24×Q8C2211yescharacteristic
12C24×D4C2211yescharacteristic
13C25×C4C2212yes—
14C22×C22⋊Q8—216no—
15C22×C22⋊Q8—216no—
16C22×C22⋊Q8—216no—
17C22×C22⋊Q8C22116yescomplemented
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C2×#386—216no—
28C2×#386C22116yes—
29C2×#385—216no—
30C2×#385C22116yescomplemented
31C22×#95C2211yescharacteristic
32C22×#95C2212yes—
33C22×#95C2212yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40#1446C2218yescomplemented
41#1444C2218yescomplemented
42#1441C22116yescomplemented
43#1439C22116yescomplemented
44#1439C22116yescomplemented
45#1438C2218yes—
46C26C2312yes—
47C23×Q8—28no—
48C23×Q8—28no—
49C23×Q8—42no—
50C23×Q8—42no—
51C23×Q8D418yescomplemented
52C23×Q8C2314yes—
53C23×D4—28no—
54C23×D4—28no—
55C23×D4—42no—
56C23×D4—42no—
57C23×D4Q818yescomplemented
58C23×D4C2314yes—
59C24×C4—28no—
60C24×C4—28no—
61C24×C4—28no—
62C24×C4—28no—
63C24×C4—28no—
64C24×C4—44no—
65C24×C4—44no—
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2318yes—
74C24×C4C2312yes—
75C2×C22⋊Q8—2256no—
76C2×C22⋊Q8—264no—
77C2×C22⋊Q8—264no—
78C2×C22⋊Q8—264no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4D4116yescomplemented
90C22×C4⋊C4C2318yes—
91C22×C4⋊C4C2318yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2318yes—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4D4116yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114#386—264no—
115#385—264no—
116C2×#95—28no—
117C2×#95—28no—
118C2×#95—28no—
119C2×#95—28no—
120C2×#95—24no—
121C2×#95—44no—
122C2×#95—42no—
123C2×#95—42no—
124C2×#95—42no—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2318yes—
133C2×#95C2314yes—
134C25—24no—
135C25—24no—
136C25—24no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—24no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25C2×D418yes—
147C25C2×Q818yes—
148C25#4118yes—
149C25C2414yes—
150C25C2411yescharacteristic
151C22×Q8—264no—
152C22×Q8—216no—
153C22×Q8—216no—
154C22×Q8—416no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8—416no—
158C22×Q8—432no—
159C22×Q8—416no—
160C22×Q8C2×D4116yescomplemented
161C22×D4—264no—
162C22×D4—216no—
163C22×D4—216no—
164C22×D4—416no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—416no—
168C22×D4—432no—
169C22×D4—416no—
170C22×D4C2×Q8116yescomplemented
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—264no—
176C23×C4—24no—
177C23×C4—24no—
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—216no—
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—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—216no—
208C23×C4—216no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—216no—
217C23×C4—216no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4—432no—
221C23×C4—432no—
222C23×C4—432no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4—432no—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D418yes—
229C23×C4C2×D418yes—
230C23×C4C2×D418yes—
231C23×C4C2×D418yes—
232C23×C4C2×D418yes—
233C23×C4C2×Q814yes—
234C23×C4#4114yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C22⋊Q8—2512no—
244C2×C4⋊C4—264no—
245C2×C4⋊C4—264no—
246C2×C4⋊C4—264no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4C2×D4132yescomplemented
259C2×C22⋊C4—264no—
260C2×C22⋊C4—264no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—264no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—432no—
277C2×C22⋊C4—432no—
278C2×C22⋊C4C2×D4132yes—
279#95—216no—
280#95—216no—
281#95—216no—
282#95—216no—
283#95—28no—
284#95—44no—
285#95—48no—
286#95—44no—
287#95—44no—
288C24—216no—
289C24—216no—
290C24—216no—
291C24—216no—
292C24—216no—
293C24—216no—
294C24—216no—
295C24—216no—
296C24—216no—
297C24—216no—
298C24—216no—
299C24—216no—
300C24—216no—
301C24—216no—
302C24—216no—
303C24—216no—
304C24—216no—
305C24—216no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24C22⋊Q8132yescomplemented
335C24C22×D418yes—
336C24C22×D412yes—
337C24C22×D414yes—
338C24C22×D414yes—
339C24C22×Q818yes—
340C24C22×Q812yes—
341C24C2×#4118yes—
342C24C2×#4114yes—
343C24C2×#4112yes—
344C24H(5,𝔽2)12yes—
345C24H(5,𝔽2)12yes—
346C24H(5,𝔽2)12yes—
347C242−1+412yes—
348C242−1+414yes—
349C24C2511yescharacteristic
350C2×Q8—264no—
351C2×Q8—416no—
352C2×Q8—4128no—
353C2×Q8—416no—
354C2×Q8—432no—
355C2×Q8—416no—
356C2×D4—264no—
357C2×D4—416no—
358C2×D4—4128no—
359C2×D4—416no—
360C2×D4—432no—
361C2×D4—416no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—264no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
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—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—432no—
426C22×C4—432no—
427C22×C4—432no—
428C22×C4—432no—
429C22×C4—4256no—
430C22×C4—432no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—432no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4#120132yes—
461C22×C4C22×D418yes—
462C22×C4C22×D418yes—
463C22×C4C22×D414yes—
464C22×C4C22×D418yes—
465C22×C4C22×D418yes—
466C22×C4C22×D418yes—
467C22×C4C22×D418yes—
468C22×C4C22×Q814yes—
469C22×C4C2×#4114yes—
470C4⋊C4—264no—
471C4⋊C4—264no—
472C4⋊C4—264no—
473C4⋊C4—416no—
474C22⋊C4—264no—
475C22⋊C4—264no—
476C22⋊C4—264no—
477C22⋊C4—432no—
478C22⋊C4—432no—
479C23—28no—
480C23—264no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—264no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—264no—
500C23—432no—
501C23—432no—
502C23—432no—
503C23—44no—
504C23—432no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—432no—
509C23—44no—
510C23—44no—
511C23—432no—
512C23—432no—
513C23—432no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—432no—
523C23—432no—
524C23—432no—
525C23—432no—
526C23—44no—
527C23—44no—
528C23C2×#12018yes—
529C23C2×C22⋊Q818yes—
530C23C2×C22⋊Q8116yescomplemented
531C23#52518yes—
532C23#53414yes—
533C23#53618yes—
534C23#53718yes—
535C23#53914yes—
536C23#53918yes—
537C23#54314yes—
538C23D4214yes—
539C23D4218yes—
540C23D4214yes—
541C23#54618yes—
542C23#54718yes—
543C23#54818yes—
544C23#54818yes—
545C23D4×Q814yes—
546C23D4×Q818yes—
547C23#55014yes—
548C23#55114yes—
549C23#55414yes—
550C23#56214yes—
551C23#56314yes—
552C23C23×D411yescharacteristic
553C23C23×D412yes—
554C23C23×D412yes—
555C23C23×Q811yescharacteristic
556C23C22×#4112yes—
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×2−1+411yescharacteristic
562C23C2×2−1+412yes—
563Q8—464no—
564D4—464no—
565C2×C4—264no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—4128no—
588C2×C4—432no—
589C2×C4—416no—
590C2×C4—432no—
591C2×C4—432no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—416no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—416no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—816no—
623C2×C4C2×#120116yes—
624C22—216no—
625C22—216no—
626C22—216no—
627C22—4128no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22#240318yes—
641C22#2407116yes—
642C22#2409116yes—
643C22#243718yes—
644C22#2449116yes—
645C22#246318yes—
646C22#268018yes—
647C22#2681116yes—
648C22#268318yes—
649C22#272818yes—
650C22#273218yes—
651C22C22×#12012yes—
652C22C22×C22⋊Q812yes—
653C22C2×#52512yes—
654C22C2×#53411yescharacteristic
655C22C2×#53612yes—
656C22C2×#53712yes—
657C22C2×#53912yes—
658C22C2×#53911yescharacteristic
659C22C2×#54311yescharacteristic
660C22C2×D4211yescharacteristic
661C22C2×D4212yes—
662C22C2×D4211yescharacteristic
663C22C2×#54612yes—
664C22C2×#54712yes—
665C22C2×#54812yes—
666C22C2×#54812yes—
667C22C2×D4×Q812yes—
668C22C2×D4×Q811yescharacteristic
669C22C2×#55011yescharacteristic
670C22C2×#55111yescharacteristic
671C22C2×#55411yescharacteristic
672C22C2×#56211yescharacteristic
673C22C2×#56311yescharacteristic
674C4—216no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—48no—
681C4—84no—
682C2—416no—
683C2#15951116yesdirect product factor
684C2C2×#240311yescharacteristic
685C2C2×#243711yescharacteristic
686C2C2×#244912yes—
687C2C2×#246311yescharacteristic
688C2C2×#240912yes—
689C2C2×#240712yes—
690C2C2×#268011yescharacteristic
691C2C2×#268112yes—
692C2C2×#268311yescharacteristic
693C2C2×#272811yescharacteristic
694C2C2×#273211yescharacteristic
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. C23×C22⋊Q8C2semidirect
3. C22×#385C2semidirect
4. #15951C2direct (N × H)
5. C2×#1446C2semidirect
6. C2×#1444C2semidirect
7. C2×#1441C2semidirect
8. C2×#1439C2semidirect
9. C2×#1439C2semidirect
17. C22×C22⋊Q8C22semidirect
30. C2×#385C22semidirect
40. #1446C22semidirect
41. #1444C22semidirect
42. #1441C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
51. C23×Q8D4semidirect
57. C23×D4Q8semidirect
89. C22×C4⋊C4D4semidirect
160. C22×Q8C2×D4semidirect
170. C22×D4C2×Q8semidirect
258. C2×C4⋊C4C2×D4semidirect
334. C24C22⋊Q8semidirect
530. C23C2×C22⋊Q8semidirect
683. C2#15951direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C22×#3862
C22×#3852
#159512
C2×#14462
C2×#14442
C2×#14412
C2×#14392
C2×#14382