Finite Group Databasebeta

← C2×#27528 #6440947

Subgroups — C2×#27528 #6440947

Order 512.

2608 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3#27528C2116yesmaximal, direct product factor
4C2×#2076C211yesmaximal, characteristic, complemented
5C2×#2075C211yesmaximal, characteristic, complemented
6C2×#2074C211yesmaximal, characteristic, complemented
7C2×#2068C211yesmaximal, characteristic, complemented
8C2×#1933C211yesmaximal, characteristic, complemented
9C2×#1928C211yesmaximal, characteristic, complemented
10C2×#1923C211yesmaximal, characteristic, complemented
11C2×#1862C211yesmaximal, characteristic, complemented
12C2×#1862C211yesmaximal, characteristic, complemented
13C2×#1804C211yesmaximal, characteristic, complemented
14C2×#1788C211yesmaximal, characteristic
15C2×#1788C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2494—24no—
35#2494C2218yescomplemented
36C2×#401—21no—
37C2×#397—21no—
38#2339—24no—
39#2339C2218yes—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#387—21no—
46C2×#386—21no—
47C2×#386—21no—
48C2×#382C2211yescharacteristic
49C2×#381—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95—21no—
56#2076C2218yescomplemented
57#2075C2218yescomplemented
58#2074C2218yescomplemented
59#2068C2218yescomplemented
60#1933C2218yescomplemented
61#1928C2218yescomplemented
62#1923C2218yescomplemented
63#1862C2218yescomplemented
64#1862C2218yescomplemented
65#1804C2218yescomplemented
66#1788C2218yes—
67#1788C2218yes—
68C4×#102C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—22no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—42no—
83C22×#41—42no—
84C22×#41C2314yes—
85C23×Q8C2311yescharacteristic
86C23×D4—21no—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C2×C4×D4C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C2×#107—42no—
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×C4⋊C8—42no—
127C2×C4⋊C8C2314yes—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—42no—
131C2×#103—42no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8—42no—
145C2×C22⋊C8—42no—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8—42no—
149C2×C4×C8C2314yes—
150#401—28no—
151#397—28no—
152#391—22no—
153#391—22no—
154#391C2314yes—
155#391C2314yes—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2314yes—
160#388—28no—
161#387—28no—
162#386—28no—
163#386—28no—
164#382—22no—
165#382C2314yes—
166#381—28no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380C2314yes—
171#380C2314yes—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4D414yes—
176C4×C22⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—28no—
190C2×#95D411yescharacteristic
191C2×#95D411yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25C2×D411yescharacteristic
202C2×#41—216no—
203C2×#41—28no—
204C2×#41—28no—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—44no—
208C2×#41—44no—
209C2×#41—48no—
210C2×#41—48no—
211C2×#41—416no—
212C2×#41—416no—
213C2×#41—416no—
214C2×#41—416no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8#4114yes—
222C22×Q8C2412yes—
223C22×D4—28no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4#4114yes—
232C22×D4#4114yes—
233C22×D4#4114yes—
234C22×D4C2412yes—
235C23×C4—28no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—82no—
259C23×C4—82no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—48no—
289C4×D4—48no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—82no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342#107—44no—
343#107—44no—
344C4⋊C8—44no—
345#103—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—28no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355C4×C8—44no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387#41—416no—
388#41—416no—
389#41—416no—
390#41—464no—
391#41—416no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×#4114yes—
405C2×D4—216no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—432no—
418C2×D4#126116yes—
419C2×D4C2×#4114yes—
420C2×D4C2×#4114yes—
421C2×D4C2×#4114yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—216no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—432no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—48no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—816no—
488C22×C4—816no—
489C22×C4#13514yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4C2511yescharacteristic
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8—44no—
517C2×C8—44no—
518C2×C8—44no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—48no—
538C4⋊C4—84no—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—84no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—216no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—42no—
570C23—48no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23#52512yes—
582C23#54612yes—
583C23#54612yes—
584C23C2×#13512yes—
585C23#57512yes—
586C23#57612yes—
587C23#57812yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—42no—
592Q8—42no—
593Q8—48no—
594D4—24no—
595D4—24no—
596D4—24no—
597D4—216no—
598D4—432no—
599D4C2×#12618yes—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—432no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—816no—
648C2×C4—816no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4#52512yes—
656C2×C4C2×#12614yes—
657C2×C4#53814yes—
658C2×C4#54612yes—
659C2×C4#54612yes—
660C2×C4#55114yes—
661C2×C4#55114yes—
662C2×C4C2×#13514yes—
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C22—28no—
674C22—44no—
675C22—432no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—88no—
689C22#248014yes—
690C22#310214yes—
691C22#312414yes—
692C22#329714yes—
693C22#330214yes—
694C22#331514yes—
695C22#332214yes—
696C22C2×#52511yescharacteristic
697C22C2×#54611yescharacteristic
698C22C2×#54611yescharacteristic
699C22C22×#13511yescharacteristic
700C22C2×#57511yescharacteristic
701C22C2×#57611yescharacteristic
702C22C2×#57811yescharacteristic
703C4—24no—
704C4—42no—
705C4—44no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#248014yes—
716C4C22×#12611yescharacteristic
717C4C2×#53811yescharacteristic
718C4C2×#55111yescharacteristic
719C4C2×#55111yescharacteristic
720C2—48no—
721C2—82no—
722C2#2752818yesdirect product factor
723C2C2×#248011yescharacteristic
724C2C2×#310211yescharacteristic
725C2C2×#312411yescharacteristic
726C2C2×#329711yescharacteristic
727C2C2×#330211yescharacteristic
728C2C2×#331511yescharacteristic
729C2C2×#332211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2494C2semidirect
3. #27528C2direct (N × H)
4. C2×#2076C2semidirect
5. C2×#2075C2semidirect
6. C2×#2074C2semidirect
7. C2×#2068C2semidirect
8. C2×#1933C2semidirect
9. C2×#1928C2semidirect
10. C2×#1923C2semidirect
11. C2×#1862C2semidirect
12. C2×#1862C2semidirect
13. C2×#1804C2semidirect
16. C2×C4×#102C2semidirect
35. #2494C22semidirect
56. #2076C22semidirect
57. #2075C22semidirect
58. #2074C22semidirect
59. #2068C22semidirect
60. #1933C22semidirect
61. #1928C22semidirect
62. #1923C22semidirect
63. #1862C22semidirect
64. #1862C22semidirect
65. #1804C22semidirect
68. C4×#102C22semidirect
722. C2#27528direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#23392
#275282
C2×#20762
C2×#20752
C2×#20742
C2×#20682
C2×#19332
C2×#19282
C2×#19232
C2×#18622
C2×#18042
C2×#17882
C2×C4×#1022