Finite Group Databasebeta

← C2×#27532 #6440959

Subgroups — C2×#27532 #6440959

Order 512.

2500 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3#27532C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic, complemented
6C2×#2073C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#1934C211yesmaximal, characteristic, complemented
9C2×#1926C211yesmaximal, characteristic, complemented
10C2×#1923C211yesmaximal, characteristic, complemented
11C2×#1865C211yesmaximal, characteristic, complemented
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1805C211yesmaximal, characteristic, complemented
14C2×#1788C211yesmaximal, characteristic
15C2×#1788C211yesmaximal, characteristic
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
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#2341—24no—
39#2341C2218yes—
40C2×#390—21no—
41C2×#390—21no—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#387—21no—
48C2×#382C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55C2×C4×C22⋊C4C2211yescharacteristic
56#2077C2218yescomplemented
57#2077C2218yescomplemented
58#2073C2218yescomplemented
59#2065C2218yescomplemented
60#1934C2218yescomplemented
61#1926C2218yescomplemented
62#1923C2218yescomplemented
63#1865C2218yescomplemented
64#1865C2218yescomplemented
65#1805C2218yescomplemented
66#1788C2218yes—
67#1788C2218yes—
68C4×#103C2218yescomplemented
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×C4C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C2×C4×Q8C2314yes—
101C2×C4×Q8C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
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⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4D411yescharacteristic
120C22×C22⋊C4D411yescharacteristic
121C22×C42—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#106—42no—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8—42no—
131C2×C4⋊C8C2314yes—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—24no—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8—42no—
149C2×C22⋊C8—42no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8—42no—
153C2×C4×C8C2314yes—
154#401—28no—
155#397—28no—
156#390—28no—
157#390—28no—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#388—28no—
167#387—28no—
168#382—22no—
169#382C2314yes—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—28no—
174#380C2314yes—
175#380C2314yes—
176#380C2314yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—28no—
180C4×C22⋊C4D414yes—
181C4×C22⋊C4C2314yes—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
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#4114yes—
220C22×Q8#4114yes—
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—22no—
230C22×D4—22no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4#4114yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—28no—
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—44no—
251C23×C4—44no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—48no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
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—24no—
338C2×C42—24no—
339C2×C42—44no—
340C2×C42—44no—
341C2×C42—82no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#106—44no—
347#106—44no—
348C4⋊C8—44no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—28no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357C22⋊C8—44no—
358C22⋊C8—44no—
359C4×C8—44no—
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—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—42no—
380C24—42no—
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—28no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8#126116yes—
402C2×Q8C2×#4114yes—
403C2×Q8C2×#4114yes—
404C2×Q8C2×#4114yes—
405C2×D4—216no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—432no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×#4114yes—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
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—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
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—22no—
455C22×C4—22no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—432no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
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#13514yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4C2511yescharacteristic
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—44no—
508C2×C8—44no—
509C2×C8—44no—
510C2×C8—44no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—48no—
532C4⋊C4—84no—
533C22⋊C4—28no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—48no—
549C42—48no—
550C42—84no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—44no—
560C23—48no—
561C23—44no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—48no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23#52512yes—
580C23#54612yes—
581C23#54612yes—
582C23C2×#13512yes—
583C23#57512yes—
584C23#57712yes—
585C23#57712yes—
586C23C23×D411yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—24no—
590Q8—216no—
591Q8C2×#12618yes—
592D4—24no—
593D4—24no—
594D4—432no—
595D4—42no—
596D4—42no—
597D4—48no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—216no—
603C2×C4—28no—
604C2×C4—22no—
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—22no—
617C2×C4—432no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
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—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4#52512yes—
652C2×C4C2×#12614yes—
653C2×C4#53814yes—
654C2×C4#54612yes—
655C2×C4#54612yes—
656C2×C4#55114yes—
657C2×C4#55114yes—
658C2×C4C2×#13514yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C8—42no—
666C8—42no—
667C8—42no—
668C8—42no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—42no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—88no—
680C22—816no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22#248014yes—
686C22#310214yes—
687C22#312514yes—
688C22#330114yes—
689C22#330414yes—
690C22#331714yes—
691C22#332614yes—
692C22C2×#52511yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#54611yescharacteristic
695C22C22×#13511yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57711yescharacteristic
699C4—28no—
700C4—24no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#248014yes—
712C4C22×#12611yescharacteristic
713C4C2×#53811yescharacteristic
714C4C2×#55111yescharacteristic
715C4C2×#55111yescharacteristic
716C2—82no—
717C2—84no—
718C2#2753218yesdirect product factor
719C2C2×#248011yescharacteristic
720C2C2×#310211yescharacteristic
721C2C2×#312511yescharacteristic
722C2C2×#330111yescharacteristic
723C2C2×#330411yescharacteristic
724C2C2×#331711yescharacteristic
725C2C2×#332611yescharacteristic
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. #27532C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2077C2semidirect
6. C2×#2073C2semidirect
7. C2×#2065C2semidirect
8. C2×#1934C2semidirect
9. C2×#1926C2semidirect
10. C2×#1923C2semidirect
11. C2×#1865C2semidirect
12. C2×#1865C2semidirect
13. C2×#1805C2semidirect
16. C2×C4×#103C2semidirect
35. #2494C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
58. #2073C22semidirect
59. #2065C22semidirect
60. #1934C22semidirect
61. #1926C22semidirect
62. #1923C22semidirect
63. #1865C22semidirect
64. #1865C22semidirect
65. #1805C22semidirect
68. C4×#103C22semidirect
718. C2#27532direct (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×#23412
#275322
C2×#20772
C2×#20732
C2×#20652
C2×#19342
C2×#19262
C2×#19232
C2×#18652
C2×#18052
C2×#17882
C2×C4×#1032