Finite Group Databasebeta

← C2×#27752 #6458540

Subgroups — C2×#27752 #6458540

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic, complemented
3#27752C2116yesmaximal, direct product factor
4C2×#2074C211yesmaximal, characteristic, complemented
5C2×#2071C211yesmaximal, characteristic
6C2×#2043C211yesmaximal, characteristic, complemented
7C2×#2042C211yesmaximal, characteristic
8C2×#1983C211yesmaximal, characteristic
9C2×#1958C211yesmaximal, characteristic
10C2×#1958C211yesmaximal, characteristic
11C2×#1956C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic
13C2×#1933C211yesmaximal, characteristic, complemented
14C2×#1805C211yesmaximal, characteristic, complemented
15C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2570—24no—
38#2570C2218yescomplemented
39#2545—24no—
40#2545C2218yes—
41C2×#400—21no—
42C2×#398—21no—
43C2×#396—21no—
44C2×#396—21no—
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#388—21no—
51C2×#387—21no—
52C2×#387—21no—
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2074C2218yescomplemented
59#2071C2218yes—
60#2043C2218yescomplemented
61#2042C2218yes—
62#1983C2218yes—
63#1958C2218yes—
64#1958C2218yes—
65#1956C2218yescomplemented
66#1934C2218yes—
67#1933C2218yescomplemented
68#1805C2218yescomplemented
69C8×C4⋊C4C2218yescomplemented
70C4×#103C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×#124—24no—
89C2×#124—22no—
90C2×#124—22no—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124—42no—
94C2×#124C2314yes—
95C2×#124C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×C4⋊C8—24no—
126C2×C4⋊C8C2314yes—
127C2×#103—24no—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—42no—
131C2×#103—42no—
132C2×#103—42no—
133C2×#103—42no—
134C2×#103—42no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102—24no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102—42no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C4×C8—24no—
152C2×C4×C8—42no—
153C2×C4×C8—42no—
154C2×C4×C8C2314yes—
155C2×C4×C8C2314yes—
156C2×C4×C8C2314yes—
157#400—28no—
158#398—28no—
159#396—28no—
160#396—28no—
161#391—22no—
162#391C2314yes—
163#390—22no—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#388—28no—
170#388—28no—
171#387—28no—
172#387—28no—
173#384—22no—
174#384C2314yes—
175#382—22no—
176#382—22no—
177#382C2314yes—
178#382C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D414yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95D411yescharacteristic
193C2×#95D411yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8#4114yes—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8#4114yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258#124—28no—
259#124—28no—
260#124—28no—
261#124—44no—
262#124—48no—
263#124—48no—
264C4×Q8—28no—
265C4×Q8—28no—
266C4×Q8—48no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—82no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42#4114yes—
329C2×C42#4114yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C4⋊C8—28no—
334#103—28no—
335#103—28no—
336#103—28no—
337#103—44no—
338#103—44no—
339#103—44no—
340#103—44no—
341#103—44no—
342#102—28no—
343#102—44no—
344#102—44no—
345#102—44no—
346C4×C8—28no—
347C4×C8—44no—
348C4×C8—44no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8C2×#4114yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4#12414yes—
451C22×C4#12414yes—
452C22×C4#13514yes—
453C22×C4#13514yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4C2511yescharacteristic
468C2×C8—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—44no—
483C2×C8C2×#4114yes—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—48no—
521C42—84no—
522C42#12418yes—
523C42C2×#4114yes—
524C42C2×#4114yes—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23C2×#12412yes—
540C23#54612yes—
541C23#55012yes—
542C23C2×#13512yes—
543C23C2×#13512yes—
544C23#57712yes—
545C23#57812yes—
546C23C23×D411yescharacteristic
547Q8—24no—
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—48no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—84no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—84no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4C2×#12414yes—
611C2×C4C2×#12414yes—
612C2×C4C2×#12412yes—
613C2×C4#52914yes—
614C2×C4#54014yes—
615C2×C4#54612yes—
616C2×C4#55012yes—
617C2×C4#55314yes—
618C2×C4#55314yes—
619C2×C4C2×#13514yes—
620C2×C4C2×#13514yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C8—24no—
628C8—42no—
629C8—42no—
630C8—42no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—168no—
639C22#249014yes—
640C22#313114yes—
641C22#313714yes—
642C22#330014yes—
643C22#330114yes—
644C22#339314yes—
645C22#339414yes—
646C22C22×#12411yescharacteristic
647C22C2×#54611yescharacteristic
648C22C2×#55011yescharacteristic
649C22C22×#13511yescharacteristic
650C22C22×#13511yescharacteristic
651C22C2×#57711yescharacteristic
652C22C2×#57811yescharacteristic
653C4—24no—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4#249014yes—
667C4C2×#52911yescharacteristic
668C4C2×#54011yescharacteristic
669C4C2×#55311yescharacteristic
670C4C2×#55311yescharacteristic
671C2—162no—
672C2#2775218yesdirect product factor
673C2C2×#249011yescharacteristic
674C2C2×#313111yescharacteristic
675C2C2×#313711yescharacteristic
676C2C2×#330011yescharacteristic
677C2C2×#330111yescharacteristic
678C2C2×#339311yescharacteristic
679C2C2×#339411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2570C2semidirect
3. #27752C2direct (N × H)
4. C2×#2074C2semidirect
6. C2×#2043C2semidirect
11. C2×#1956C2semidirect
13. C2×#1933C2semidirect
14. C2×#1805C2semidirect
15. C2×C8×C4⋊C4C2semidirect
16. C2×C4×#103C2semidirect
38. #2570C22semidirect
58. #2074C22semidirect
60. #2043C22semidirect
65. #1956C22semidirect
67. #1933C22semidirect
68. #1805C22semidirect
69. C8×C4⋊C4C22semidirect
70. C4×#103C22semidirect
672. C2#27752direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25452
C2×#25702
#277522
C2×#20742
C2×#20712
C2×#20432
C2×#20422
C2×#19832
C2×#19582
C2×#19562
C2×#19342
C2×#19332
C2×#18052
C2×C8×C4⋊C42
C2×C4×#1032