Finite Group Databasebeta

← C2×#27347 #6449447

Subgroups — C2×#27347 #6449447

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2491C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic, complemented
3#27347C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2067C211yesmaximal, characteristic, complemented
6C2×#2058C211yesmaximal, characteristic, complemented
7C2×#2034C211yesmaximal, characteristic
8C2×#1987C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic, complemented
11C2×#1925C211yesmaximal, characteristic, complemented
12C2×#1924C211yesmaximal, characteristic, complemented
13C2×#1921C211yesmaximal, characteristic, complemented
14C2×#1874C211yesmaximal, characteristic
15C2×#1852C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124—21no—
19C22×#121C2211yescharacteristic
20C22×#117—21no—
21C23×C4⋊C4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2491—24no—
35#2491C2218yes—
36C2×#394—21no—
37C2×#393—21no—
38#2344—24no—
39#2344C2218yescomplemented
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×#381—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2077C2218yescomplemented
55#2067C2218yescomplemented
56#2058C2218yescomplemented
57#2034C2218yes—
58#1987C2218yescomplemented
59#1936C2218yescomplemented
60#1926C2218yescomplemented
61#1925C2218yescomplemented
62#1924C2218yescomplemented
63#1921C2218yescomplemented
64#1874C2218yes—
65#1852C2218yescomplemented
66#1776C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8C2314yes—
90C2×#124—28no—
91C2×#124—48no—
92C2×#121—22no—
93C2×#121—24no—
94C2×#121—42no—
95C2×#121C2314yes—
96C2×#117—28no—
97C2×#117—48no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×#107—24no—
128C2×#107—24no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106—24no—
132C2×#106—42no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—42no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8—24no—
146C2×C22⋊C8—42no—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×#97—24no—
150C2×#97C2314yes—
151#394—28no—
152#393—28no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2314yes—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#388—28no—
162#387—28no—
163#387—28no—
164#381—28no—
165#380—22no—
166#380—22no—
167#380C2314yes—
168#380C2314yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
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×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—21no—
197C25C2×D411yescharacteristic
198C22×Q8—24no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8—82no—
202C22×D4—24no—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—82no—
213C22×D4C2412yes—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2412yes—
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—432no—
259#121—28no—
260#121—44no—
261#121—48no—
262#117—432no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4#4114yes—
293C2×C4⋊C4#4114yes—
294C2×C4⋊C4#4114yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
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×C22⋊C4—82no—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334#107—28no—
335#107—28no—
336#106—28no—
337#106—44no—
338C4⋊C8—44no—
339#102—28no—
340#102—28no—
341#102—28no—
342#102—44no—
343C22⋊C8—28no—
344C22⋊C8—44no—
345#97—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95C2×D414yes—
354#95C2×D414yes—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24AGL(1,ℤ/8)14yes—
370C24C22×D412yes—
371C24C22×D411yescharacteristic
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—84no—
375C2×Q8—816no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—84no—
392C2×D4—816no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
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—48no—
428C22×C4—432no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
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—42no—
451C22×C4—42no—
452C22×C4AGL(1,ℤ/8)14yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466C2×C8—28no—
467C2×C8—22no—
468C2×C8—22no—
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—44no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8C2×#4114yes—
482C4⋊C4—216no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
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—22no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4C2×#4114yes—
505C4⋊C4C2×#4114yes—
506C4⋊C4C2×#4114yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—48no—
523C42—48no—
524C42—48no—
525C23—28no—
526C23—28no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23#52512yes—
553C23#54612yes—
554C23#54712yes—
555C23C2×AGL(1,ℤ/8)14yes—
556C23C2×AGL(1,ℤ/8)12yes—
557C23C2×AGL(1,ℤ/8)12yes—
558C23#57712yes—
559C23#57812yes—
560C23C23×D411yescharacteristic
561Q8—816no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—816no—
567D4—84no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—432no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
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—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#52512yes—
622C2×C4#52914yes—
623C2×C4#54214yes—
624C2×C4#54612yes—
625C2×C4#54712yes—
626C2×C4#55514yes—
627C2×C4#55614yes—
628C2×C4C2×AGL(1,ℤ/8)14yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C8—24no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—416no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—168no—
655C22#247414yes—
656C22#310914yes—
657C22#312114yes—
658C22#330814yes—
659C22#331014yes—
660C22#334314yes—
661C22#335414yes—
662C22C2×#52511yescharacteristic
663C22C2×#54611yescharacteristic
664C22C2×#54711yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C2×#57711yescharacteristic
668C22C2×#57811yescharacteristic
669C4—24no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#247414yes—
682C4C2×#52911yescharacteristic
683C4C2×#54211yescharacteristic
684C4C2×#55511yescharacteristic
685C4C2×#55611yescharacteristic
686C2—44no—
687C2—162no—
688C2#2734718yesdirect product factor
689C2C2×#247411yescharacteristic
690C2C2×#310911yescharacteristic
691C2C2×#312111yescharacteristic
692C2C2×#331011yescharacteristic
693C2C2×#330811yescharacteristic
694C2C2×#334311yescharacteristic
695C2C2×#335411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C2×#2344C2semidirect
3. #27347C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2067C2semidirect
6. C2×#2058C2semidirect
8. C2×#1987C2semidirect
9. C2×#1936C2semidirect
10. C2×#1926C2semidirect
11. C2×#1925C2semidirect
12. C2×#1924C2semidirect
13. C2×#1921C2semidirect
15. C2×#1852C2semidirect
16. C2×#1776C2semidirect
39. #2344C22semidirect
54. #2077C22semidirect
55. #2067C22semidirect
56. #2058C22semidirect
58. #1987C22semidirect
59. #1936C22semidirect
60. #1926C22semidirect
61. #1925C22semidirect
62. #1924C22semidirect
63. #1921C22semidirect
65. #1852C22semidirect
66. #1776C22semidirect
688. C2#27347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24912
C2×#23442
#273472
C2×#20772
C2×#20672
C2×#20582
C2×#20342
C2×#19872
C2×#19362
C2×#19262
C2×#19252
C2×#19242
C2×#19212
C2×#18742
C2×#18522
C2×#17762