Finite Group Databasebeta

← C2×#27349 #6449449

Subgroups — C2×#27349 #6449449

Order 512.

2082 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#27349C2116yesmaximal, direct product factor
4C2×#2076C211yesmaximal, characteristic
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2057C211yesmaximal, characteristic, complemented
7C2×#2036C211yesmaximal, characteristic
8C2×#1988C211yesmaximal, characteristic, complemented
9C2×#1935C211yesmaximal, characteristic, complemented
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1927C211yesmaximal, characteristic, complemented
12C2×#1924C211yesmaximal, characteristic
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1872C211yesmaximal, characteristic
15C2×#1853C211yesmaximal, characteristic
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117—21no—
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2492—24no—
34#2492C2218yes—
35C2×#400—21no—
36C2×#393—21no—
37#2344—24no—
38#2344C2218yes—
39C2×#391—21no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
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#2076C2218yes—
55#2066C2218yescomplemented
56#2057C2218yescomplemented
57#2036C2218yes—
58#1988C2218yescomplemented
59#1935C2218yescomplemented
60#1928C2218yescomplemented
61#1927C2218yescomplemented
62#1924C2218yes—
63#1920C2218yescomplemented
64#1872C2218yes—
65#1853C2218yes—
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×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8C2314yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8C2314yes—
93C2×#117—28no—
94C2×#117—48no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C2×#107—24no—
122C2×#107—42no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106—24no—
126C2×#106—24no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8—42no—
130C2×C4⋊C8C2314yes—
131C2×#103—24no—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—42no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×C22⋊C8—24no—
140C2×C22⋊C8—42no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97—24no—
144C2×#97C2314yes—
145#400—28no—
146#393—28no—
147#391—22no—
148#391—22no—
149#391—28no—
150#391C2314yes—
151#391C2314yes—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#388—28no—
157#387—28no—
158#387—28no—
159#381—28no—
160#380—22no—
161#380—22no—
162#380C2314yes—
163#380C2314yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C22⋊C4D414yes—
171C4×C22⋊C4C2314yes—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25C2×D411yescharacteristic
196C22×Q8—21no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—82no—
207C22×Q8C2412yes—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8#4114yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4⋊Q8—28no—
251C4⋊Q8—44no—
252C4⋊Q8—48no—
253C22⋊Q8—28no—
254C22⋊Q8—44no—
255C22⋊Q8—48no—
256#117—432no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
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—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—82no—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—82no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—28no—
323#107—44no—
324#106—28no—
325#106—28no—
326C4⋊C8—44no—
327#103—28no—
328#103—28no—
329#103—28no—
330#103—44no—
331C22⋊C8—28no—
332C22⋊C8—44no—
333#97—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95C2×D414yes—
346#95C2×D414yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24#13714yes—
354C24C22×D412yes—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—48no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—84no—
372C2×Q8—816no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—48no—
409C22×C4—432no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4AGL(1,ℤ/8)14yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C22×C4C2511yescharacteristic
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—44no—
463C2×C8—44no—
464C2×C8—44no—
465C2×C8C2×#4114yes—
466C4⋊C4—216no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4C2×#4114yes—
490C4⋊C4C2×#4114yes—
491C4⋊C4C2×#4114yes—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—84no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—48no—
504C42—48no—
505C42—84no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23#52512yes—
520C23#54612yes—
521C23#54712yes—
522C23C2×AGL(1,ℤ/8)12yes—
523C23C2×#13714yes—
524C23C2×#13712yes—
525C23#57712yes—
526C23#57812yes—
527C23C23×D411yescharacteristic
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—816no—
533Q8—84no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—432no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4#52512yes—
592C2×C4#52914yes—
593C2×C4#54214yes—
594C2×C4#54612yes—
595C2×C4#54712yes—
596C2×C4#55514yes—
597C2×C4#55614yes—
598C2×C4C2×AGL(1,ℤ/8)14yes—
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C8—24no—
606C8—42no—
607C8—42no—
608C8—42no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—416no—
617C22#247414yes—
618C22#311014yes—
619C22#312114yes—
620C22#330614yes—
621C22#331214yes—
622C22#334414yes—
623C22#335314yes—
624C22C2×#52511yescharacteristic
625C22C2×#54611yescharacteristic
626C22C2×#54711yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×#13711yescharacteristic
629C22C2×#57711yescharacteristic
630C22C2×#57811yescharacteristic
631C4—24no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#247414yes—
645C4C2×#52911yescharacteristic
646C4C2×#54211yescharacteristic
647C4C2×#55511yescharacteristic
648C4C2×#55611yescharacteristic
649C2—44no—
650C2#2734918yesdirect product factor
651C2C2×#247411yescharacteristic
652C2C2×#311011yescharacteristic
653C2C2×#312111yescharacteristic
654C2C2×#330611yescharacteristic
655C2C2×#331211yescharacteristic
656C2C2×#334411yescharacteristic
657C2C2×#335311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #27349C2direct (N × H)
5. C2×#2066C2semidirect
6. C2×#2057C2semidirect
8. C2×#1988C2semidirect
9. C2×#1935C2semidirect
10. C2×#1928C2semidirect
11. C2×#1927C2semidirect
13. C2×#1920C2semidirect
16. C2×#1776C2semidirect
55. #2066C22semidirect
56. #2057C22semidirect
58. #1988C22semidirect
59. #1935C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
63. #1920C22semidirect
66. #1776C22semidirect
650. C2#27349direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#23442
#273492
C2×#20762
C2×#20662
C2×#20572
C2×#20362
C2×#19882
C2×#19352
C2×#19282
C2×#19272
C2×#19242
C2×#19202
C2×#18722
C2×#18532
C2×#17762