Finite Group Databasebeta

← C2×#27504 #6451200

Subgroups — C2×#27504 #6451200

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#27504C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic, complemented
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2054C211yesmaximal, characteristic
8C2×#1972C211yesmaximal, characteristic
9C2×#1949C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic, complemented
12C2×#1925C211yesmaximal, characteristic, complemented
13C2×#1917C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2513—24no—
34#2513C2218yescomplemented
35C2×#397—21no—
36C2×#397—21no—
37#2354—24no—
38#2354C2218yescomplemented
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#388—21no—
47C2×#388—21no—
48C2×#388—21no—
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2093C2218yescomplemented
57#2092C2218yescomplemented
58#2084C2218yescomplemented
59#2054C2218yes—
60#1972C2218yes—
61#1949C2218yescomplemented
62#1926C2218yescomplemented
63#1926C2218yescomplemented
64#1925C2218yescomplemented
65#1917C2218yescomplemented
66#1879C2218yes—
67#1864C2218yescomplemented
68#1776C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×D8—24no—
81C22×D8—24no—
82C22×D8—42no—
83C22×D8C2314yes—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—22no—
91C2×C4×D4C2314yes—
92C2×#117—28no—
93C2×#117—48no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C2×#107—24no—
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106—24no—
127C2×#106—42no—
128C2×#106—42no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97—24no—
147C2×#97C2314yes—
148#397—28no—
149#397—28no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2314yes—
157#390C2314yes—
158#390C2314yes—
159#390C2314yes—
160#389—22no—
161#389C2314yes—
162#388—28no—
163#388—28no—
164#388—28no—
165#388—28no—
166#384—22no—
167#384C2314yes—
168#382—22no—
169#382C2314yes—
170#380—28no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D414yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95D411yescharacteristic
186C2×#95D411yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4#4114yes—
211C22×D4C2412yes—
212C22×D4C2412yes—
213C23×C4—28no—
214C23×C4—21no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2411yescharacteristic
230C2×D8—28no—
231C2×D8—28no—
232C2×D8—44no—
233C2×D8—416no—
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#4114yes—
245C22×C8#4114yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4×D4—216no—
251C4×D4—28no—
252C4×D4—28no—
253#117—432no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4—82no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—82no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328#107—28no—
329#107—44no—
330#106—28no—
331#106—44no—
332#106—44no—
333C4⋊C8—28no—
334#102—28no—
335#102—28no—
336#102—28no—
337#102—28no—
338#102—44no—
339#102—44no—
340#97—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95C2×D414yes—
348#95C2×D414yes—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24C22×D411yescharacteristic
366C2×D4—24no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4C2×#4114yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—22no—
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—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—432no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
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×C4—42no—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C4C2511yescharacteristic
452D8—416no—
453C2×C8—28no—
454C2×C8—28no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8C2×#4114yes—
468C2×C8C2×#4114yes—
469C4⋊C4—24no—
470C4⋊C4—28no—
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—22no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4C2×#4114yes—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—48no—
513C42—48no—
514C42—84no—
515C23—24no—
516C23—48no—
517C23—48no—
518C23—42no—
519C23—42no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
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—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23#52512yes—
544C23#54712yes—
545C23#55012yes—
546C23C2×AGL(1,ℤ/8)12yes—
547C23#57512yes—
548C23#57612yes—
549C23#57712yes—
550C23C23×D411yescharacteristic
551D4—24no—
552D4—24no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—48no—
560D4—84no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
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—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—432no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—44no—
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#52512yes—
610C2×C4#52914yes—
611C2×C4#54214yes—
612C2×C4#54712yes—
613C2×C4#55012yes—
614C2×C4#55114yes—
615C2×C4#55214yes—
616C2×C4C2×AGL(1,ℤ/8)14yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C8—24no—
624C8—24no—
625C8—42no—
626C8—42no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—816no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—168no—
643C22#247914yes—
644C22#311414yes—
645C22#311914yes—
646C22#334714yes—
647C22#335414yes—
648C22#339914yes—
649C22#340114yes—
650C22C2×#52511yescharacteristic
651C22C2×#54711yescharacteristic
652C22C2×#55011yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57611yescharacteristic
656C22C2×#57711yescharacteristic
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#247914yes—
670C4C2×#52911yescharacteristic
671C4C2×#54211yescharacteristic
672C4C2×#55111yescharacteristic
673C4C2×#55211yescharacteristic
674C2—84no—
675C2—162no—
676C2#2750418yesdirect product factor
677C2C2×#247911yescharacteristic
678C2C2×#311411yescharacteristic
679C2C2×#311911yescharacteristic
680C2C2×#334711yescharacteristic
681C2C2×#335411yescharacteristic
682C2C2×#339911yescharacteristic
683C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. C2×#2513C2semidirect
2. C2×#2354C2semidirect
3. #27504C2direct (N × H)
4. C2×#2093C2semidirect
5. C2×#2092C2semidirect
6. C2×#2084C2semidirect
9. C2×#1949C2semidirect
10. C2×#1926C2semidirect
11. C2×#1926C2semidirect
12. C2×#1925C2semidirect
13. C2×#1917C2semidirect
15. C2×#1864C2semidirect
16. C2×#1776C2semidirect
34. #2513C22semidirect
38. #2354C22semidirect
56. #2093C22semidirect
57. #2092C22semidirect
58. #2084C22semidirect
61. #1949C22semidirect
62. #1926C22semidirect
63. #1926C22semidirect
64. #1925C22semidirect
65. #1917C22semidirect
67. #1864C22semidirect
68. #1776C22semidirect
676. C2#27504direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#23542
#275042
C2×#20932
C2×#20922
C2×#20842
C2×#20542
C2×#19722
C2×#19492
C2×#19262
C2×#19252
C2×#19172
C2×#18792
C2×#18642
C2×#17762