Finite Group Databasebeta

← C2×#27535 #6440969

Subgroups — C2×#27535 #6440969

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3#27535C2116yesmaximal, direct product factor
4C2×#2075C211yesmaximal, characteristic
5C2×#2075C211yesmaximal, characteristic
6C2×#2072C211yesmaximal, characteristic, complemented
7C2×#2068C211yesmaximal, characteristic, complemented
8C2×#1935C211yesmaximal, characteristic, complemented
9C2×#1928C211yesmaximal, characteristic, complemented
10C2×#1924C211yesmaximal, characteristic
11C2×#1863C211yesmaximal, characteristic, complemented
12C2×#1863C211yesmaximal, characteristic, complemented
13C2×#1803C211yesmaximal, characteristic, complemented
14C2×#1787C211yesmaximal, characteristic
15C2×#1787C211yesmaximal, characteristic
16C2×#1763C211yesmaximal, characteristic, complemented
17C24×Q8C2211yescharacteristic
18C22×#126—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2496—24no—
36#2496C2218yes—
37C2×#400—21no—
38C2×#397—21no—
39Q8×C22⋊C4—24no—
40Q8×C22⋊C4C2218yes—
41C2×#391—21no—
42C2×#391—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55C2×C4×C22⋊C4C2211yescharacteristic
56#2075C2218yes—
57#2075C2218yes—
58#2072C2218yescomplemented
59#2068C2218yescomplemented
60#1935C2218yescomplemented
61#1928C2218yescomplemented
62#1924C2218yes—
63#1863C2218yescomplemented
64#1863C2218yescomplemented
65#1803C2218yescomplemented
66#1787C2218yes—
67#1787C2218yes—
68#1763C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—22no—
80C23×Q8—21no—
81C23×Q8—24no—
82C23×Q8—24no—
83C23×Q8—42no—
84C23×Q8—42no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2314yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×#126—28no—
95C2×#126—48no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
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×#107—42no—
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×C4⋊C8—42no—
131C2×C4⋊C8C2314yes—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—24no—
136C2×#103—24no—
137C2×#103—24no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8—42no—
149C2×C22⋊C8—42no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97—42no—
153C2×#97C2314yes—
154#400—28no—
155#397—28no—
156#391—22no—
157#391—22no—
158#391—28no—
159#391—28no—
160#391C2314yes—
161#391C2314yes—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#387—28no—
167#384—22no—
168#384C2314yes—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—28no—
173#380C2314yes—
174#380C2314yes—
175#380C2314yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2314yes—
178C4×C22⋊C4—28no—
179C4×C22⋊C4D414yes—
180C4×C22⋊C4C2314yes—
181C2×#95—21no—
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×#95—21no—
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
202C22×Q8—28no—
203C22×Q8—216no—
204C22×Q8—28no—
205C22×Q8—28no—
206C22×Q8—21no—
207C22×Q8—24no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—28no—
214C22×Q8—28no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8—48no—
224C22×Q8—48no—
225C22×Q8—416no—
226C22×Q8—416no—
227C22×Q8—416no—
228C22×Q8—416no—
229C22×Q8#4114yes—
230C22×Q8#4114yes—
231C22×Q8#4114yes—
232C22×Q8#4114yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C23×C4—21no—
236C23×C4—28no—
237C23×C4—24no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
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×C8—22no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277#126—432no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×Q8—48no—
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—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—44no—
345#107—44no—
346C4⋊C8—44no—
347#103—28no—
348#103—28no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#97—44no—
358#95—24no—
359#95—24no—
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—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24AGL(1,ℤ/8)14yes—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C2×Q8—216no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—28no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—432no—
400C2×Q8—416no—
401C2×Q8—416no—
402C2×Q8—416no—
403C2×Q8—464no—
404C2×Q8—416no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8#126116yescomplemented
410C2×Q8C2×#4114yes—
411C2×Q8C2×#4114yes—
412C2×Q8C2×#4114yes—
413C2×Q8C2×#4114yes—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—24no—
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—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—432no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
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#13714yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
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—44no—
504C2×C8—44no—
505C2×C8—44no—
506C2×C8—44no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
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—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—48no—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—48no—
543C42—48no—
544C42—48no—
545C23—28no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—28no—
550C23—28no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23#52512yes—
559C23#54612yes—
560C23#54612yes—
561C23C2×AGL(1,ℤ/8)14yes—
562C23C2×AGL(1,ℤ/8)12yes—
563C23C2×#13712yes—
564C23#57812yes—
565C23#57812yes—
566C23C23×D411yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—24no—
570Q8—216no—
571Q8—24no—
572Q8—24no—
573Q8—432no—
574Q8—42no—
575Q8—42no—
576Q8—48no—
577Q8C2×#12618yescomplemented
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—216no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—432no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4#52512yes—
636C2×C4C2×#12614yes—
637C2×C4#53814yes—
638C2×C4#54612yes—
639C2×C4#54612yes—
640C2×C4#55114yes—
641C2×C4#55114yes—
642C2×C4C2×#13714yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C8—42no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—416no—
659C22—42no—
660C22—42no—
661C22#248014yes—
662C22#311214yes—
663C22#312714yes—
664C22#330814yes—
665C22#330814yes—
666C22#331214yes—
667C22#332814yes—
668C22C2×#52511yescharacteristic
669C22C2×#54611yescharacteristic
670C22C2×#54611yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C22×#13711yescharacteristic
673C22C2×#57811yescharacteristic
674C22C2×#57811yescharacteristic
675C4—28no—
676C4—24no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#248014yes—
689C4C22×#12611yescharacteristic
690C4C2×#53811yescharacteristic
691C4C2×#55111yescharacteristic
692C4C2×#55111yescharacteristic
693C2—44no—
694C2#2753518yesdirect product factor
695C2C2×#248011yescharacteristic
696C2C2×#311211yescharacteristic
697C2C2×#312711yescharacteristic
698C2C2×#330811yescharacteristic
699C2C2×#330811yescharacteristic
700C2C2×#331211yescharacteristic
701C2C2×#332811yescharacteristic
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. #27535C2direct (N × H)
6. C2×#2072C2semidirect
7. C2×#2068C2semidirect
8. C2×#1935C2semidirect
9. C2×#1928C2semidirect
11. C2×#1863C2semidirect
12. C2×#1863C2semidirect
13. C2×#1803C2semidirect
16. C2×#1763C2semidirect
58. #2072C22semidirect
59. #2068C22semidirect
60. #1935C22semidirect
61. #1928C22semidirect
63. #1863C22semidirect
64. #1863C22semidirect
65. #1803C22semidirect
68. #1763C22semidirect
409. C2×Q8#126semidirect
577. Q8C2×#126semidirect
694. C2#27535direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×Q8×C22⋊C42
#275352
C2×#20752
C2×#20722
C2×#20682
C2×#19352
C2×#19282
C2×#19242
C2×#18632
C2×#18032
C2×#17872
C2×#17632