Finite Group Databasebeta

← C2×#32112 #7507736

Subgroups — C2×#32112 #7507736

Order 512.

2264 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic, complemented
2C2×#2491C211yesmaximal, characteristic
3#32112C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2055C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#2023C211yesmaximal, characteristic, complemented
9C2×#2000C211yesmaximal, characteristic
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1987C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1893C211yesmaximal, characteristic
15C2×#1892C211yesmaximal, characteristic, complemented
16C2×#1850C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#124—21no—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2492—24no—
35#2492C2218yescomplemented
36#2491—24no—
37#2491C2218yes—
38C2×#400—21no—
39C2×#394—21no—
40C2×#393—21no—
41C2×#393—21no—
42C2×#391—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2077C2218yescomplemented
55#2055C2218yescomplemented
56#2055C2218yescomplemented
57#2024C2218yescomplemented
58#2023C2218yescomplemented
59#2000C2218yes—
60#1988C2218yescomplemented
61#1987C2218yescomplemented
62#1927C2218yescomplemented
63#1926C2218yescomplemented
64#1893C2218yes—
65#1892C2218yescomplemented
66#1850C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8—21no—
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16—42no—
84C22×SD16—42no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C23×C8—42no—
88C23×C8—42no—
89C23×C8—42no—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2314yes—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2314yes—
103C2×#124—28no—
104C2×#124—48no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—42no—
108C2×C22⋊Q8C2314yes—
109C2×#121—22no—
110C2×#121—24no—
111C2×#121—42no—
112C2×#121C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C2×#106—42no—
137C2×#106C2314yes—
138C2×C4⋊C8—42no—
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8—42no—
155C2×C22⋊C8C2314yes—
156#400—28no—
157#394—28no—
158#393—28no—
159#393—28no—
160#391—22no—
161#391—22no—
162#391—28no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#388—28no—
170#388—28no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174#380C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
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×#95D411yescharacteristic
190C2×#95D411yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—82no—
217C22×Q8—82no—
218C22×Q8C2412yes—
219C22×D4—24no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—82no—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—416no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8—416no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—48no—
281C4⋊Q8—48no—
282#124—432no—
283C22⋊Q8—28no—
284C22⋊Q8—44no—
285C22⋊Q8—48no—
286#121—28no—
287#121—44no—
288#121—48no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
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#4114yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4SD1618yescomplemented
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—44no—
346C2×C42—82no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351#106—44no—
352C4⋊C8—44no—
353C4⋊C8—44no—
354#103—28no—
355#103—44no—
356#103—44no—
357#102—28no—
358#102—44no—
359#102—44no—
360C22⋊C8—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C2×SD1614yes—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—84no—
406C2×Q8—84no—
407C2×Q8—816no—
408C2×Q8—816no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—84no—
425C2×D4—816no—
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—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
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—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C2×SD1614yes—
481C22×C4C2×SD1614yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495SD16—416no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—416no—
509C2×C8—44no—
510C2×C8—44no—
511C2×C8—44no—
512C2×C8—44no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4C2×#4114yes—
533C4⋊C4C2×#4114yes—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4C2×SD16116yescomplemented
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—84no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23#53712yes—
580C23#54612yes—
581C23#54612yes—
582C23C22×SD1614yes—
583C23C22×SD1612yes—
584C23#57512yes—
585C23#57612yes—
586C23#57812yes—
587C23C23×D411yescharacteristic
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—816no—
593Q8—816no—
594Q8—84no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—816no—
600D4—84no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4#53712yes—
653C2×C4#54214yes—
654C2×C4#54214yes—
655C2×C4#54612yes—
656C2×C4#54612yes—
657C2×C4#55914yes—
658C2×C4H(3,𝔽4)14yes—
659C2×C4C22×SD1614yes—
660C2×C4C22×SD1614yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C8—42no—
668C8—42no—
669C8—42no—
670C8—42no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—416no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—168no—
687C22#271214yes—
688C22#324614yes—
689C22#326214yes—
690C22#329514yes—
691C22#329814yes—
692C22#331314yes—
693C22#331914yes—
694C22C2×#53711yescharacteristic
695C22C2×#54611yescharacteristic
696C22C2×#54611yescharacteristic
697C22C23×SD1611yescharacteristic
698C22C2×#57511yescharacteristic
699C22C2×#57611yescharacteristic
700C22C2×#57811yescharacteristic
701C4—24no—
702C4—24no—
703C4—42no—
704C4—44no—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#271214yes—
714C4C2×#54211yescharacteristic
715C4C2×#54211yescharacteristic
716C4C2×#55911yescharacteristic
717C4C2×H(3,𝔽4)11yescharacteristic
718C2—44no—
719C2—162no—
720C2#3211218yesdirect product factor
721C2C2×#271211yescharacteristic
722C2C2×#324611yescharacteristic
723C2C2×#326211yescharacteristic
724C2C2×#329511yescharacteristic
725C2C2×#329811yescharacteristic
726C2C2×#331911yescharacteristic
727C2C2×#331311yescharacteristic
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
1. C2×#2492C2semidirect
3. #32112C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2055C2semidirect
6. C2×#2055C2semidirect
7. C2×#2024C2semidirect
8. C2×#2023C2semidirect
10. C2×#1988C2semidirect
11. C2×#1987C2semidirect
12. C2×#1926C2semidirect
13. C2×#1927C2semidirect
15. C2×#1892C2semidirect
16. C2×#1850C2semidirect
35. #2492C22semidirect
54. #2077C22semidirect
55. #2055C22semidirect
56. #2055C22semidirect
57. #2024C22semidirect
58. #2023C22semidirect
60. #1988C22semidirect
61. #1987C22semidirect
62. #1927C22semidirect
63. #1926C22semidirect
65. #1892C22semidirect
66. #1850C22semidirect
336. C2×C22⋊C4SD16semidirect
543. C22⋊C4C2×SD16semidirect
720. C2#32112direct (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×#24912
#321122
C2×#20772
C2×#20552
C2×#20242
C2×#20232
C2×#20002
C2×#19882
C2×#19872
C2×#19262
C2×#19272
C2×#18932
C2×#18922
C2×#18502