Finite Group Databasebeta

← C2×#32697 #7513938

Subgroups — C2×#32697 #7513938

Order 512.

1447 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32697C2116yesmaximal, direct product factor
2C2×#2102C211yesmaximal, characteristic, complemented
3C2×#2082C211yesmaximal, characteristic, complemented
4C2×#2082C211yesmaximal, characteristic, complemented
5C2×#2070C211yesmaximal, characteristic, complemented
6C2×#2049C211yesmaximal, characteristic
7C2×#2005C211yesmaximal, characteristic
8C2×#1966C211yesmaximal, characteristic, complemented
9C2×#1932C211yesmaximal, characteristic
10C2×#1915C211yesmaximal, characteristic
11C2×#1912C211yesmaximal, characteristic
12C2×#1902C211yesmaximal, characteristic
13C2×#1886C211yesmaximal, characteristic, complemented
14C2×#1861C211yesmaximal, characteristic
15C2×#1841C211yesmaximal, characteristic, complemented
16C2×#1830C211yesmaximal, characteristic, complemented
17C22×#137C2211yescharacteristic
18C2×#568C2211yescharacteristic
19C2×#536C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#430C2211yescharacteristic
23C2×#430C2211yescharacteristic
24C2×#430C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C2×#421C2211yescharacteristic
28C22×#104C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#413C2211yescharacteristic
33C22×#101C2211yescharacteristic
34C22×#101C2211yescharacteristic
35C22×#101C2211yescharacteristic
36C22×#101C2211yescharacteristic
37C22×#101C2211yescharacteristic
38C22×#101C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2102C2218yescomplemented
41#2082C2218yescomplemented
42#2082C2218yescomplemented
43#2070C2218yescomplemented
44#2049C2218yes—
45#2005C2218yes—
46#1966C2218yescomplemented
47#1932C2218yes—
48#1915C2218yes—
49#1912C2218yes—
50#1902C2218yes—
51#1886C2218yescomplemented
52#1861C2218yes—
53#1841C2218yescomplemented
54#1830C2218yescomplemented
55C2×#344C2211yescharacteristic
56C2×#344C2211yescharacteristic
57C2×#344C2211yescharacteristic
58C2×#344C2211yescharacteristic
59C2×#344C2211yescharacteristic
60C2×#344C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C2×#339C2211yescharacteristic
64C2×#338C2211yescharacteristic
65C2×H(3,ℤ/4)C2211yescharacteristic
66C2×H(3,ℤ/4)C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×Q8C2311yescharacteristic
70C2×#137—22no—
71C2×#137—41no—
72C2×#137—41no—
73C2×#137—41no—
74C2×#137—41no—
75C2×#137C2314yes—
76C22×Q16—21no—
77C22×SD16—21no—
78#568—41no—
79#568—41no—
80#568—41no—
81#568C2314yes—
82C2×#131—41no—
83C2×#131—41no—
84C22×M16—41no—
85C22×M16C2311yescharacteristic
86C22×M16C2311yescharacteristic
87C22×M16C2311yescharacteristic
88C22×M16C2311yescharacteristic
89C22×M16C2311yescharacteristic
90C22×M16C2311yescharacteristic
91C22×M16C2311yescharacteristic
92C22×M16C2311yescharacteristic
93#536—22no—
94#536—22no—
95#536—41no—
96#536C2314yes—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—41no—
103C2×C4⋊Q8—41no—
104C2×C4⋊Q8C2314yes—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—41no—
108#519—22no—
109#519—41no—
110#519—41no—
111#519C2314yes—
112C2×C4×Q8—41no—
113C2×C4×Q8—41no—
114C2×#117—22no—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—22no—
119C22×C42C2311yescharacteristic
120#430—41no—
121#430—41no—
122#430—41no—
123#430C2314yes—
124#430C2314yes—
125#430C2314yes—
126C2×#108—41no—
127C2×#108—41no—
128C2×#108—41no—
129#423—41no—
130#423C2314yes—
131C2×C4⋊C8—41no—
132#421—41no—
133#421—41no—
134#421C2314yes—
135#421C2314yes—
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—22no—
139C2×#104—41no—
140C2×#104—41no—
141C2×#104C2314yes—
142#419—41no—
143#419—41no—
144#419—41no—
145#419C2314yes—
146#419C2314yes—
147#419C2314yes—
148C2×#103—41no—
149C2×#103—41no—
150C2×#103—41no—
151#413—22no—
152#413C2314yes—
153C2×#101—21no—
154C2×#101—41no—
155C2×#101—41no—
156C2×#101—41no—
157C2×#101—41no—
158C2×#101—41no—
159C2×#101—41no—
160C2×#101—41no—
161C2×#101—41no—
162C2×#101—41no—
163C2×#101—41no—
164C2×#101—41no—
165C2×#101—41no—
166C2×#101C2314yes—
167C2×#101C2314yes—
168C2×#101C2314yes—
169C2×#101C2314yes—
170C2×#101C2314yes—
171C2×#101C2314yes—
172C2×#100—21no—
173C4×M16—22no—
174C4×M16C2314yes—
175C2×#97—21no—
176C2×C4×C8—21no—
177#344C2314yes—
178#344C2314yes—
179#344C2314yes—
180#344C2314yes—
181#344C2314yes—
182#344C2314yes—
183#339C2314yes—
184#339C2314yes—
185#339C2314yes—
186#338C2314yes—
187H(3,ℤ/4)C2314yes—
188H(3,ℤ/4)C2314yes—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41#4112yes—
200C2×#41C2412yes—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8#4111yescharacteristic
227C22×Q8#4112yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C22×D4—21no—
231C22×D4#4111yescharacteristic
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4#4112yes—
236C23×C4C2411yescharacteristic
237#137—44no—
238#137—44no—
239#137—44no—
240#137—44no—
241C2×Q16—24no—
242C2×Q16—42no—
243C2×Q16—42no—
244C2×SD16—24no—
245C2×SD16—42no—
246C2×SD16—42no—
247#131—44no—
248#131—44no—
249#131—44no—
250#131—44no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—21no—
264C2×M16—21no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—21no—
273C2×M16—21no—
274C2×M16—21no—
275C2×M16—21no—
276C2×M16—21no—
277C2×M16—21no—
278C2×M16—21no—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—21no—
282C2×M16—21no—
283C2×M16—21no—
284C2×M16—21no—
285C2×M16—21no—
286C2×M16—21no—
287C2×M16—21no—
288C2×M16—21no—
289C2×M16—21no—
290C2×M16—21no—
291C2×M16—44no—
292C2×M16—42no—
293C2×M16—44no—
294C2×M16C2412yes—
295C2×M16C2412yes—
296C2×M16C2412yes—
297C2×M16C2412yes—
298C2×M16C2412yes—
299C2×M16C2412yes—
300C2×M16C2412yes—
301C2×M16C2412yes—
302C22×C8—21no—
303C22×C8—21no—
304C22×C8—21no—
305C22×C8—21no—
306C22×C8—21no—
307C22×C8—21no—
308C22×C8—21no—
309C22×C8—21no—
310C22×C8—42no—
311C4⋊Q8—24no—
312C4⋊Q8—24no—
313C4⋊Q8—24no—
314C4⋊Q8—44no—
315C4⋊Q8—42no—
316C4⋊Q8—44no—
317C4⋊Q8—42no—
318C4⋊Q8—42no—
319#124—24no—
320#124—42no—
321#123—28no—
322#123—44no—
323C22⋊Q8—44no—
324C22⋊Q8—44no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8—44no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117—28no—
333#117—21no—
334#117—21no—
335#117—21no—
336#117—21no—
337#117—21no—
338#117—21no—
339#117—44no—
340#117C2412yes—
341#117C2412yes—
342#117C2412yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—42no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—42no—
363C2×C42#4111yescharacteristic
364C2×C42#4111yescharacteristic
365C2×C42#4112yes—
366C2×C42C2412yes—
367#108—44no—
368#108—44no—
369#108—44no—
370C4⋊C8—44no—
371#104—24no—
372#104—24no—
373#104—44no—
374#104—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#101—24no—
379#101—44no—
380#101—44no—
381#101—44no—
382#101—44no—
383#101—44no—
384#101—44no—
385#100—24no—
386#97—24no—
387C4×C8—24no—
388C24—21no—
389C24C2×#4111yescharacteristic
390#41—22no—
391#41—22no—
392#41—22no—
393#41—22no—
394#41—22no—
395#41—22no—
396#41—22no—
397#41—22no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403#41—41no—
404#41—41no—
405#41—41no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—41no—
436C2×Q8—41no—
437C2×Q8—41no—
438C2×Q8—41no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—41no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—41no—
456C2×Q8—41no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—41no—
460C2×Q8—41no—
461C2×Q8—41no—
462C2×Q8—41no—
463C2×Q8—41no—
464C2×Q8—41no—
465C2×Q8—41no—
466C2×Q8—41no—
467C2×Q8C2×#4112yes—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4C2×#4112yes—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—44no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4C2×#4114yes—
509C22×C4C2×#4111yescharacteristic
510C22×C4C2×#4111yescharacteristic
511C22×C4C2×#4111yescharacteristic
512C22×C4C2×#4111yescharacteristic
513C22×C4C2×#4111yescharacteristic
514C22×C4H(5,𝔽2)11yescharacteristic
515C22×C4H(5,𝔽2)11yescharacteristic
516C22×C4H(5,𝔽2)11yescharacteristic
517C22×C4H(5,𝔽2)11yescharacteristic
518C22×C4H(5,𝔽2)11yescharacteristic
519C22×C4H(5,𝔽2)11yescharacteristic
520C22×C4H(5,𝔽2)11yescharacteristic
521C22×C4H(5,𝔽2)11yescharacteristic
522C22×C4C2511yescharacteristic
523Q16—48no—
524SD16—48no—
525M16—22no—
526M16—22no—
527M16—22no—
528M16—22no—
529M16—22no—
530M16—22no—
531M16—22no—
532M16—22no—
533M16—22no—
534M16—22no—
535M16—22no—
536M16—22no—
537M16—22no—
538M16—22no—
539M16—22no—
540M16—22no—
541M16—48no—
542M16—41no—
543M16—41no—
544M16—41no—
545M16—41no—
546M16—41no—
547M16—41no—
548M16—41no—
549M16—41no—
550M16—41no—
551M16—41no—
552M16—41no—
553M16—41no—
554M16—41no—
555M16—41no—
556M16—41no—
557M16—41no—
558C2×C8—22no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—22no—
562C2×C8—22no—
563C2×C8—22no—
564C2×C8—22no—
565C2×C8—22no—
566C2×C8—48no—
567C2×C8—41no—
568C2×C8—41no—
569C2×C8—41no—
570C2×C8—41no—
571C2×C8—41no—
572C2×C8—41no—
573C2×C8—41no—
574C2×C8—41no—
575C2×C8—41no—
576C2×C8—41no—
577C2×C8—41no—
578C2×C8—41no—
579C2×C8—41no—
580C2×C8—41no—
581C2×C8—41no—
582C2×C8—41no—
583C4⋊C4—22no—
584C4⋊C4—22no—
585C4⋊C4—22no—
586C4⋊C4—48no—
587C4⋊C4—48no—
588C4⋊C4—48no—
589C4⋊C4—48no—
590C4⋊C4—41no—
591C4⋊C4—41no—
592C4⋊C4—41no—
593C22⋊C4—22no—
594C22⋊C4—22no—
595C22⋊C4—22no—
596C22⋊C4—48no—
597C22⋊C4—41no—
598C22⋊C4—41no—
599C22⋊C4—41no—
600C42—21no—
601C42—22no—
602C42—21no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—48no—
608C42—41no—
609C42—41no—
610C42—41no—
611C42C2×#4112yes—
612C42C2×#4112yes—
613C23—22no—
614C23—42no—
615C23—41no—
616C23—41no—
617C23—82no—
618C23#53811yescharacteristic
619C23#53811yescharacteristic
620C23#55111yescharacteristic
621C23#55111yescharacteristic
622C23#55111yescharacteristic
623C23#55111yescharacteristic
624C23C22×#4111yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—42no—
628Q8—42no—
629Q8—42no—
630Q8—42no—
631Q8—42no—
632Q8—42no—
633Q8—42no—
634Q8—41no—
635Q8—41no—
636Q8—42no—
637Q8—42no—
638Q8—42no—
639Q8—42no—
640Q8—42no—
641Q8—81no—
642Q8—81no—
643Q8—81no—
644Q8—81no—
645D4—22no—
646D4—22no—
647D4—42no—
648D4—42no—
649D4—42no—
650D4—42no—
651D4—41no—
652D4—41no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—82no—
680C2×C4—82no—
681C2×C4—82no—
682C2×C4—81no—
683C2×C4—81no—
684C2×C4—81no—
685C2×C4—81no—
686C2×C4#53811yescharacteristic
687C2×C4#53811yescharacteristic
688C2×C4#53811yescharacteristic
689C2×C4#53811yescharacteristic
690C2×C4#53811yescharacteristic
691C2×C4#53811yescharacteristic
692C2×C4#54011yescharacteristic
693C2×C4#54011yescharacteristic
694C2×C4#54011yescharacteristic
695C2×C4#54011yescharacteristic
696C2×C4#55111yescharacteristic
697C2×C4#55111yescharacteristic
698C2×C4#55111yescharacteristic
699C2×C4#55111yescharacteristic
700C2×C4#55311yescharacteristic
701C2×C4#55311yescharacteristic
702C2×C4#55311yescharacteristic
703C2×C4#55311yescharacteristic
704C2×C4#56011yescharacteristic
705C2×C4#56011yescharacteristic
706C2×C4#56011yescharacteristic
707C2×C4#56011yescharacteristic
708C2×C4C22×#4111yescharacteristic
709C2×C4C22×#4111yescharacteristic
710C2×C4C2×H(5,𝔽2)11yescharacteristic
711C2×C4C2×H(5,𝔽2)11yescharacteristic
712C2×C4C2×H(5,𝔽2)11yescharacteristic
713C2×C4C2×H(5,𝔽2)11yescharacteristic
714C8—42no—
715C8—42no—
716C8—42no—
717C8—42no—
718C8—42no—
719C8—42no—
720C8—42no—
721C8—42no—
722C22—22no—
723C22—22no—
724C22—22no—
725C22—41no—
726C22—41no—
727C22—41no—
728C22—41no—
729C22—42no—
730C22—82no—
731C22—82no—
732C22—161no—
733C22#274911yescharacteristic
734C22#274911yescharacteristic
735C22#274911yescharacteristic
736C22#274911yescharacteristic
737C22C2×#53811yescharacteristic
738C22C2×#55111yescharacteristic
739C22C2×#55111yescharacteristic
740C4—24no—
741C4—42no—
742C4—44no—
743C4—42no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#274911yescharacteristic
749C4#274911yescharacteristic
750C4#274911yescharacteristic
751C4#274911yescharacteristic
752C4C2×#53811yescharacteristic
753C4C2×#54011yescharacteristic
754C4C2×#55311yescharacteristic
755C4C2×#56011yescharacteristic
756C2—21no—
757C2—21no—
758C2—21no—
759C2—21no—
760C2—21no—
761C2—21no—
762C2—82no—
763C2#3269712yesdirect product factor
764C2C2×#274911yescharacteristic
subgroupG/H
center, socleC22#2749
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #32697C2direct (N × H)
2. C2×#2102C2semidirect
3. C2×#2082C2semidirect
4. C2×#2082C2semidirect
5. C2×#2070C2semidirect
8. C2×#1966C2semidirect
13. C2×#1886C2semidirect
15. C2×#1841C2semidirect
16. C2×#1830C2semidirect
40. #2102C22semidirect
41. #2082C22semidirect
42. #2082C22semidirect
43. #2070C22semidirect
46. #1966C22semidirect
51. #1886C22semidirect
53. #1841C22semidirect
54. #1830C22semidirect
763. C2#32697direct (N × H)

Maximal subgroups

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

subgroupindex
#326972
C2×#21022
C2×#20822
C2×#20702
C2×#20492
C2×#20052
C2×#19662
C2×#19322
C2×#19152
C2×#19122
C2×#19022
C2×#18862
C2×#18612
C2×#18412
C2×#18302