Finite Group Databasebeta

← C2×#20229 #6359583

Subgroups — C2×#20229 #6359583

Order 512.

1591 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic, complemented
2C2×#3369C211yesmaximal, characteristic, complemented
3C2×#3357C211yesmaximal, characteristic
4C2×#3340C211yesmaximal, characteristic
5#20229C2116yesmaximal, direct product factor
6C2×#1723C211yesmaximal, characteristic, complemented
7C2×#1696C211yesmaximal, characteristic, complemented
8C2×#1692C211yesmaximal, characteristic, complemented
9C2×#1703C211yesmaximal, characteristic, complemented
10C2×#1675C211yesmaximal, characteristic, complemented
11C2×#1634C211yesmaximal, characteristic
12C2×#1633C211yesmaximal, characteristic, complemented
13C2×#1638C211yesmaximal, characteristic, complemented
14C2×#1644C211yesmaximal, characteristic, complemented
15C2×#1592C211yesmaximal, characteristic, complemented
16C2×#1582C211yesmaximal, characteristic, complemented
17C2×Q82C2211yescharacteristic
18C2×#556C2211yescharacteristic
19C2×#555C2211yescharacteristic
20C2×#554C2211yescharacteristic
21#3385—24no—
22#3385C2218yescomplemented
23#3369—24no—
24#3369C2218yescomplemented
25#3357—24no—
26#3357C2218yes—
27#3340—24no—
28#3340C2218yes—
29C2×#501—21no—
30C2×#500C2211yescharacteristic
31C2×#500C2211yescharacteristic
32C2×#498C2211yescharacteristic
33C2×#494C2211yescharacteristic
34C2×#489—21no—
35C2×#487C2211yescharacteristic
36C2×#487C2211yescharacteristic
37C2×#485—21no—
38C2×#485—21no—
39C2×#482—21no—
40C2×#480—21no—
41C2×#479—21no—
42C2×#477C2211yescharacteristic
43C2×#477C2211yescharacteristic
44C2×#477C2211yescharacteristic
45C2×#475—21no—
46C2×#476C2211yescharacteristic
47C2×#476C2211yescharacteristic
48C2×#476C2211yescharacteristic
49C2×#474—21no—
50C2×#474C2211yescharacteristic
51C2×#468—21no—
52C2×#465—21no—
53C2×#464—21no—
54C2×#462C2211yescharacteristic
55C2×#446C2211yescharacteristic
56C2×#446C2211yescharacteristic
57C2×#441—21no—
58C2×#440—21no—
59C2×C4×SD16C2211yescharacteristic
60C2×C4×SD16C2211yescharacteristic
61C2×#436C2211yescharacteristic
62C2×C8×D4C2211yescharacteristic
63C2×#428—21no—
64C2×#427—21no—
65#1723C2218yescomplemented
66#1703C2218yescomplemented
67#1696C2218yescomplemented
68#1692C2218yescomplemented
69#1675C2218yescomplemented
70#1644C2218yescomplemented
71#1638C2218yescomplemented
72#1634C2218yes—
73#1633C2218yescomplemented
74#1592C2218yescomplemented
75#1582C2218yescomplemented
76C2×#335C2211yescharacteristic
77C2×#334C2211yescharacteristic
78C2×#333C2211yescharacteristic
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×Q8⋊C8C2211yescharacteristic
83C2×Q8⋊C8C2211yescharacteristic
84C2×D4⋊C8C2211yescharacteristic
85C2×D4⋊C8C2211yescharacteristic
86C2×#322C2211yescharacteristic
87C22×Q16—21no—
88C22×Q16—41no—
89C22×SD16D411yescharacteristic
90C22×SD16D411yescharacteristic
91C22×M16—21no—
92C23×C8—21no—
93Q82—22no—
94Q82—22no—
95Q82—41no—
96Q82C2314yes—
97#556—22no—
98#556—22no—
99#556—41no—
100#556C2314yes—
101#555—22no—
102#555—22no—
103#555—41no—
104#555C2314yes—
105#554—22no—
106#554—22no—
107#554—41no—
108#554C2314yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—41no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×#126—41no—
117C2×#125—21no—
118C2×#124—21no—
119C2×#124—41no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—41no—
123C2×#121—21no—
124C2×#121—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117—21no—
135C2×#117—21no—
136C22×C4⋊C4—21no—
137#501—28no—
138#500—22no—
139#500—22no—
140#500C2314yes—
141#500C2314yes—
142#498—22no—
143#498C2314yes—
144#494—41no—
145#494C2314yes—
146#489—28no—
147#487—22no—
148#487—41no—
149#487C2314yes—
150#487C2314yes—
151#485—28no—
152#485—28no—
153#482—28no—
154#480—28no—
155#479—28no—
156#477—22no—
157#477—22no—
158#477—22no—
159#477C2314yes—
160#477C2314yes—
161#477C2314yes—
162#476—22no—
163#476—22no—
164#476—22no—
165#476C2314yes—
166#476C2314yes—
167#476C2314yes—
168#475—28no—
169#474—22no—
170#474—28no—
171#474C2314yes—
172#468—28no—
173#465—28no—
174#464—28no—
175#462—22no—
176#462C2314yes—
177#446—22no—
178#446—22no—
179#446C2314yes—
180#446C2314yes—
181#441—28no—
182#440—28no—
183C4×SD16D414yes—
184C4×SD16D414yes—
185C4×SD16C2314yes—
186C4×SD16C2314yes—
187#436—22no—
188#436C2314yes—
189C8×D4—22no—
190C8×D4C2314yes—
191#428—28no—
192#427—28no—
193C2×#107—21no—
194C2×#107—21no—
195C2×#107—21no—
196C2×#107—21no—
197C2×#107—21no—
198C2×#107—21no—
199C2×#107—21no—
200C2×#107—21no—
201C2×#106—21no—
202C2×#106—21no—
203C2×#106—21no—
204C2×#106—21no—
205C2×#106—21no—
206C2×#106D411yescharacteristic
207C2×#106D411yescharacteristic
208C2×C4⋊C8—21no—
209C2×C4⋊C8—21no—
210C2×C4⋊C8C2311yescharacteristic
211C2×C4⋊C8C2311yescharacteristic
212C2×C4⋊C8C2311yescharacteristic
213C2×C4⋊C8C2311yescharacteristic
214C2×#103—21no—
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#103—21no—
220C2×#103—41no—
221C2×#103D411yescharacteristic
222C2×#103D411yescharacteristic
223C2×#102—21no—
224C2×#102—21no—
225C2×#102—21no—
226C2×#102—21no—
227C2×#102D411yescharacteristic
228C2×#102D411yescharacteristic
229C2×C22⋊C8—21no—
230C2×C22⋊C8—21no—
231C2×#97—21no—
232C2×#97—21no—
233C2×C4×C8C2311yescharacteristic
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236#335C2314yes—
237#334C2314yes—
238#333C2314yes—
239#333C2314yes—
240#333C2314yes—
241#332C2314yes—
242Q8⋊C8C2314yes—
243Q8⋊C8C2314yes—
244D4⋊C8C2314yes—
245D4⋊C8C2314yes—
246#322C2314yes—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—41no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2×D411yescharacteristic
254C22×D4—41no—
255C22×D4C2×D411yescharacteristic
256C22×D4C2×D411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—41no—
260C2×Q16—24no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16—84no—
264C2×SD16—24no—
265C2×SD16—42no—
266C2×SD16C2×D414yes—
267C2×SD16C2×D414yes—
268C2×M16—24no—
269C2×M16—44no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—24no—
276C22×C8—44no—
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8C2412yes—
300C4⋊Q8C2412yes—
301C4⋊Q8C2412yes—
302C4⋊Q8C2412yes—
303#126—44no—
304#126—44no—
305#125—24no—
306#125—42no—
307#124—24no—
308#124—42no—
309#124—44no—
310#124—44no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—42no—
316C22⋊Q8—42no—
317#121—24no—
318#121—24no—
319#121—42no—
320#121—42no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—42no—
329C4×Q8—42no—
330C4×Q8—42no—
331C4×Q8C2×D412yes—
332C4×Q8C2×D412yes—
333C4×Q8C2412yes—
334C4×Q8C2412yes—
335C4×D4—24no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—42no—
341C4×D4C2×D412yes—
342C4×D4C2×D412yes—
343C4×D4C2412yes—
344C4×D4C2412yes—
345#117—24no—
346#117—24no—
347#117—24no—
348#117—42no—
349#117—42no—
350#117—42no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—41no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—41no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—41no—
382C2×C42—41no—
383C2×C42—41no—
384C2×C42C2411yescharacteristic
385#107—24no—
386#107—24no—
387#107—24no—
388#107—24no—
389#107—24no—
390#107—24no—
391#107—24no—
392#107—24no—
393#106—24no—
394#106—24no—
395#106—24no—
396#106—24no—
397#106—24no—
398#106C2×D414yes—
399#106C2×D414yes—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8C2412yes—
407C4⋊C8C2412yes—
408C4⋊C8C2412yes—
409C4⋊C8C2412yes—
410#103—24no—
411#103—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—44no—
417#103C2×D414yes—
418#103C2×D414yes—
419#102—24no—
420#102—24no—
421#102—24no—
422#102—24no—
423#102C2×D414yes—
424#102C2×D414yes—
425C22⋊C8—24no—
426C22⋊C8—24no—
427#97—24no—
428#97—24no—
429C4×C8—21no—
430C4×C8C2×D412yes—
431C4×C8C2×D412yes—
432C4×C8C2412yes—
433C4×C8C2412yes—
434C4×C8C2412yes—
435C24—21no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—41no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8—162no—
458C2×Q8C22×D412yes—
459C2×Q8C22×D412yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—44no—
467C2×D4—162no—
468C2×D4C22×D412yes—
469C2×D4C22×D412yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—162no—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4H(5,𝔽2)11yescharacteristic
493Q16—48no—
494Q16—88no—
495SD16—28no—
496SD16—44no—
497M16—48no—
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—42no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—48no—
512C2×C8#13714yes—
513C2×C8C22×D412yes—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C2×C8C22×D412yes—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4C22×D412yes—
547C4⋊C4C22×D412yes—
548C4⋊C4C22×D412yes—
549C4⋊C4C22×D412yes—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42C22×D411yescharacteristic
567C42C22×D411yescharacteristic
568C42H(5,𝔽2)11yescharacteristic
569C42C2511yescharacteristic
570C23—22no—
571C23—22no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23D4211yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—82no—
588Q8—82no—
589Q8—82no—
590Q8—82no—
591Q8—161no—
592Q8—161no—
593Q8—161no—
594Q8—164no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—22no—
599D4—42no—
600D4—42no—
601D4—164no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
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—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—164no—
630C2×C4D4211yescharacteristic
631C2×C4D4211yescharacteristic
632C2×C4D4211yescharacteristic
633C2×C4C2×#13712yes—
634C2×C4#57512yes—
635C2×C4#57812yes—
636C2×C4#57812yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C23×D411yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C8—22no—
641C8—22no—
642C8—22no—
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C8C2×#13714yes—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—84no—
656C22—84no—
657C22#326912yes—
658C22#327612yes—
659C22#328812yes—
660C22C2×D4211yescharacteristic
661C4—21no—
662C4—21no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#326912yes—
674C4#327612yes—
675C4C22×#13711yescharacteristic
676C4C2×#57511yescharacteristic
677C4C2×#57811yescharacteristic
678C4C2×#57811yescharacteristic
679C2—82no—
680C2—82no—
681C2#2022914yesdirect product factor
682C2C2×#326911yescharacteristic
683C2C2×#327611yescharacteristic
684C2C2×#328811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC42C25
prime radicalC2×#555C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3385C2semidirect
2. C2×#3369C2semidirect
5. #20229C2direct (N × H)
6. C2×#1723C2semidirect
7. C2×#1696C2semidirect
8. C2×#1692C2semidirect
9. C2×#1703C2semidirect
10. C2×#1675C2semidirect
12. C2×#1633C2semidirect
13. C2×#1638C2semidirect
14. C2×#1644C2semidirect
15. C2×#1592C2semidirect
16. C2×#1582C2semidirect
22. #3385C22semidirect
24. #3369C22semidirect
65. #1723C22semidirect
66. #1703C22semidirect
67. #1696C22semidirect
68. #1692C22semidirect
69. #1675C22semidirect
70. #1644C22semidirect
71. #1638C22semidirect
73. #1633C22semidirect
74. #1592C22semidirect
75. #1582C22semidirect
681. C2#20229direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33692
C2×#33572
C2×#33402
#202292
C2×#17232
C2×#16962
C2×#16922
C2×#17032
C2×#16752
C2×#16342
C2×#16332
C2×#16382
C2×#16442
C2×#15922
C2×#15822