Finite Group Databasebeta

← C2×#20181 #6359532

Subgroups — C2×#20181 #6359532

Order 512.

1651 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic, complemented
2C2×#3385C211yesmaximal, characteristic, complemented
3C2×#3335C211yesmaximal, characteristic
4C2×#3331C211yesmaximal, characteristic
5#20181C2116yesmaximal, direct product factor
6C2×#1723C211yesmaximal, characteristic, complemented
7C2×#1704C211yesmaximal, characteristic, complemented
8C2×#1692C211yesmaximal, characteristic, complemented
9C2×#1695C211yesmaximal, characteristic, complemented
10C2×#1675C211yesmaximal, characteristic, complemented
11C2×#1645C211yesmaximal, characteristic, complemented
12C2×#1635C211yesmaximal, characteristic
13C2×#1632C211yesmaximal, characteristic, complemented
14C2×#1632C211yesmaximal, characteristic, complemented
15C2×#1592C211yesmaximal, characteristic, complemented
16C2×#1582C211yesmaximal, characteristic, complemented
17C2×Q82C2211yescharacteristic
18C2×#554C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#547C2211yescharacteristic
21#3385—24no—
22#3385—24no—
23#3385C2218yescomplemented
24#3385C2218yescomplemented
25#3335—24no—
26#3335C2218yes—
27#3331—24no—
28#3331C2218yes—
29C2×#501—21no—
30C2×#501—21no—
31C2×#500C2211yescharacteristic
32C2×#500C2211yescharacteristic
33C2×#498C2211yescharacteristic
34C2×#494C2211yescharacteristic
35C2×#487C2211yescharacteristic
36C2×#487C2211yescharacteristic
37C2×#481—21no—
38C2×#483—21no—
39C2×#483—21no—
40C2×#479—21no—
41C2×#479—21no—
42C2×#477C2211yescharacteristic
43C2×#477C2211yescharacteristic
44C2×#475C2211yescharacteristic
45C2×#476C2211yescharacteristic
46C2×#476C2211yescharacteristic
47C2×#476C2211yescharacteristic
48C2×#476C2211yescharacteristic
49C2×#474—21no—
50C2×#474—21no—
51C2×#470—21no—
52C2×#465—21no—
53C2×#462C2211yescharacteristic
54C2×#452—21no—
55C2×#446C2211yescharacteristic
56C2×#446C2211yescharacteristic
57C2×#440—21no—
58C2×#440—21no—
59C2×C4×SD16C2211yescharacteristic
60C2×C4×SD16C2211yescharacteristic
61C2×#436C2211yescharacteristic
62C2×C8×D4C2211yescharacteristic
63C22×#106—21no—
64C2×#417—21no—
65#1723C2218yescomplemented
66#1704C2218yescomplemented
67#1695C2218yescomplemented
68#1692C2218yescomplemented
69#1675C2218yescomplemented
70#1645C2218yescomplemented
71#1635C2218yes—
72#1632C2218yescomplemented
73#1632C2218yescomplemented
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×#41—21no—
88C22×Q16—21no—
89C22×Q16—41no—
90C22×SD16D411yescharacteristic
91C22×SD16D411yescharacteristic
92C22×M16—21no—
93C23×C8—21no—
94Q82—22no—
95Q82—22no—
96Q82—41no—
97Q82C2314yes—
98#554—22no—
99#554—22no—
100#554—22no—
101#554—22no—
102#554—41no—
103#554—41no—
104#554C2314yes—
105#554C2314yes—
106#547—22no—
107#547—22no—
108#547—41no—
109#547C2314yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—41no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#123—41no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—41no—
124C2×C22⋊Q8—41no—
125C2×#121—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137#501—28no—
138#501—28no—
139#500—22no—
140#500—22no—
141#500C2314yes—
142#500C2314yes—
143#498—22no—
144#498C2314yes—
145#494—41no—
146#494C2314yes—
147#487—22no—
148#487—41no—
149#487C2314yes—
150#487C2314yes—
151#483—28no—
152#483—28no—
153#481—28no—
154#479—28no—
155#479—28no—
156#477—22no—
157#477—22no—
158#477C2314yes—
159#477C2314yes—
160#476—22no—
161#476—22no—
162#476—22no—
163#476—22no—
164#476C2314yes—
165#476C2314yes—
166#476C2314yes—
167#476C2314yes—
168#475—22no—
169#475C2314yes—
170#474—28no—
171#474—28no—
172#470—28no—
173#465—28no—
174#462—22no—
175#462C2314yes—
176#452—28no—
177#446—22no—
178#446—22no—
179#446C2314yes—
180#446C2314yes—
181#440—28no—
182#440—28no—
183C4×SD16D414yes—
184C4×SD16D414yes—
185C4×SD16C2314yes—
186C4×SD16C2314yes—
187#436—22no—
188#436C2314yes—
189C8×D4—22no—
190C8×D4C2314yes—
191C2×#107—21no—
192C2×#107—21no—
193C2×#107—21no—
194C2×#107—21no—
195C2×#107—21no—
196C2×#107—21no—
197C2×#106—21no—
198C2×#106—21no—
199C2×#106—21no—
200C2×#106—21no—
201C2×#106—21no—
202C2×#106—21no—
203C2×#106—21no—
204C2×#106—28no—
205C2×#106D411yescharacteristic
206C2×#106D411yescharacteristic
207C2×C4⋊C8—21no—
208C2×C4⋊C8—21no—
209C2×C4⋊C8C2311yescharacteristic
210C2×C4⋊C8C2311yescharacteristic
211C2×C4⋊C8C2311yescharacteristic
212C2×C4⋊C8C2311yescharacteristic
213#417—28no—
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—
247C2×#41—24no—
248C2×#41—42no—
249C2×#41—44no—
250C2×#41—82no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—41no—
256C22×Q8C2×D411yescharacteristic
257C22×Q8C2×D411yescharacteristic
258C22×D4—41no—
259C22×D4C2×D411yescharacteristic
260C22×D4C2×D411yescharacteristic
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—41no—
264C23×C4—41no—
265C23×C4—41no—
266C2×Q16—24no—
267C2×Q16—44no—
268C2×Q16—82no—
269C2×Q16—84no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16C2×D414yes—
273C2×SD16C2×D414yes—
274C2×M16—24no—
275C2×M16—82no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—24no—
282C22×C8—44no—
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8C2412yes—
306C4⋊Q8C2412yes—
307C4⋊Q8C2412yes—
308C4⋊Q8C2412yes—
309#125—24no—
310#125—24no—
311#125—42no—
312#125—42no—
313#123—44no—
314#123—44no—
315C22⋊Q8—24no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—44no—
319C22⋊Q8—44no—
320C22⋊Q8—44no—
321C22⋊Q8—44no—
322C22⋊Q8—42no—
323C22⋊Q8—42no—
324C22⋊Q8—42no—
325#121—24no—
326#121—42no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—24no—
332C4×Q8—24no—
333C4×Q8—42no—
334C4×Q8—42no—
335C4×Q8C2×D412yes—
336C4×Q8C2×D412yes—
337C4×Q8C2412yes—
338C4×Q8C2412yes—
339C4×D4—24no—
340C4×D4—24no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—42no—
346C4×D4—42no—
347C4×D4C2×D412yes—
348C4×D4C2×D412yes—
349C4×D4C2412yes—
350C4×D4C2412yes—
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—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—41no—
381C2×C22⋊C4—41no—
382C2×C22⋊C4—41no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—41no—
386C2×C42C2411yescharacteristic
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#106—24no—
399#106—24no—
400#106C2×D414yes—
401#106C2×D414yes—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8C2412yes—
409C4⋊C8C2412yes—
410C4⋊C8C2412yes—
411C4⋊C8C2412yes—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—44no—
419#103C2×D414yes—
420#103C2×D414yes—
421#102—24no—
422#102—24no—
423#102—24no—
424#102—24no—
425#102C2×D414yes—
426#102C2×D414yes—
427C22⋊C8—24no—
428C22⋊C8—24no—
429#97—24no—
430#97—24no—
431C4×C8—21no—
432C4×C8C2×D412yes—
433C4×C8C2×D412yes—
434C4×C8C2412yes—
435C4×C8C2412yes—
436C4×C8C2412yes—
437C24—21no—
438C24—21no—
439#41—48no—
440#41—84no—
441#41—88no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—44no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—41no—
456C2×Q8—41no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×Q8—82no—
462C2×Q8—162no—
463C2×Q8C22×D412yes—
464C2×Q8C22×D412yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—44no—
471C2×D4—84no—
472C2×D4#13714yes—
473C2×D4C22×D412yes—
474C2×D4C22×D412yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—84no—
497C22×C4—162no—
498C22×C4—162no—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4H(5,𝔽2)11yescharacteristic
502Q16—84no—
503Q16—88no—
504SD16—28no—
505SD16—28no—
506M16—84no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—42no—
516C2×C8—42no—
517C2×C8—42no—
518C2×C8—42no—
519C2×C8—42no—
520C2×C8—48no—
521C2×C8#13714yes—
522C2×C8C22×D412yes—
523C2×C8C22×D412yes—
524C2×C8C22×D412yes—
525C2×C8C22×D412yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4C22×D412yes—
558C4⋊C4C22×D412yes—
559C4⋊C4C22×D412yes—
560C4⋊C4C22×D412yes—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—44no—
573C42C22×D411yescharacteristic
574C42C22×D411yescharacteristic
575C42H(5,𝔽2)11yescharacteristic
576C42C2511yescharacteristic
577C23—22no—
578C23—24no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23D4211yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8—22no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—82no—
595Q8—82no—
596Q8—82no—
597Q8—82no—
598Q8—161no—
599Q8—161no—
600Q8—161no—
601Q8—164no—
602D4—24no—
603D4—22no—
604D4—22no—
605D4—22no—
606D4—42no—
607D4—88no—
608D4C2×#13714yes—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—88no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—164no—
638C2×C4—164no—
639C2×C4D4211yescharacteristic
640C2×C4D4211yescharacteristic
641C2×C4D4211yescharacteristic
642C2×C4C2×#13712yes—
643C2×C4C2×#13712yes—
644C2×C4#57712yes—
645C2×C4#57812yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C23×D411yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C8—22no—
650C8—22no—
651C8—22no—
652C8—42no—
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C8C2×#13714yes—
658C22—24no—
659C22—42no—
660C22—42no—
661C22—48no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—84no—
666C22#326612yes—
667C22#327612yes—
668C22#328812yes—
669C22C2×D4211yescharacteristic
670C4—21no—
671C4—21no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#326612yes—
683C4#327612yes—
684C4C22×#13711yescharacteristic
685C4C22×#13711yescharacteristic
686C4C2×#57711yescharacteristic
687C4C2×#57811yescharacteristic
688C2—44no—
689C2—82no—
690C2#2018114yesdirect product factor
691C2C2×#326611yescharacteristic
692C2C2×#327611yescharacteristic
693C2C2×#328811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC42C25
prime radicalC2×#547C22
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×#3385C2semidirect
5. #20181C2direct (N × H)
6. C2×#1723C2semidirect
7. C2×#1704C2semidirect
8. C2×#1692C2semidirect
9. C2×#1695C2semidirect
10. C2×#1675C2semidirect
11. C2×#1645C2semidirect
13. C2×#1632C2semidirect
14. C2×#1632C2semidirect
15. C2×#1592C2semidirect
16. C2×#1582C2semidirect
23. #3385C22semidirect
24. #3385C22semidirect
65. #1723C22semidirect
66. #1704C22semidirect
67. #1695C22semidirect
68. #1692C22semidirect
69. #1675C22semidirect
70. #1645C22semidirect
72. #1632C22semidirect
73. #1632C22semidirect
74. #1592C22semidirect
75. #1582C22semidirect
690. C2#20181direct (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×#33352
C2×#33312
#201812
C2×#17232
C2×#17042
C2×#16922
C2×#16952
C2×#16752
C2×#16452
C2×#16352
C2×#16322
C2×#15922
C2×#15822