Finite Group Databasebeta

← C2×#62369 #10578399

Subgroups — C2×#62369 #10578399

Order 512.

6116 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3523C212yesmaximal, complemented
2#62369C2132yesmaximal, direct product factor
3C2×#3402C212yesmaximal, complemented
4C2×#3398C212yesmaximal, complemented
5C2×#3390C212yesmaximal, complemented
6C2×#3386C212yesmaximal, complemented
7C2×#3377C214yesmaximal, complemented
8C2×#3374C212yesmaximal, complemented
9C2×#3367C212yesmaximal, complemented
10C2×#3250C212yesmaximal
11C2×#3249C212yesmaximal, complemented
12C2×#3248C212yesmaximal, complemented
13C2×#3181C214yesmaximal, complemented
14C2×#3170C211yesmaximal, characteristic, complemented
15C2×#3168C211yesmaximal, characteristic
16C22×#446C211yesmaximal, characteristic, complemented
17#3523—216no—
18#3523C22132yescomplemented
19C2×#557C2212yes—
20C2×#557C2212yes—
21C2×#557C2214yes—
22C2×#556C2212yes—
23C2×#556C2214yes—
24C2×#552—22no—
25C2×#552—24no—
26C2×D4×Q8—22no—
27C2×D4×Q8C2212yes—
28C2×#541—22no—
29C2×#541—24no—
30C2×#533C2212yes—
31C2×#533C2214yes—
32C2×#529C2214yes—
33C2×#529C2212yes—
34C2×#519—22no—
35C22×C4×Q8C2211yescharacteristic
36#3402C22132yescomplemented
37#3398C22132yescomplemented
38#3390C22132yescomplemented
39#3386C22132yescomplemented
40#3377C22164yescomplemented
41#3374C22132yescomplemented
42#3367C22132yescomplemented
43#3250C22132yes—
44#3249C22132yescomplemented
45#3248C22132yescomplemented
46#3181C22164yescomplemented
47#3170C22116yescomplemented
48#3168C22116yes—
49C2×#501C2214yes—
50C2×#500C2211yescharacteristic
51C2×#499C2211yescharacteristic
52C2×#497C2214yes—
53C2×#495C2211yescharacteristic
54C2×#494C2211yescharacteristic
55C2×#491C2214yes—
56C2×#489C2214yes—
57C2×#488C2212yes—
58C2×#487C2212yes—
59C2×#479C2214yes—
60C2×#479C2214yes—
61C2×#477C2214yes—
62C2×#477C2214yes—
63C2×#475C2212yes—
64C2×#478C2214yes—
65C2×#474C2212yes—
66C2×#467C2212yes—
67C2×#467C2212yes—
68C2×#467C2214yes—
69C2×#467C2211yescharacteristic
70C2×#466C2211yescharacteristic
71C2×#465C2212yes—
72C2×#465C2214yes—
73C2×#464C2214yes—
74C2×#464C2214yes—
75C2×#462C2212yes—
76C2×#462C2214yes—
77C2×#462C2214yes—
78C2×#463C2214yes—
79C2×#461C2212yes—
80C2×#446C2212yes—
81C2×#446C2212yes—
82C2×#446C2212yes—
83C2×#446C2212yes—
84C2×#446C22116yescomplemented
85C2×#441C2214yes—
86C2×#441C2214yes—
87C2×#440C2214yes—
88C2×#440C2214yes—
89C2×C4×Q16C2212yes—
90C2×C4×Q16C2212yes—
91C2×C4×Q16C2211yescharacteristic
92C2×C4×SD16C2212yes—
93C2×C4×D8C2211yescharacteristic
94C22×C4⋊C8C2211yescharacteristic
95C22×C4⋊C8C2212yes—
96C22×#97C2212yes—
97C22×C4×C8C2211yescharacteristic
98C23×Q8—22no—
99C23×Q8D411yescharacteristic
100C22×Q16C2312yes—
101C22×Q16C2312yes—
102C22×Q16C2311yescharacteristic
103C22×Q16C2314yes—
104C22×SD16C2312yes—
105C22×SD16C2314yes—
106C22×D8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109C23×C8C2312yes—
110#557—28no—
111#557—28no—
112#557—216no—
113#557C23116yes—
114#557C23116yes—
115#557C23132yes—
116#556—28no—
117#556—216no—
118#556C23116yes—
119#556C23132yes—
120#552—232no—
121#552—264no—
122D4×Q8—232no—
123D4×Q8—28no—
124D4×Q8C23116yes—
125#541—232no—
126#541—264no—
127#533—28no—
128#533—216no—
129#533C23116yes—
130#533C23132yes—
131C2×C4⋊Q8—22no—
132C2×C4⋊Q8—24no—
133C2×C4⋊Q8C2312yes—
134C2×C4⋊Q8C2314yes—
135C2×C4⋊Q8C2312yes—
136C2×C4⋊Q8C2314yes—
137#529—216no—
138#529—28no—
139#529C23116yes—
140#529C23132yes—
141C2×#126—24no—
142C2×#126—24no—
143C2×#126—24no—
144C2×#126—24no—
145C2×#126—22no—
146C2×#125—24no—
147C2×#125—24no—
148C2×#125—24no—
149C2×#125—22no—
150C2×#125—24no—
151C2×#125—24no—
152C2×#125—22no—
153C2×#125C2314yes—
154C2×#125C2312yes—
155C2×#124C2312yes—
156C2×#124C2314yes—
157C2×#123—24no—
158C2×#123—22no—
159C2×C22⋊Q8—22no—
160C2×C22⋊Q8—24no—
161C2×C22⋊Q8—22no—
162C2×C22⋊Q8—24no—
163C2×C22⋊Q8—22no—
164C2×C22⋊Q8—24no—
165C2×C22⋊Q8C2314yes—
166C2×C22⋊Q8C2312yes—
167C2×#121C2312yes—
168#519—232no—
169C2×C4×Q8—22no—
170C2×C4×Q8—22no—
171C2×C4×Q8—24no—
172C2×C4×Q8—22no—
173C2×C4×Q8—24no—
174C2×C4×Q8D412yes—
175C2×C4×Q8D412yes—
176C2×C4×Q8D418yes—
177C2×C4×Q8C2312yes—
178C2×C4×Q8C2314yes—
179C2×C4×Q8C2314yes—
180C2×C4×Q8C2314yes—
181C2×C4×Q8C2314yes—
182C2×C4×Q8C2312yes—
183C2×C4×Q8C2312yes—
184C2×C4×Q8C2318yes—
185C2×C4×D4—22no—
186C2×C4×D4—24no—
187C2×C4×D4C2314yes—
188C2×C4×D4C2312yes—
189C2×#117—22no—
190C2×#117—24no—
191C2×#117—22no—
192C2×#117—24no—
193C2×#117—22no—
194C2×#117—24no—
195C22×C4⋊C4D411yescharacteristic
196C22×C4⋊C4D412yes—
197C22×C42C2311yescharacteristic
198C22×C42C2312yes—
199#501C23132yes—
200#500C2318yes—
201#499C2318yes—
202#497C23132yes—
203#495C2318yes—
204#494C2318yes—
205#491C23132yes—
206#489C23132yes—
207#488C23116yes—
208#487C23116yes—
209#479C23132yes—
210#479C23132yes—
211#478C23132yes—
212#477C23132yes—
213#477C23132yes—
214#475C23116yes—
215#474C23116yes—
216#467C2318yes—
217#467C23116yes—
218#467C23116yes—
219#467C23132yes—
220#466C2318yes—
221#465C23116yes—
222#465C23132yes—
223#464C23132yes—
224#464C23132yes—
225#463C23132yes—
226#462C23116yes—
227#462C23132yes—
228#462C23132yes—
229#461C23116yes—
230#446C23116yes—
231#446C23116yes—
232#446C23116yes—
233#446C23116yes—
234#441C23132yes—
235#441C23132yes—
236#440C23132yes—
237#440C23132yes—
238C4×Q16C2318yes—
239C4×Q16C23116yes—
240C4×Q16C23116yes—
241C4×SD16C23116yes—
242C4×D8C2318yes—
243C2×#107C2312yes—
244C2×#107C2311yescharacteristic
245C2×#107C2314yes—
246C2×#107C2311yescharacteristic
247C2×#107C2312yes—
248C2×#106C2314yes—
249C2×#106C2312yes—
250C2×C4⋊C8C2314yes—
251C2×C4⋊C8C2312yes—
252C2×C4⋊C8C2314yes—
253C2×C4⋊C8C2312yes—
254C2×C4⋊C8C2318yes—
255C2×C4⋊C8C23116yes—
256C2×#103C2314yes—
257C2×#103C2312yes—
258C2×#103C2314yes—
259C2×#103C2314yes—
260C2×#103C2314yes—
261C2×#103C2314yes—
262C2×#103C2312yes—
263C2×#102C2312yes—
264C2×#102C2314yes—
265C2×#102C2312yes—
266C2×#97C2314yes—
267C2×#97C2314yes—
268C2×#97C23116yes—
269C2×C4×C8C2311yescharacteristic
270C2×C4×C8C2311yescharacteristic
271C2×C4×C8C2311yescharacteristic
272C2×C4×C8C2311yescharacteristic
273C2×C4×C8C2318yes—
274C22×Q8—216no—
275C22×Q8—24no—
276C22×Q8—22no—
277C22×Q8—44no—
278C22×Q8—416no—
279C22×Q8C2×D412yes—
280C22×Q8C2×D412yes—
281C22×Q8C2×D418yes—
282C22×Q8C2412yes—
283C22×Q8C2414yes—
284C22×D4—22no—
285C22×D4C2412yes—
286C23×C4—24no—
287C23×C4—22no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D412yes—
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D412yes—
292C23×C4C2411yescharacteristic
293C2×Q16—28no—
294C2×Q16—42no—
295C2×Q16—48no—
296C2×Q16—44no—
297C2×Q16C2418yes—
298C2×Q16C2414yes—
299C2×Q16C2418yes—
300C2×Q16C24116yes—
301C2×SD16—28no—
302C2×SD16—48no—
303C2×SD16C2418yes—
304C2×SD16C24116yes—
305C2×D8—42no—
306C2×D8C2414yes—
307C22×C8—24no—
308C22×C8—42no—
309C22×C8—44no—
310C22×C8C2411yescharacteristic
311C22×C8C2414yes—
312C22×C8C2411yescharacteristic
313C22×C8C2411yescharacteristic
314C22×C8C2414yes—
315C22×C8C2414yes—
316C22×C8C2418yes—
317C22×C8C2411yescharacteristic
318C4⋊Q8—216no—
319C4⋊Q8—232no—
320C4⋊Q8—24no—
321C4⋊Q8—28no—
322C4⋊Q8—24no—
323C4⋊Q8—28no—
324C4⋊Q8C2418yes—
325C4⋊Q8C24116yes—
326C4⋊Q8C2418yes—
327C4⋊Q8C24116yes—
328#126—232no—
329#126—232no—
330#126—232no—
331#126—232no—
332#126—216no—
333#125—232no—
334#125—232no—
335#125—232no—
336#125—216no—
337#125—232no—
338#125—232no—
339#125—216no—
340#125—28no—
341#125—24no—
342#125C24116yes—
343#125C2418yes—
344#124—28no—
345#124—24no—
346#124C24116yes—
347#124C2418yes—
348#123—216no—
349#123—232no—
350C22⋊Q8—216no—
351C22⋊Q8—232no—
352C22⋊Q8—216no—
353C22⋊Q8—232no—
354C22⋊Q8—28no—
355C22⋊Q8—24no—
356C22⋊Q8—216no—
357C22⋊Q8—232no—
358C22⋊Q8C2418yes—
359C22⋊Q8C24116yes—
360#121—24no—
361#121C2418yes—
362C4×Q8—216no—
363C4×Q8—216no—
364C4×Q8—232no—
365C4×Q8—216no—
366C4×Q8—232no—
367C4×Q8—24no—
368C4×Q8—28no—
369C4×Q8—28no—
370C4×Q8—28no—
371C4×Q8—28no—
372C4×Q8C2×D4116yes—
373C4×Q8C2×D4116yes—
374C4×Q8C2×D418yes—
375C4×Q8C2×D418yes—
376C4×Q8C2418yes—
377C4×Q8C24116yes—
378C4×Q8C24116yes—
379C4×Q8C24116yes—
380C4×Q8C24116yes—
381C4×Q8C2418yes—
382C4×Q8C2418yes—
383C4×D4—216no—
384C4×D4—232no—
385C4×D4—24no—
386C4×D4—28no—
387C4×D4C2418yes—
388C4×D4C24116yes—
389#117—216no—
390#117—232no—
391#117—216no—
392#117—232no—
393#117—216no—
394#117—232no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—24no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—24no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4C2×D412yes—
413C2×C4⋊C4C2×D414yes—
414C2×C4⋊C4C2×D414yes—
415C2×C4⋊C4C2×D412yes—
416C2×C4⋊C4C2×D418yes—
417C2×C4⋊C4C2×D4116yes—
418C2×C4⋊C4C2414yes—
419C2×C4⋊C4C2412yes—
420C2×C4⋊C4C2412yes—
421C2×C22⋊C4—24no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—24no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—24no—
429C2×C22⋊C4—22no—
430C2×C42—24no—
431C2×C42—24no—
432C2×C42—24no—
433C2×C42—24no—
434C2×C42—22no—
435C2×C42C2×D411yescharacteristic
436C2×C42C2×D414yes—
437C2×C42C2×D411yescharacteristic
438C2×C42C2×D414yes—
439C2×C42C2×D418yes—
440C2×C42C2411yescharacteristic
441C2×C42C2414yes—
442C2×C42C2411yescharacteristic
443C2×C42C2414yes—
444C2×C42C2418yes—
445#107C2414yes—
446#107C2418yes—
447#107C2414yes—
448#107C2418yes—
449#107C24116yes—
450#106C2418yes—
451#106C24116yes—
452C4⋊C8C2418yes—
453C4⋊C8C24116yes—
454C4⋊C8C24116yes—
455C4⋊C8C2418yes—
456#103C2418yes—
457#103C24116yes—
458#103C2418yes—
459#103C24116yes—
460#103C24116yes—
461#103C24116yes—
462#103C24116yes—
463#102C2418yes—
464#102C2418yes—
465#102C24116yes—
466#97C24116yes—
467#97C24116yes—
468C4×C8C2414yes—
469C4×C8C2414yes—
470C4×C8C2414yes—
471C4×C8C2414yes—
472C24—22no—
473C24C22×D411yescharacteristic
474C2×Q8—28no—
475C2×Q8—22no—
476C2×Q8—24no—
477C2×Q8—216no—
478C2×Q8—28no—
479C2×Q8—24no—
480C2×Q8—28no—
481C2×Q8—28no—
482C2×Q8—24no—
483C2×Q8—44no—
484C2×Q8—416no—
485C2×Q8—416no—
486C2×Q8—464no—
487C2×Q8—48no—
488C2×Q8C22×D418yes—
489C2×Q8C22×D418yes—
490C2×Q8C2514yes—
491C2×Q8C2518yes—
492C2×D4—28no—
493C2×D4—22no—
494C2×D4—24no—
495C2×D4—24no—
496C2×D4—48no—
497C2×D4C2514yes—
498C22×C4—24no—
499C22×C4—28no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—216no—
509C22×C4—28no—
510C22×C4—44no—
511C22×C4—416no—
512C22×C4—48no—
513C22×C4C22×D412yes—
514C22×C4C22×D414yes—
515C22×C4C22×D418yes—
516C22×C4C22×D414yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D414yes—
519C22×C4C22×D414yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D418yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C2512yes—
526C22×C4C2511yescharacteristic
527Q16—216no—
528Q16—44no—
529Q16—416no—
530Q16—48no—
531SD16—216no—
532SD16—416no—
533D8—44no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—28no—
537C2×C8—22no—
538C2×C8—28no—
539C2×C8—44no—
540C2×C8—48no—
541C2×C82−1+414yes—
542C2×C8C2512yes—
543C2×C8C2512yes—
544C2×C8C2512yes—
545C2×C8C2518yes—
546C2×C8C2512yes—
547C4⋊C4—216no—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—216no—
551C4⋊C4—28no—
552C4⋊C4—216no—
553C4⋊C4—216no—
554C4⋊C4—28no—
555C4⋊C4—216no—
556C4⋊C4—28no—
557C4⋊C4—216no—
558C4⋊C4—28no—
559C4⋊C4—216no—
560C4⋊C4—216no—
561C4⋊C4—216no—
562C4⋊C4—216no—
563C4⋊C4—28no—
564C4⋊C4—24no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4C22×D418yes—
568C4⋊C4C22×D4116yes—
569C4⋊C4C22×D4116yes—
570C4⋊C4C22×D418yes—
571C4⋊C4C2514yes—
572C4⋊C4C2514yes—
573C4⋊C4C2518yes—
574C22⋊C4—216no—
575C22⋊C4—28no—
576C22⋊C4—28no—
577C22⋊C4—28no—
578C22⋊C4—216no—
579C22⋊C4—216no—
580C22⋊C4—28no—
581C22⋊C4—216no—
582C22⋊C4—28no—
583C42—216no—
584C42—216no—
585C42—216no—
586C42—216no—
587C42—28no—
588C42C22×D414yes—
589C42C22×D4116yes—
590C42C22×D412yes—
591C42C22×D418yes—
592C42C22×D412yes—
593C42C22×D414yes—
594C42C2512yes—
595C42C2512yes—
596C42C2518yes—
597C23—22no—
598C23—22no—
599C23—24no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23#57814yes—
604C23C23×D412yes—
605C23C23×D411yescharacteristic
606Q8—216no—
607Q8—24no—
608Q8—28no—
609Q8—28no—
610Q8—24no—
611Q8—48no—
612Q8—432no—
613Q8—44no—
614Q8—48no—
615Q8—416no—
616D4—24no—
617D4—24no—
618D4—416no—
619D4—44no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—216no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—48no—
638C2×C4—432no—
639C2×C4—44no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—416no—
645C2×C4#57514yes—
646C2×C4C23×D412yes—
647C2×C4C23×D418yes—
648C2×C4C23×D418yes—
649C2×C4C23×D412yes—
650C2×C4C23×D412yes—
651C2×C4C23×D412yes—
652C2×C4C23×D411yescharacteristic
653C2×C4C23×D412yes—
654C2×C4C2×2−1+412yes—
655C2×C4#58512yes—
656C2×C4C2611yescharacteristic
657C8—22no—
658C8—22no—
659C8—28no—
660C8—22no—
661C8C2×2−1+414yes—
662C22—22no—
663C22—22no—
664C22—28no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—88no—
669C22#349712yes—
670C22C2×#57512yes—
671C22C2×#57814yes—
672C22C2×#57812yes—
673C22C24×D411yescharacteristic
674C4—22no—
675C4—22no—
676C4—28no—
677C4—22no—
678C4—28no—
679C4—22no—
680C4—44no—
681C4—48no—
682C4—48no—
683C4—44no—
684C4—44no—
685C4#349712yes—
686C4C2×#57514yes—
687C4C22×2−1+411yescharacteristic
688C4C2×#58511yescharacteristic
689C2—24no—
690C2—84no—
691C2#6236914yesdirect product factor
692C2C2×#349711yescharacteristic
693C2C22×#57511yescharacteristic
694C2C22×#57811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC2×#466C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3523C2semidirect
2. #62369C2direct (N × H)
3. C2×#3402C2semidirect
4. C2×#3398C2semidirect
5. C2×#3390C2semidirect
6. C2×#3386C2semidirect
7. C2×#3377C2semidirect
8. C2×#3374C2semidirect
9. C2×#3367C2semidirect
11. C2×#3249C2semidirect
12. C2×#3248C2semidirect
13. C2×#3181C2semidirect
14. C2×#3170C2semidirect
16. C22×#446C2semidirect
18. #3523C22semidirect
36. #3402C22semidirect
37. #3398C22semidirect
38. #3390C22semidirect
39. #3386C22semidirect
40. #3377C22semidirect
41. #3374C22semidirect
42. #3367C22semidirect
44. #3249C22semidirect
45. #3248C22semidirect
46. #3181C22semidirect
47. #3170C22semidirect
84. C2×#446C22semidirect
691. C2#62369direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35232
#623692
C2×#34022
C2×#33982
C2×#33902
C2×#33862
C2×#33772
C2×#33742
C2×#33672
C2×#32502
C2×#32492
C2×#32482
C2×#31812
C2×#31702
C2×#31682
C22×#4462