Finite Group Databasebeta

← C2×#82735 #11661569

Subgroups — C2×#82735 #11661569

Order 768.

5078 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#235C211yesmaximal, characteristic
2C22×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
3C22×#942C212yesmaximal, complemented
4C2×#5207C211yesmaximal, characteristic, complemented
5C2×#5050C212yesmaximal, complemented
6#82735C2116yesmaximal, direct product factor
7C2×#5047C214yesmaximal, complemented
8C2×#5046C212yesmaximal, complemented
9C2×#4966C212yesmaximal, complemented
10C2×#3007—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic, complemented
12C24×C3×C4C2211yescharacteristic
13C23×#67—21no—
14C22×#235—21no—
15C22×#235—21no—
16C22×#235—24no—
17C22×#235—216no—
18C22×#235C2212yes—
19C22×#235C2218yes—
20C22×#235C2212yes—
21C23×D12C2211yescharacteristic
22C23×C4×D3—21no—
23C23×Dic6C2211yescharacteristic
24C2×C3×C22⋊Q8—21no—
25C2×C3×C22⋊Q8—21no—
26C2×C3×C22⋊Q8—24no—
27C2×C3×C22⋊Q8C2212yes—
28C2×C3×C22⋊Q8C2212yescomplemented
29C2×C3×C22⋊Q8C2218yescomplemented
30C22×C3×C4⋊C4—21no—
31C22×C3×C4⋊C4C2211yescharacteristic, complemented
32C22×C3×C22⋊C4—21no—
33C22×#216C2214yescomplemented
34C2×#942—216no—
35C2×#942C22116yescomplemented
36C22×#215C2214yescomplemented
37C22×#208C2212yescomplemented
38#5207C2218yescomplemented
39C2×#830C2212yes—
40C2×#830C2212yescomplemented
41#5050C22116yescomplemented
42#5047C22132yescomplemented
43#5046C22116yescomplemented
44C2×#810C2214yescomplemented
45#4966C22116yescomplemented
46C2×#809C2214yescomplemented
47C23×#41—31no—
48C22×#137—32no—
49C22×C22⋊Q8D311yescharacteristic, complemented
50#3007—316no—
51C2×#452—34no—
52C2×#451—32no—
53C2×#450—32no—
54C2×#417—32no—
55C2×#407—31no—
56C25×C3D411yescharacteristic
57C24×D3—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—28no—
61C22×C3×Q8C2312yes—
62C22×C3×Q8C2314yescomplemented
63C23×C3×C4—28no—
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D414yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2314yes—
73C23×C3×C4C2311yescharacteristic
74C22×#67—216no—
75C22×#67—22no—
76C22×#67—28no—
77C22×#67—22no—
78C23×Dic3—21no—
79C2×#235—216no—
80C2×#235—264no—
81C2×#235—216no—
82C2×#235—28no—
83C2×#235—28no—
84C2×#235—24no—
85C2×#235—28no—
86C2×#235—24no—
87C2×#235—264no—
88C2×#235—28no—
89C2×#235—28no—
90C2×#235D4116yes—
91C2×#235C2318yes—
92C2×#235C2318yes—
93C22×D12—21no—
94C22×D12—22no—
95C22×D12—28no—
96C22×D12C2314yes—
97C22×D12C2312yes—
98C22×C4×D3—216no—
99C22×C4×D3—22no—
100C22×C4×D3—28no—
101C22×C4×D3—22no—
102C22×Dic6—21no—
103C22×Dic6—22no—
104C22×Dic6—28no—
105C22×Dic6C2314yes—
106C22×Dic6C2312yes—
107C3×C22⋊Q8—24no—
108C3×C22⋊Q8—28no—
109C3×C22⋊Q8—28no—
110C3×C22⋊Q8—24no—
111C3×C22⋊Q8C2318yes—
112C3×C22⋊Q8C2318yescomplemented
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—28no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C4⋊C4C2314yescomplemented
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4—28no—
122C2×C3×C22⋊C4—22no—
123C2×#216—216no—
124C2×#216C23116yescomplemented
125#942—264no—
126C2×#215—216no—
127C2×#215C23116yescomplemented
128C2×#208—28no—
129C2×#208C2318yescomplemented
130C22×C3⋊C8C2314yes—
131#830C2318yes—
132#830C2318yescomplemented
133#810C23116yescomplemented
134#809C23116yescomplemented
135C22×#41—32no—
136C22×#41—32no—
137C22×#41—38no—
138C22×#41—616no—
139C22×#41—61no—
140C22×#41—64no—
141C22×#41—61no—
142C23×Q8—31no—
143C23×Q8C2×D311yescharacteristic, complemented
144C23×D4—31no—
145C23×D4—61no—
146C24×C4—61no—
147C24×C4C2×D311yescharacteristic
148C2×#137—316no—
149C2×#137—616no—
150C22×Q16—34no—
151C22×SD16—34no—
152C22×M16—32no—
153C2×C22⋊Q8—21no—
154C2×C22⋊Q8—21no—
155C2×C22⋊Q8—24no—
156C2×C22⋊Q8C2×D312yes—
157C2×C22⋊Q8C2×D312yescomplemented
158C2×C22⋊Q8C2×D318yescomplemented
159C22×C4⋊C4—21no—
160C22×C4⋊C4C2×D311yescharacteristic, complemented
161C22×C22⋊C4—21no—
162#452—332no—
163#451—316no—
164#450—316no—
165#417—316no—
166C2×#103—34no—
167C2×#102—34no—
168#407—38no—
169C2×C22⋊C8—32no—
170C2×C22⋊C8—32no—
171C24×C3—24no—
172C24×C3—24no—
173C24×C3—42no—
174C24×C3C2×D411yescharacteristic
175C24×C3C2×D414yes—
176C24×C3C2×D411yescharacteristic
177C24×C3C2×D411yescharacteristic
178C23×D3—28no—
179C23×D3—24no—
180C23×D3—22no—
181C23×D3—44no—
182C23×D3—44no—
183C2×C3×Q8—22no—
184C2×C3×Q8—24no—
185C2×C3×Q8—28no—
186C2×C3×Q8—216no—
187C2×C3×Q8—44no—
188C2×C3×Q8—48no—
189C2×C3×Q8C2414yes—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—216no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—216no—
197C22×C3×C4—22no—
198C22×C3×C4—24no—
199C22×C3×C4—22no—
200C22×C3×C4—24no—
201C22×C3×C4—44no—
202C22×C3×C4—44no—
203C22×C3×C4C2×D418yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D414yes—
209C22×C3×C4C2×D414yes—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2412yes—
214C22×C3×C4C2412yes—
215C22×C3×C4C2411yescharacteristic
216C22×C3×C4C2412yes—
217C2×#67—264no—
218C2×#67—216no—
219C2×#67—216no—
220C2×#67—28no—
221C2×#67—28no—
222C22×Dic3—28no—
223C22×Dic3—24no—
224C22×Dic3—22no—
225#235—264no—
226#235—264no—
227#235—232no—
228#235—232no—
229#235—216no—
230#235—216no—
231#235—216no—
232#235C2×D4132yes—
233C2×D12—28no—
234C2×D12—216no—
235C2×D12—24no—
236C2×D12—24no—
237C2×D12—22no—
238C2×D12—216no—
239C2×D12C2×D418yes—
240C2×D12C2414yes—
241C2×C4×D3—216no—
242C2×C4×D3—264no—
243C2×C4×D3—216no—
244C2×C4×D3—28no—
245C2×C4×D3—28no—
246C2×Dic6—28no—
247C2×Dic6—216no—
248C2×Dic6—24no—
249C2×Dic6—24no—
250C2×Dic6—22no—
251C2×Dic6—216no—
252C2×Dic6C2×D418yes—
253C2×Dic6C2414yes—
254C3×C4⋊C4—28no—
255C3×C4⋊C4—28no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4C2414yes—
259C3×C22⋊C4—28no—
260C3×C22⋊C4—28no—
261#216—232no—
262#215—232no—
263#208—216no—
264C2×C3⋊C8—28no—
265C2×C3⋊C8C2418yes—
266C25—61no—
267C25#6711yescharacteristic
268C2×#41—316no—
269C2×#41—38no—
270C2×#41—38no—
271C2×#41—68no—
272C2×#41—68no—
273C2×#41—664no—
274C2×#41—616no—
275C2×#41—664no—
276C2×#41—616no—
277C2×#41—68no—
278C2×#41—68no—
279C2×#41—64no—
280C2×#41—68no—
281C2×#41—64no—
282C22×Q8—22no—
283C22×Q8—21no—
284C22×Q8—28no—
285C22×Q8—32no—
286C22×Q8—34no—
287C22×Q8—68no—
288C22×Q8—62no—
289C22×Q8—61no—
290C22×Q8C22×D312yes—
291C22×Q8C22×D314yescomplemented
292C22×D4—32no—
293C22×D4—34no—
294C22×D4—68no—
295C22×D4—616no—
296C22×D4—68no—
297C22×D4—62no—
298C22×D4—62no—
299C22×D4—62no—
300C22×D4—61no—
301C23×C4—28no—
302C23×C4—21no—
303C23×C4—21no—
304C23×C4—616no—
305C23×C4—61no—
306C23×C4—68no—
307C23×C4—62no—
308C23×C4—62no—
309C23×C4#6711yescharacteristic
310C23×C4#6711yescharacteristic
311C23×C4#6714yes—
312C23×C4#6711yescharacteristic
313C23×C4C22×D311yescharacteristic
314C23×C4C22×D311yescharacteristic
315C23×C4C22×D314yes—
316C23×C4C22×D311yescharacteristic
317#137—664no—
318C2×Q16—316no—
319C2×Q16—616no—
320C2×SD16—316no—
321C2×SD16—616no—
322C2×M16—38no—
323C2×M16—68no—
324C22×C8—34no—
325C22⋊Q8—24no—
326C22⋊Q8—28no—
327C22⋊Q8—28no—
328C22⋊Q8—24no—
329C22⋊Q8C22×D318yes—
330C22⋊Q8C22×D318yescomplemented
331C2×C4⋊C4—22no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C22×D312yes—
337C2×C4⋊C4C22×D314yescomplemented
338C2×C22⋊C4—22no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—22no—
341#103—316no—
342#102—316no—
343C22⋊C8—38no—
344C22⋊C8—38no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—28no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—28no—
353C23×C3—22no—
354C23×C3—44no—
355C23×C3—416no—
356C23×C3#12014yes—
357C23×C3#13714yescomplemented
358C23×C3C22×D412yes—
359C23×C3C22×D412yes—
360C23×C3C22×D412yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—216no—
363C22×D3—28no—
364C22×D3—24no—
365C22×D3—432no—
366C22×D3—48no—
367C22×D3—44no—
368C22×D3—48no—
369C22×D3—44no—
370C3×Q8—28no—
371C3×Q8—44no—
372C3×Q8—48no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—28no—
376C2×C3×C4—24no—
377C2×C3×C4—28no—
378C2×C3×C4—28no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—24no—
384C2×C3×C4—48no—
385C2×C3×C4—48no—
386C2×C3×C4—44no—
387C2×C3×C4—44no—
388C2×C3×C4#12014yes—
389C2×C3×C4#12018yes—
390C2×C3×C4C22×D414yes—
391C2×C3×C4C22×D418yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D412yes—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×D412yes—
396C2×C3×C4C2511yescharacteristic
397#67—232no—
398#67—232no—
399C2×Dic3—216no—
400C2×Dic3—28no—
401C2×Dic3—24no—
402D12—216no—
403D12—28no—
404D12—24no—
405D12C22×D418yes—
406C4×D3—232no—
407C4×D3—232no—
408Dic6—216no—
409Dic6—28no—
410Dic6—24no—
411Dic6C22×D418yes—
412C3⋊C8—28no—
413C24—24no—
414C24—24no—
415C24—42no—
416C24—68no—
417C24—64no—
418C24—62no—
419C24—124no—
420C24—124no—
421C24C2×#6711yescharacteristic
422C24C2×#6714yes—
423C24C2×#6711yescharacteristic
424C24C2×#6711yescharacteristic
425#41—332no—
426#41—616no—
427#41—616no—
428#41—664no—
429#41—632no—
430#41—664no—
431#41—632no—
432#41—616no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—28no—
436C2×Q8—216no—
437C2×Q8—38no—
438C2×Q8—34no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—616no—
442C2×Q8—68no—
443C2×Q8—616no—
444C2×Q8—64no—
445C2×Q8—64no—
446C2×Q8—62no—
447C2×Q8C23×D314yes—
448C2×D4—38no—
449C2×D4—34no—
450C2×D4—616no—
451C2×D4—68no—
452C2×D4—616no—
453C2×D4—664no—
454C2×D4—616no—
455C2×D4—616no—
456C2×D4—68no—
457C2×D4—68no—
458C2×D4—64no—
459C2×D4—64no—
460C2×D4—62no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—216no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—616no—
475C22×C4—664no—
476C22×C4—68no—
477C22×C4—64no—
478C22×C4—62no—
479C22×C4—616no—
480C22×C4—68no—
481C22×C4—68no—
482C22×C4D3×D418yes—
483C22×C4C2×#6714yes—
484C22×C4C2×#6714yes—
485C22×C4C2×#6712yes—
486C22×C4C2×#6712yes—
487C22×C4C2×#6714yes—
488C22×C4C2×#6711yescharacteristic
489C22×C4C2×#6711yescharacteristic
490C22×C4C2×#6712yes—
491C22×C4C2×#6711yescharacteristic
492C22×C4C23×D312yes—
493C22×C4C23×D312yes—
494C22×C4C23×D312yes—
495C22×C4C23×D311yescharacteristic
496Q16—632no—
497SD16—632no—
498M16—616no—
499C2×C8—38no—
500C2×C8—68no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4C23×D314yes—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22×C3—22no—
509C22×C3—28no—
510C22×C3—28no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—416no—
518C22×C3—44no—
519C22×C3C2×#12014yes—
520C22×C3C2×#12012yes—
521C22×C3C2×#13712yes—
522C22×C3C2×#13714yescomplemented
523C22×C3#57512yes—
524C22×C3C23×D411yescharacteristic
525C2×D3—28no—
526C2×D3—432no—
527C2×D3—416no—
528C2×D3—44no—
529C2×D3—44no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—44no—
533C3×C4—44no—
534C3×C4C2×#12012yes—
535C3×C4C2×#12014yes—
536C3×C4C23×D412yes—
537Dic3—28no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—28no—
543C23—22no—
544C23—22no—
545C23—28no—
546C23—22no—
547C23—44no—
548C23—416no—
549C23—616no—
550C23—64no—
551C23—68no—
552C23—1232no—
553C23—128no—
554C23—124no—
555C23—128no—
556C23—124no—
557C23#93714yes—
558C23#94214yescomplemented
559C23C22×#6712yes—
560C23C22×#6712yes—
561C23C22×#6712yes—
562C23C22×#6711yescharacteristic
563Q8—28no—
564Q8—38no—
565Q8—48no—
566Q8—44no—
567Q8—616no—
568Q8—64no—
569Q8—68no—
570D4—38no—
571D4—616no—
572D4—632no—
573D4—632no—
574D4—64no—
575D4—68no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—632no—
592C2×C4—616no—
593C2×C4—64no—
594C2×C4—68no—
595C2×C4—632no—
596C2×C4#93714yes—
597C2×C4#93718yes—
598C2×C4C2×D3×D414yes—
599C2×C4C2×D3×D418yes—
600C2×C4C22×#6712yes—
601C2×C4C22×#6712yes—
602C2×C4C22×#6712yes—
603C2×C4C22×#6711yescharacteristic
604C2×C4C24×D311yescharacteristic
605C8—68no—
606C2×C3—24no—
607C2×C3—24no—
608C2×C3—42no—
609C2×C3#300714yescomplemented
610C2×C3C22×#12011yescharacteristic
611C2×C3C22×#13711yescharacteristic
612C2×C3C2×#57511yescharacteristic
613D3—48no—
614C22—28no—
615C22—28no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—416no—
624C22—44no—
625C22—68no—
626C22—1232no—
627C22—1216no—
628C22—124no—
629C22—124no—
630C22C2×#93714yes—
631C22C2×#93712yes—
632C22C2×#94212yes—
633C22C2×#94214yescomplemented
634C22#582212yes—
635C22C23×#6711yescharacteristic
636C4—22no—
637C4—22no—
638C4—44no—
639C4—44no—
640C4—68no—
641C4C2×#93714yes—
642C4C2×#93712yes—
643C4C22×D3×D412yes—
644C3C2×#300711yescharacteristic, complemented, Sylow
645C2—24no—
646C2—24no—
647C2—42no—
648C2—128no—
649C2#8273514yesdirect product factor
650C2C22×#93711yescharacteristic
651C2C22×#94211yescharacteristic
652C2C2×#582211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC23×#235C2
hypercenterC22×C22⋊Q8D3
hypocenterC3C2×#3007
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×C3×C22⋊Q8C2semidirect
3. C22×#942C2semidirect
4. C2×#5207C2semidirect
5. C2×#5050C2semidirect
6. #82735C2direct (N × H)
7. C2×#5047C2semidirect
8. C2×#5046C2semidirect
9. C2×#4966C2semidirect
11. C23×C3×Q8C22semidirect
28. C2×C3×C22⋊Q8C22semidirect
29. C2×C3×C22⋊Q8C22semidirect
31. C22×C3×C4⋊C4C22semidirect
33. C22×#216C22semidirect
35. C2×#942C22semidirect
36. C22×#215C22semidirect
37. C22×#208C22semidirect
38. #5207C22semidirect
40. C2×#830C22semidirect
41. #5050C22semidirect
42. #5047C22semidirect
43. #5046C22semidirect
44. C2×#810C22semidirect
45. #4966C22semidirect
46. C2×#809C22semidirect
49. C22×C22⋊Q8D3semidirect
62. C22×C3×Q8C23semidirect
112. C3×C22⋊Q8C23semidirect
119. C2×C3×C4⋊C4C23semidirect
124. C2×#216C23semidirect
127. C2×#215C23semidirect
129. C2×#208C23semidirect
132. #830C23semidirect
133. #810C23semidirect
134. #809C23semidirect
143. C23×Q8C2×D3semidirect
157. C2×C22⋊Q8C2×D3semidirect
158. C2×C22⋊Q8C2×D3semidirect
160. C22×C4⋊C4C2×D3semidirect
291. C22×Q8C22×D3semidirect
330. C22⋊Q8C22×D3semidirect
337. C2×C4⋊C4C22×D3semidirect
357. C23×C3#137semidirect
522. C22×C3C2×#137semidirect
558. C23#942semidirect
609. C2×C3#3007semidirect
633. C22C2×#942semidirect
644. C3C2×#3007semidirect
649. C2#82735direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2352
C22×C3×C22⋊Q82
C22×#9422
C2×#52072
C2×#50502
#827352
C2×#50472
C2×#50462
C2×#49662
C2×#30073