Finite Group Databasebeta

← C2×#84084 #11661571

Subgroups — C2×#84084 #11661571

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×C3×#41C211yesmaximal, characteristic
2C22×#951C212yesmaximal, complemented
3C22×#924C211yesmaximal, characteristic, complemented
4#84084C2116yesmaximal, direct product factor
5C2×#5239C214yesmaximal, complemented
6C2×#5232C212yesmaximal, complemented
7C2×#5228C212yesmaximal, complemented
8C2×#5218C212yesmaximal, complemented
9C2×#5207C211yesmaximal, characteristic, complemented
10C2×#3007—31nomaximal, Sylow
11C22×C3×#41—216no—
12C22×C3×#41—21no—
13C22×C3×#41—24no—
14C22×C3×#41—21no—
15C22×C3×#41C2212yes—
16C22×C3×#41C2218yes—
17C22×C3×#41C2212yes—
18C23×C3×Q8C2211yescharacteristic
19C23×C3×D4—21no—
20C23×C3×D4C2211yescharacteristic
21C24×C3×C4—21no—
22C24×C3×C4C2211yescharacteristic
23C23×Dic6C2211yescharacteristic, complemented
24C22×#217—21no—
25C2×#951—216no—
26C2×#951C22116yescomplemented
27C22×#216C2214yescomplemented
28C22×#214C2214yescomplemented
29C22×C4⋊Dic3C2211yescharacteristic, complemented
30C2×#924—21no—
31C2×#924—24no—
32C2×#924—21no—
33C2×#924C2212yes—
34C2×#924C2218yescomplemented
35C2×#924C2212yescomplemented
36C22×Dic3⋊C4—21no—
37C22×#208C2212yescomplemented
38#5239C22132yescomplemented
39#5232C22116yescomplemented
40#5228C22116yescomplemented
41C2×#835C2214yescomplemented
42#5218C22116yescomplemented
43C2×#832C2214yescomplemented
44#5207C2218yescomplemented
45C2×#830C2212yes—
46C2×#830C2212yescomplemented
47C23×#41D311yescharacteristic
48C22×#137—32no—
49C22×C22⋊Q8—31no—
50#3007—316no—
51C2×#452—34no—
52C2×#451—32no—
53C2×#450—32no—
54C2×#417—32no—
55C2×#407—31no—
56C25×C3—21no—
57C25×C3D411yescharacteristic
58C2×C3×#41—216no—
59C2×C3×#41—264no—
60C2×C3×#41—216no—
61C2×C3×#41—264no—
62C2×C3×#41—28no—
63C2×C3×#41—28no—
64C2×C3×#41—28no—
65C2×C3×#41—28no—
66C2×C3×#41—24no—
67C2×C3×#41—28no—
68C2×C3×#41—24no—
69C2×C3×#41D4116yes—
70C2×C3×#41C2318yes—
71C2×C3×#41C2318yes—
72C22×C3×Q8—28no—
73C22×C3×Q8—21no—
74C22×C3×Q8—22no—
75C22×C3×Q8C2314yes—
76C22×C3×Q8C2312yes—
77C22×C3×D4—216no—
78C22×C3×D4—22no—
79C22×C3×D4—28no—
80C22×C3×D4—22no—
81C22×C3×D4—28no—
82C22×C3×D4—21no—
83C22×C3×D4—22no—
84C22×C3×D4C2314yes—
85C22×C3×D4C2312yes—
86C23×C3×C4—216no—
87C23×C3×C4—21no—
88C23×C3×C4—22no—
89C23×C3×C4—28no—
90C23×C3×C4—22no—
91C23×C3×C4—28no—
92C23×C3×C4D411yescharacteristic
93C23×C3×C4D411yescharacteristic
94C23×C3×C4D411yescharacteristic
95C23×C3×C4D414yes—
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2311yescharacteristic
98C23×C3×C4C2314yes—
99C23×C3×C4C2311yescharacteristic
100C23×Dic3—21no—
101C23×Dic3—21no—
102C22×Dic6—28no—
103C22×Dic6—22no—
104C22×Dic6—21no—
105C22×Dic6C2312yes—
106C22×Dic6C2314yescomplemented
107C2×#217—22no—
108C2×#217—28no—
109C2×#217—22no—
110#951—264no—
111C2×#216—216no—
112C2×#216C23116yescomplemented
113C2×#214—216no—
114C2×#214C23116yescomplemented
115C2×C4⋊Dic3—22no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2314yescomplemented
119#924—24no—
120#924—28no—
121#924—24no—
122#924—28no—
123#924C2318yes—
124#924C2318yescomplemented
125C2×Dic3⋊C4—22no—
126C2×Dic3⋊C4—28no—
127C2×Dic3⋊C4—22no—
128C2×#208—28no—
129C2×#208C2318yescomplemented
130C22×C3⋊C8C2314yes—
131#835C23116yescomplemented
132#832C23116yescomplemented
133#830C2318yes—
134#830C2318yescomplemented
135C22×#41—216no—
136C22×#41—21no—
137C22×#41—24no—
138C22×#41—21no—
139C22×#41C2×D312yes—
140C22×#41C2×D318yes—
141C22×#41C2×D312yes—
142C23×Q8—31no—
143C23×Q8C2×D311yescharacteristic
144C23×D4—21no—
145C23×D4C2×D311yescharacteristic
146C24×C4—21no—
147C24×C4C2×D311yescharacteristic
148C2×#137—316no—
149C2×#137—616no—
150C22×Q16—34no—
151C22×SD16—34no—
152C22×M16—32no—
153C2×C22⋊Q8—32no—
154C2×C22⋊Q8—32no—
155C2×C22⋊Q8—38no—
156C2×C22⋊Q8—61no—
157C2×C22⋊Q8—64no—
158C2×C22⋊Q8—61no—
159C22×C4⋊C4—31no—
160C22×C4⋊C4—61no—
161C22×C22⋊C4—61no—
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—28no—
172C24×C3—24no—
173C24×C3—24no—
174C24×C3—24no—
175C24×C3—22no—
176C24×C3—42no—
177C24×C3—44no—
178C24×C3—44no—
179C24×C3C2×D411yescharacteristic
180C24×C3C2×D414yes—
181C24×C3C2×D411yescharacteristic
182C24×C3C2×D411yescharacteristic
183C3×#41—264no—
184C3×#41—232no—
185C3×#41—264no—
186C3×#41—216no—
187C3×#41—216no—
188C3×#41—232no—
189C3×#41—216no—
190C3×#41C2×D4132yes—
191C2×C3×Q8—216no—
192C2×C3×Q8—28no—
193C2×C3×Q8—216no—
194C2×C3×Q8—24no—
195C2×C3×Q8—24no—
196C2×C3×Q8—22no—
197C2×C3×Q8C2×D418yes—
198C2×C3×Q8C2414yes—
199C2×C3×D4—264no—
200C2×C3×D4—216no—
201C2×C3×D4—28no—
202C2×C3×D4—216no—
203C2×C3×D4—216no—
204C2×C3×D4—28no—
205C2×C3×D4—28no—
206C2×C3×D4—216no—
207C2×C3×D4—24no—
208C2×C3×D4—22no—
209C2×C3×D4—24no—
210C2×C3×D4C2×D418yes—
211C2×C3×D4C2414yes—
212C22×C3×C4—28no—
213C22×C3×C4—264no—
214C22×C3×C4—216no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—22no—
218C22×C3×C4—24no—
219C22×C3×C4—216no—
220C22×C3×C4—24no—
221C22×C3×C4—216no—
222C22×C3×C4—28no—
223C22×C3×C4—28no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—216no—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D414yes—
230C22×C3×C4C2×D414yes—
231C22×C3×C4C2×D412yes—
232C22×C3×C4C2×D412yes—
233C22×C3×C4C2×D414yes—
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D412yes—
236C22×C3×C4C2×D418yes—
237C22×C3×C4C2412yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2412yes—
241C22×Dic3—22no—
242C22×Dic3—24no—
243C22×Dic3—22no—
244C22×Dic3—24no—
245C22×Dic3—44no—
246C22×Dic3—44no—
247C2×Dic6—28no—
248C2×Dic6—216no—
249C2×Dic6—22no—
250C2×Dic6—24no—
251C2×Dic6—44no—
252C2×Dic6—48no—
253C2×Dic6C2414yes—
254#217—28no—
255#217—28no—
256#216—232no—
257#214—232no—
258C4⋊Dic3—22no—
259C4⋊Dic3—24no—
260C4⋊Dic3C2414yes—
261Dic3⋊C4—28no—
262Dic3⋊C4—28no—
263#208—216no—
264C2×C3⋊C8—28no—
265C2×C3⋊C8C2418yes—
266C25—21no—
267C25#6711yescharacteristic
268C2×#41—216no—
269C2×#41—264no—
270C2×#41—216no—
271C2×#41—264no—
272C2×#41—28no—
273C2×#41—28no—
274C2×#41—28no—
275C2×#41—28no—
276C2×#41—24no—
277C2×#41—28no—
278C2×#41—24no—
279C2×#41#67116yes—
280C2×#41C22×D318yes—
281C2×#41C22×D318yes—
282C22×Q8—28no—
283C22×Q8—22no—
284C22×Q8—21no—
285C22×Q8—32no—
286C22×Q8—34no—
287C22×Q8—62no—
288C22×Q8—61no—
289C22×Q8—68no—
290C22×Q8C22×D314yes—
291C22×Q8C22×D312yes—
292C22×D4—216no—
293C22×D4—28no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4—28no—
297C22×D4—22no—
298C22×D4—21no—
299C22×D4C22×D314yes—
300C22×D4C22×D312yes—
301C23×C4—216no—
302C23×C4—21no—
303C23×C4—28no—
304C23×C4—22no—
305C23×C4—22no—
306C23×C4—28no—
307C23×C4—61no—
308C23×C4—61no—
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—38no—
326C22⋊Q8—38no—
327C22⋊Q8—68no—
328C22⋊Q8—64no—
329C22⋊Q8—64no—
330C22⋊Q8—68no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—68no—
336C2×C4⋊C4—62no—
337C2×C4⋊C4—62no—
338C2×C22⋊C4—68no—
339C2×C22⋊C4—62no—
340C2×C22⋊C4—62no—
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—216no—
350C23×C3—28no—
351C23×C3—22no—
352C23×C3—28no—
353C23×C3—22no—
354C23×C3—28no—
355C23×C3—22no—
356C23×C3—24no—
357C23×C3—44no—
358C23×C3—416no—
359C23×C3—432no—
360C23×C3—48no—
361C23×C3—44no—
362C23×C3—48no—
363C23×C3—44no—
364C23×C3#12014yes—
365C23×C3#13714yescomplemented
366C23×C3C22×D412yes—
367C23×C3C22×D412yes—
368C23×C3C22×D412yes—
369C23×C3C22×D411yescharacteristic
370C3×Q8—216no—
371C3×Q8—28no—
372C3×Q8—24no—
373C3×Q8C22×D418yes—
374C3×D4—232no—
375C3×D4—216no—
376C3×D4—232no—
377C3×D4—28no—
378C3×D4—24no—
379C3×D4C22×D418yes—
380C2×C3×C4—216no—
381C2×C3×C4—232no—
382C2×C3×C4—22no—
383C2×C3×C4—28no—
384C2×C3×C4—22no—
385C2×C3×C4—28no—
386C2×C3×C4—24no—
387C2×C3×C4—28no—
388C2×C3×C4—232no—
389C2×C3×C4—24no—
390C2×C3×C4—24no—
391C2×C3×C4—28no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4#12014yes—
395C2×C3×C4#12018yes—
396C2×C3×C4C22×D412yes—
397C2×C3×C4C22×D412yes—
398C2×C3×C4C22×D412yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D414yes—
401C2×C3×C4C22×D418yes—
402C2×C3×C4C2511yescharacteristic
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—48no—
406C2×Dic3—48no—
407C2×Dic3—44no—
408C2×Dic3—44no—
409Dic6—28no—
410Dic6—48no—
411Dic6—44no—
412C3⋊C8—28no—
413C24—28no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—22no—
418C24—42no—
419C24—44no—
420C24—44no—
421C24C2×#6711yescharacteristic
422C24C2×#6714yes—
423C24C2×#6711yescharacteristic
424C24C2×#6711yescharacteristic
425#41—264no—
426#41—232no—
427#41—264no—
428#41—216no—
429#41—216no—
430#41—232no—
431#41—216no—
432#41C2×#67132yes—
433C2×Q8—216no—
434C2×Q8—28no—
435C2×Q8—216no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—34no—
440C2×Q8—64no—
441C2×Q8—62no—
442C2×Q8—68no—
443C2×Q8—616no—
444C2×Q8—124no—
445C2×Q8—128no—
446C2×Q8C2×#6718yes—
447C2×Q8C23×D314yes—
448C2×D4—264no—
449C2×D4—216no—
450C2×D4—28no—
451C2×D4—216no—
452C2×D4—216no—
453C2×D4—28no—
454C2×D4—28no—
455C2×D4—216no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4C2×#6718yes—
460C2×D4C23×D314yes—
461C22×C4—28no—
462C22×C4—264no—
463C22×C4—216no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—216no—
469C22×C4—24no—
470C22×C4—216no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—216no—
476C22×C4—64no—
477C22×C4—64no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4C2×#6714yes—
483C22×C4C2×#6714yes—
484C22×C4C2×#6712yes—
485C22×C4C2×#6712yes—
486C22×C4C2×#6714yes—
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6711yescharacteristic
489C22×C4C2×#6712yes—
490C22×C4C2×#6711yescharacteristic
491C22×C4C2×#6718yes—
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—34no—
502C4⋊C4—64no—
503C4⋊C4—62no—
504C4⋊C4—68no—
505C4⋊C4—68no—
506C22⋊C4—68no—
507C22⋊C4—68no—
508C22×C3—22no—
509C22×C3—28no—
510C22×C3—28no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—28no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—416no—
519C22×C3—432no—
520C22×C3—44no—
521C22×C3—416no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3C2×#12014yes—
525C22×C3C2×#12012yes—
526C22×C3C2×#13712yes—
527C22×C3C2×#13714yescomplemented
528C22×C3#57512yes—
529C22×C3C23×D411yescharacteristic
530C3×C4—22no—
531C3×C4—28no—
532C3×C4—22no—
533C3×C4C2×#12014yes—
534C3×C4C2×#12012yes—
535C3×C4C23×D412yes—
536Dic3—44no—
537Dic3—44no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—216no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—28no—
547C23—28no—
548C23—22no—
549C23—28no—
550C23—44no—
551C23—416no—
552C23—432no—
553C23—48no—
554C23—44no—
555C23—44no—
556C23—48no—
557C23#95114yescomplemented
558C23#95314yes—
559C23C22×#6712yes—
560C23C22×#6712yes—
561C23C22×#6712yes—
562C23C22×#6711yescharacteristic
563Q8—216no—
564Q8—24no—
565Q8—28no—
566Q8—68no—
567Q8—128no—
568Q8—124no—
569Q8C22×#6718yes—
570D4—232no—
571D4—216no—
572D4—232no—
573D4—24no—
574D4—28no—
575D4C22×#6718yes—
576C2×C4—216no—
577C2×C4—232no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—232no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—128no—
593C2×C4—124no—
594C2×C4—128no—
595C2×C4—124no—
596C2×C4#95314yes—
597C2×C4#95318yes—
598C2×C4C22×#6712yes—
599C2×C4C22×#6712yes—
600C2×C4C22×#6714yes—
601C2×C4C22×#6718yes—
602C2×C4C22×#6712yes—
603C2×C4C22×#6711yescharacteristic
604C2×C4C24×D311yescharacteristic
605C8—68no—
606C2×C3—24no—
607C2×C3—24no—
608C2×C3—42no—
609C2×C3—48no—
610C2×C3#300714yescomplemented
611C2×C3C22×#12011yescharacteristic
612C2×C3C22×#13711yescharacteristic
613C2×C3C2×#57511yescharacteristic
614C22—22no—
615C22—28no—
616C22—28no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—28no—
622C22—22no—
623C22—22no—
624C22—416no—
625C22—432no—
626C22—44no—
627C22—416no—
628C22—44no—
629C22—44no—
630C22#582212yes—
631C22C2×#95112yes—
632C22C2×#95114yescomplemented
633C22C2×#95314yes—
634C22C2×#95312yes—
635C22C23×#6711yescharacteristic
636C4—22no—
637C4—28no—
638C4—22no—
639C4—124no—
640C4—124no—
641C4C2×#95314yes—
642C4C2×#95312yes—
643C4C23×#6712yes—
644C3C2×#300711yescharacteristic, complemented, Sylow
645C2—24no—
646C2—24no—
647C2—42no—
648C2—48no—
649C2#8408414yesdirect product factor
650C2C2×#582211yescharacteristic
651C2C22×#95111yescharacteristic
652C2C22×#95311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC23×C3×#41C2
socleC23×C3C22×D4
hypercenterC23×#41D3
hypocenterC3C2×#3007
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C22×#924C2semidirect
4. #84084C2direct (N × H)
5. C2×#5239C2semidirect
6. C2×#5232C2semidirect
7. C2×#5228C2semidirect
8. C2×#5218C2semidirect
9. C2×#5207C2semidirect
23. C23×Dic6C22semidirect
26. C2×#951C22semidirect
27. C22×#216C22semidirect
28. C22×#214C22semidirect
29. C22×C4⋊Dic3C22semidirect
34. C2×#924C22semidirect
35. C2×#924C22semidirect
37. C22×#208C22semidirect
38. #5239C22semidirect
39. #5232C22semidirect
40. #5228C22semidirect
41. C2×#835C22semidirect
42. #5218C22semidirect
43. C2×#832C22semidirect
44. #5207C22semidirect
46. C2×#830C22semidirect
106. C22×Dic6C23semidirect
112. C2×#216C23semidirect
114. C2×#214C23semidirect
118. C2×C4⋊Dic3C23semidirect
124. #924C23semidirect
129. C2×#208C23semidirect
131. #835C23semidirect
132. #832C23semidirect
134. #830C23semidirect
365. C23×C3#137semidirect
527. C22×C3C2×#137semidirect
557. C23#951semidirect
610. C2×C3#3007semidirect
632. C22C2×#951semidirect
644. C3C2×#3007semidirect
649. C2#84084direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#412
C22×#9512
C22×#9242
#840842
C2×#52392
C2×#52322
C2×#52282
C2×#52182
C2×#52072
C2×#30073