Finite Group Databasebeta

← C2×#82351 #10963601

Subgroups — C2×#82351 #10963601

Order 768.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C22×#924C211yesmaximal, characteristic, complemented
3C22×#924C211yesmaximal, characteristic, complemented
4C22×#926C211yesmaximal, characteristic, complemented
5C2×#5222C212yesmaximal
6C2×#5212C211yesmaximal, characteristic, complemented
7#82351C2116yesmaximal, direct product factor
8C2×#4976C212yesmaximal, complemented
9C2×#4952C212yesmaximal, complemented
10C2×#4945C212yesmaximal, complemented
11C2×#4929C211yesmaximal, characteristic, complemented
12C2×#4928C211yesmaximal, characteristic, complemented
13C2×#2415—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C24×C3×C4C2211yescharacteristic
17C23×Dic6C2211yescharacteristic
18C2×C3×C4×D4—28no—
19C2×C3×C4×D4C2218yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C2212yes—
28C2×#926—28no—
29C2×#926C2218yescomplemented
30C22×C4⋊Dic3C2211yescharacteristic
31C22×C4⋊Dic3C2211yescharacteristic
32C22×C4⋊Dic3C2211yescharacteristic
33C2×#924—28no—
34C2×#924—28no—
35C2×#924C2218yescomplemented
36C2×#924C2218yescomplemented
37C22×Dic3⋊C4C2212yes—
38C22×Dic3⋊C4C2212yes—
39C22×Dic3⋊C4C2212yes—
40C22×C4×Dic3C2212yes—
41#5222C22116yes—
42#5212C2218yescomplemented
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46C2×C3⋊#95C2212yes—
47#4976C22116yescomplemented
48#4952C22116yescomplemented
49#4945C22116yescomplemented
50#4929C2218yescomplemented
51#4928C2218yescomplemented
52C22×C22⋊Q8—31no—
53C22×C22⋊Q8—31no—
54C22×C4×D4D311yescharacteristic
55C22×#117—31no—
56#2415—316no—
57C2×#397—32no—
58C2×#394—32no—
59C2×#391—31no—
60C2×#389—31no—
61C2×#389—32no—
62C2×#388—32no—
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C25×C3C2311yescharacteristic
66C22×C3×D4—24no—
67C22×C3×D4—24no—
68C22×C3×D4—24no—
69C22×C3×D4C2314yes—
70C23×C3×C4—24no—
71C23×C3×C4—24no—
72C23×C3×C4—24no—
73C23×C3×C4—24no—
74C23×C3×C4—24no—
75C23×C3×C4D418yes—
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2314yes—
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2314yes—
81C23×C3×C4C2311yescharacteristic
82C23×C3×C4C2311yescharacteristic
83C23×Dic3C2312yes—
84C23×Dic3C2312yes—
85C23×Dic3C2312yes—
86C23×Dic3C2312yes—
87C22×Dic6—24no—
88C22×Dic6—24no—
89C22×Dic6—24no—
90C22×Dic6C2314yes—
91C3×C4×D4—232no—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4C2314yes—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4—24no—
96C2×C3×C22⋊C4C2314yes—
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C42—24no—
99C2×C3×C42C2314yes—
100C2×#217—28no—
101C2×#217—28no—
102C2×#217—28no—
103C2×#217—28no—
104C2×#217C2318yes—
105C2×#217C2318yes—
106C2×#217C2318yes—
107C2×#217C2318yes—
108#926—232no—
109C2×C4⋊Dic3—24no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3D418yescomplemented
112C2×C4⋊Dic3C2314yes—
113C2×C4⋊Dic3C2314yes—
114C2×C4⋊Dic3C2314yes—
115#924—232no—
116#924—232no—
117C2×Dic3⋊C4—28no—
118C2×Dic3⋊C4—28no—
119C2×Dic3⋊C4—28no—
120C2×Dic3⋊C4C2318yes—
121C2×Dic3⋊C4C2318yes—
122C2×Dic3⋊C4C2318yes—
123C2×C4×Dic3—28no—
124C2×C4×Dic3C2318yes—
125C3⋊#95C2318yes—
126C3⋊#95C2318yes—
127C3⋊#95C2318yes—
128C3⋊#95C2318yes—
129C23×Q8—31no—
130C23×D4C2×D311yescharacteristic
131C24×C4C2×D311yescharacteristic
132C24×C4C2×D311yescharacteristic
133C2×C22⋊Q8—38no—
134C2×C22⋊Q8—38no—
135C2×C22⋊Q8—68no—
136C2×C22⋊Q8—68no—
137C2×C4×D4—28no—
138C2×C4×D4C2×D318yes—
139C2×#117—38no—
140C2×#117—68no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—32no—
143C22×C4⋊C4—32no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4—32no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148C22×C22⋊C4—32no—
149C22×C22⋊C4—32no—
150C22×C22⋊C4—32no—
151C22×C22⋊C4—32no—
152C22×C22⋊C4C2×D311yescharacteristic
153C22×C22⋊C4C2×D311yescharacteristic
154C22×C42—32no—
155C22×C42C2×D311yescharacteristic
156#397—316no—
157#394—316no—
158#391—38no—
159#389—316no—
160#389—38no—
161#388—316no—
162#380—38no—
163C2×#95—32no—
164C2×#95—32no—
165C2×#95—32no—
166C2×#95—32no—
167C24×C3—22no—
168C24×C3—22no—
169C24×C3—22no—
170C24×C3—22no—
171C24×C3—22no—
172C24×C3—22no—
173C24×C3—22no—
174C24×C3—22no—
175C24×C3—22no—
176C24×C3—22no—
177C24×C3—22no—
178C24×C3—22no—
179C24×C3C2×D414yes—
180C24×C3#4114yes—
181C24×C3C2412yes—
182C24×C3C2412yes—
183C24×C3C2411yescharacteristic
184C2×C3×D4—232no—
185C2×C3×D4—28no—
186C2×C3×D4—28no—
187C2×C3×D4—28no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—232no—
193C22×C3×C4—28no—
194C22×C3×C4—28no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—28no—
202C22×C3×C4—28no—
203C22×C3×C4—28no—
204C22×C3×C4—416no—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D4116yes—
208C22×C3×C4C2×D414yes—
209C22×C3×C4#4114yes—
210C22×C3×C4#4114yes—
211C22×C3×C4C2412yes—
212C22×C3×C4C2412yes—
213C22×C3×C4C2412yes—
214C22×C3×C4C2412yes—
215C22×C3×C4C2412yes—
216C22×Dic3—24no—
217C22×Dic3—24no—
218C22×Dic3—24no—
219C22×Dic3—24no—
220C22×Dic3—24no—
221C22×Dic3—24no—
222C22×Dic3—24no—
223C22×Dic3—24no—
224C22×Dic3—24no—
225C22×Dic3—24no—
226C22×Dic3—24no—
227C22×Dic3C2×D418yes—
228C22×Dic3C2414yes—
229C22×Dic3C2414yes—
230C22×Dic3C2414yes—
231C22×Dic3C2414yes—
232C2×Dic6—28no—
233C2×Dic6—28no—
234C2×Dic6—28no—
235C2×Dic6—416no—
236C3×C4⋊C4—28no—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—28no—
239C3×C42—28no—
240#217—216no—
241#217—216no—
242#217—216no—
243#217—216no—
244C4⋊Dic3—28no—
245C4⋊Dic3—28no—
246C4⋊Dic3C2×D4116yescomplemented
247Dic3⋊C4—216no—
248Dic3⋊C4—216no—
249Dic3⋊C4—216no—
250C4×Dic3—216no—
251C25C22×D311yescharacteristic
252C25C22×D311yescharacteristic
253C22×Q8—34no—
254C22×Q8—64no—
255C22×Q8—64no—
256C22×Q8—64no—
257C22×D4—24no—
258C22×D4—24no—
259C22×D4—24no—
260C22×D4C22×D314yes—
261C23×C4—24no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—24no—
265C23×C4—24no—
266C23×C4—32no—
267C23×C4—32no—
268C23×C4—32no—
269C23×C4—32no—
270C23×C4#6718yes—
271C23×C4C22×D314yes—
272C23×C4C22×D311yescharacteristic
273C23×C4C22×D311yescharacteristic
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D314yes—
276C23×C4C22×D311yescharacteristic
277C23×C4C22×D311yescharacteristic
278C22⋊Q8—632no—
279C22⋊Q8—632no—
280C4×D4—232no—
281#117—632no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—38no—
284C2×C4⋊C4—34no—
285C2×C4⋊C4—34no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4—38no—
288C2×C4⋊C4—38no—
289C2×C4⋊C4—38no—
290C2×C4⋊C4—68no—
291C2×C4⋊C4—68no—
292C2×C4⋊C4—64no—
293C2×C4⋊C4—68no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4C22×D314yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—38no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—38no—
302C2×C22⋊C4—68no—
303C2×C22⋊C4—68no—
304C2×C22⋊C4—68no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4C22×D314yes—
307C2×C22⋊C4C22×D314yes—
308C2×C42—24no—
309C2×C42—38no—
310C2×C42—68no—
311C2×C42C22×D314yes—
312#95—38no—
313#95—38no—
314#95—38no—
315#95—38no—
316C23×C3—28no—
317C23×C3—28no—
318C23×C3—28no—
319C23×C3—28no—
320C23×C3—28no—
321C23×C3—28no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—44no—
335C23×C3—44no—
336C23×C3—44no—
337C23×C3—44no—
338C23×C3—44no—
339C23×C3—44no—
340C23×C3—44no—
341C23×C3—44no—
342C23×C3C22×D412yes—
343C23×C3C22×D414yes—
344C23×C3C22×D412yes—
345C23×C3C2×#4114yes—
346C23×C3C2×#4112yes—
347C23×C3C2×#4112yes—
348C23×C3C2×#4112yes—
349C23×C32−1+412yes—
350C23×C32−1+412yes—
351C23×C3C2511yescharacteristic
352C3×D4—232no—
353C2×C3×C4—28no—
354C2×C3×C4—28no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—232no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—44no—
367C2×C3×C4—44no—
368C2×C3×C4—416no—
369C2×C3×C4#121116yes—
370C2×C3×C4C22×D414yes—
371C2×C3×C4C22×D414yes—
372C2×C3×C4C22×D414yes—
373C2×C3×C4C2×#4114yes—
374C2×C3×C4C2×#4114yes—
375C2×Dic3—216no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—48no—
388C2×Dic3—48no—
389C2×Dic3—48no—
390C2×Dic3C22×D418yes—
391Dic6—416no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24#23514yes—
405C24C2×#6714yes—
406C24C23×D312yes—
407C24C23×D312yes—
408C24C23×D311yescharacteristic
409C2×Q8—68no—
410C2×Q8—68no—
411C2×Q8—68no—
412C2×Q8—1216no—
413C2×D4—232no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—232no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—38no—
434C22×C4—34no—
435C22×C4—34no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—416no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4#23514yes—
451C22×C4D3×D414yes—
452C22×C4#23714yes—
453C22×C4C2×#6714yes—
454C22×C4C2×#67116yes—
455C22×C4C2×#6714yes—
456C22×C4C23×D312yes—
457C22×C4C23×D312yes—
458C22×C4C23×D312yes—
459C22×C4C23×D312yes—
460C22×C4C23×D312yes—
461C4⋊C4—28no—
462C4⋊C4—316no—
463C4⋊C4—616no—
464C4⋊C4—616no—
465C4⋊C4—68no—
466C4⋊C4—616no—
467C4⋊C4—68no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—616no—
471C22⋊C4—616no—
472C22⋊C4—616no—
473C22⋊C4—616no—
474C42—28no—
475C42—616no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—24no—
482C22×C3—42no—
483C22×C3—416no—
484C22×C3—416no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3C2×#12114yes—
493C22×C3#52514yes—
494C22×C3#53614yes—
495C22×C3#54714yes—
496C22×C3#54814yes—
497C22×C3#55214yes—
498C22×C3#55614yes—
499C22×C3C23×D411yescharacteristic
500C22×C3C23×D411yescharacteristic
501C22×C3C22×#4111yescharacteristic
502C22×C3C22×#4111yescharacteristic
503C22×C3C22×#4111yescharacteristic
504C22×C3C2×2−1+411yescharacteristic
505C22×C3C2×2−1+411yescharacteristic
506C3×C4—24no—
507C3×C4—24no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4C2×#12118yes—
511Dic3—28no—
512Dic3—44no—
513Dic3—44no—
514Dic3—44no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C2×#23514yes—
542C23C2×#23512yes—
543C23C2×#23512yes—
544C23C2×D3×D412yes—
545C23C2×#23712yes—
546C23#101012yes—
547C23#101012yes—
548C23C22×#6714yes—
549C23C22×#6712yes—
550C23C24×D311yescharacteristic
551Q8—1216no—
552D4—232no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—232no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—38no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—416no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—616no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—128no—
583C2×C4—128no—
584C2×C4—128no—
585C2×C4#939116yes—
586C2×C4C2×#23514yes—
587C2×C4C2×D3×D414yes—
588C2×C4C2×#23714yes—
589C2×C4C22×#6714yes—
590C2×C4C22×#6714yes—
591C2×C3—44no—
592C2×C3—44no—
593C2×C3#241518yescomplemented
594C2×C3C22×#12111yescharacteristic
595C2×C3C2×#52511yescharacteristic
596C2×C3C2×#53611yescharacteristic
597C2×C3C2×#54711yescharacteristic
598C2×C3C2×#54811yescharacteristic
599C2×C3C2×#55211yescharacteristic
600C2×C3C2×#55611yescharacteristic
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—416no—
612C22—416no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22#554114yes—
618C22#554214yes—
619C22#555114yes—
620C22#555214yes—
621C22#579114yes—
622C22C2×#93914yes—
623C22#582514yes—
624C22C22×#23511yescharacteristic
625C22C22×#23511yescharacteristic
626C22C22×D3×D411yescharacteristic
627C22C22×#23711yescharacteristic
628C22C2×#101011yescharacteristic
629C22C2×#101011yescharacteristic
630C22C23×#6711yescharacteristic
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—68no—
636C4—124no—
637C4—124no—
638C4—124no—
639C4C2×#93918yes—
640C3C2×#241511yescharacteristic, complemented, Sylow
641C2—44no—
642C2—44no—
643C2#8235118yesdirect product factor
644C2C2×#555211yescharacteristic
645C2C2×#554111yescharacteristic
646C2C2×#554211yescharacteristic
647C2C2×#555111yescharacteristic
648C2C2×#579111yescharacteristic
649C2C22×#93911yescharacteristic
650C2C2×#582511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×C4×D4C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC22×C4×D4D3
hypocenterC3C2×#2415
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×#924C2semidirect
3. C22×#924C2semidirect
4. C22×#926C2semidirect
6. C2×#5212C2semidirect
7. #82351C2direct (N × H)
8. C2×#4976C2semidirect
9. C2×#4952C2semidirect
10. C2×#4945C2semidirect
11. C2×#4929C2semidirect
12. C2×#4928C2semidirect
29. C2×#926C22semidirect
35. C2×#924C22semidirect
36. C2×#924C22semidirect
42. #5212C22semidirect
47. #4976C22semidirect
48. #4952C22semidirect
49. #4945C22semidirect
50. #4929C22semidirect
51. #4928C22semidirect
111. C2×C4⋊Dic3D4semidirect
246. C4⋊Dic3C2×D4semidirect
593. C2×C3#2415semidirect
640. C3C2×#2415semidirect
643. C2#82351direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C22×#9242
C22×#9262
C2×#52222
C2×#52122
#823512
C2×#49762
C2×#49522
C2×#49452
C2×#49292
C2×#49282
C2×#24153