Finite Group Databasebeta

← C2×#82662 #10964291

Subgroups — C2×#82662 #10964291

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×#936C212yesmaximal, complemented
3C22×#888C211yesmaximal, characteristic, complemented
4C2×#5222C212yesmaximal, complemented
5C2×#5221C211yesmaximal, characteristic, complemented
6C2×#5221C211yesmaximal, characteristic, complemented
7C2×#5212C211yesmaximal, characteristic, complemented
8C2×C4×#217C211yesmaximal, characteristic, complemented
9#82662C2116yesmaximal, direct product factor
10C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
11C2×#4947C212yesmaximal, complemented
12C2×#4945C212yesmaximal, complemented
13C2×#2491—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×C3×D4C2212yes—
16C24×C3×C4C2211yescharacteristic
17C23×Dic6C2212yes—
18C2×C3×#121—28no—
19C2×C3×#121C2218yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C2212yes—
28C22×#217C2211yescharacteristic
29C2×#936—216no—
30C2×#936C22116yescomplemented
31C22×C4⋊Dic3C2211yescharacteristic
32C22×C4⋊Dic3C2211yescharacteristic
33C22×Dic3⋊C4C2212yes—
34C22×C4×Dic3C2211yescharacteristic
35C22×C4×Dic3C2211yescharacteristic
36C22×C4×Dic3C2212yes—
37C2×#888—28no—
38C2×#888C2218yescomplemented
39#5222C22116yescomplemented
40#5221C2218yescomplemented
41#5221C2218yescomplemented
42#5212C2218yescomplemented
43C4×#217C2218yescomplemented
44C2×C3⋊#95C2211yescharacteristic
45C2×C3⋊#95C2211yescharacteristic
46C2×C3⋊#95C2212yes—
47C2×C3⋊#95C2212yes—
48Dic3×C4⋊C4C2218yescomplemented
49#4947C22116yescomplemented
50#4945C22116yescomplemented
51C22×C4⋊Q8—31no—
52C22×#124—32no—
53C22×#121D311yescharacteristic
54#2491—316no—
55C2×#394—32no—
56C2×#393—32no—
57C2×#390—31no—
58C2×#390—31no—
59C2×#388—32no—
60C2×#380—31no—
61C2×C4×C4⋊C4—31no—
62C2×C4×C22⋊C4—31no—
63C25×C3C2312yes—
64C25×C3C2311yescharacteristic
65C22×C3×D4—28no—
66C22×C3×D4—28no—
67C22×C3×D4—24no—
68C22×C3×D4—24no—
69C22×C3×D4—24no—
70C22×C3×D4—44no—
71C22×C3×D4C2314yes—
72C22×C3×D4C2318yes—
73C23×C3×C4—24no—
74C23×C3×C4—24no—
75C23×C3×C4—24no—
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2314yes—
78C23×C3×C4C2312yes—
79C23×C3×C4C2311yescharacteristic
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2312yes—
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2311yescharacteristic
85C23×Dic3C2311yescharacteristic
86C22×Dic6—28no—
87C22×Dic6—28no—
88C22×Dic6—44no—
89C22×Dic6C2318yes—
90C3×#121—232no—
91C2×C3×C4⋊C4D418yes—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C22⋊C4—28no—
94C2×C3×C22⋊C4C2318yes—
95C2×#217—28no—
96C2×#217—28no—
97C2×#217—28no—
98C2×#217—24no—
99C2×#217—28no—
100C2×#217—44no—
101C2×#217D418yescomplemented
102C2×#217C2318yes—
103C2×#217C2318yes—
104C2×#217C2318yes—
105C2×#217C2314yes—
106C2×#217C2318yes—
107C2×#217C2318yes—
108C2×#217C2314yes—
109#936—264no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3—24no—
112C2×C4⋊Dic3C2314yes—
113C2×C4⋊Dic3C2314yes—
114C2×Dic3⋊C4—28no—
115C2×Dic3⋊C4C2318yes—
116C2×C4×Dic3—24no—
117C2×C4×Dic3—24no—
118C2×C4×Dic3—28no—
119C2×C4×Dic3C2318yes—
120C2×C4×Dic3C2314yes—
121C2×C4×Dic3C2314yes—
122#888—232no—
123C3⋊#95C2314yes—
124C3⋊#95C2318yes—
125C3⋊#95C2314yes—
126C3⋊#95C2318yes—
127C23×Q8—32no—
128C23×D4C2×D311yescharacteristic
129C23×D4C2×D312yes—
130C24×C4C2×D311yescharacteristic
131C2×C4⋊Q8—38no—
132C2×C4⋊Q8—68no—
133C2×#124—316no—
134C2×#124—616no—
135C2×#121—28no—
136C2×#121C2×D318yes—
137C22×C4⋊C4—32no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—31no—
144C22×C22⋊C4—32no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—32no—
148C22×C22⋊C4C2×D312yes—
149C22×C42—32no—
150C22×C42—31no—
151C22×C42—31no—
152#394—316no—
153#393—316no—
154#390—38no—
155#390—38no—
156#388—316no—
157#380—38no—
158C4×C4⋊C4—38no—
159C4×C22⋊C4—38no—
160C2×#95—31no—
161C2×#95—32no—
162C2×#95—32no—
163C2×#95—31no—
164C24×C3—22no—
165C24×C3—22no—
166C24×C3—22no—
167C24×C3—22no—
168C24×C3—22no—
169C24×C3—22no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3—24no—
173C24×C3—24no—
174C24×C3—24no—
175C24×C3—24no—
176C24×C3—24no—
177C24×C3C2×D414yes—
178C24×C3C2414yes—
179C24×C3C2412yes—
180C24×C3C2411yescharacteristic
181C2×C3×D4—216no—
182C2×C3×D4—216no—
183C2×C3×D4—28no—
184C2×C3×D4—28no—
185C2×C3×D4—28no—
186C2×C3×D4—48no—
187C2×C3×D4—432no—
188C2×C3×D4—416no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—28no—
195C22×C3×C4—28no—
196C22×C3×C4—28no—
197C22×C3×C4—416no—
198C22×C3×C4C2×D418yes—
199C22×C3×C4C2×D414yes—
200C22×C3×C4#4114yes—
201C22×C3×C4#4114yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2414yes—
204C22×C3×C4C2412yes—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—22no—
215C22×Dic3—24no—
216C22×Dic3—24no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—24no—
220C22×Dic3C2×D414yes—
221C22×Dic3C2×D414yes—
222C22×Dic3#4114yes—
223C22×Dic3C2412yes—
224C22×Dic3C2414yes—
225C22×Dic3C2414yes—
226C22×Dic3C2412yes—
227C22×Dic3C2412yes—
228C22×Dic3C2412yes—
229C2×Dic6—216no—
230C2×Dic6—216no—
231C2×Dic6—48no—
232C2×Dic6—432no—
233C3×C4⋊C4C2×D4116yes—
234C3×C22⋊C4—216no—
235#217—216no—
236#217—216no—
237#217—216no—
238#217—28no—
239#217—216no—
240#217—48no—
241#217C2×D4116yescomplemented
242C4⋊Dic3—28no—
243C4⋊Dic3—28no—
244Dic3⋊C4—216no—
245C4×Dic3—28no—
246C4×Dic3—28no—
247C4×Dic3—216no—
248C25C22×D312yes—
249C25C22×D311yescharacteristic
250C22×Q8—38no—
251C22×Q8—68no—
252C22×Q8—68no—
253C22×Q8—124no—
254C22×D4—28no—
255C22×D4—24no—
256C22×D4—24no—
257C22×D4—28no—
258C22×D4—24no—
259C22×D4—44no—
260C22×D4C22×D314yes—
261C22×D4C22×D318yes—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—24no—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—32no—
269C23×C4—32no—
270C23×C4—31no—
271C23×C4C22×D314yes—
272C23×C4C22×D311yescharacteristic
273C23×C4C22×D312yes—
274C23×C4C22×D311yescharacteristic
275C4⋊Q8—632no—
276#124—664no—
277#121—232no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—38no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—68no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4#6718yes—
285C2×C4⋊C4C22×D314yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—38no—
288C2×C22⋊C4—34no—
289C2×C22⋊C4—38no—
290C2×C22⋊C4—38no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—34no—
293C2×C22⋊C4—38no—
294C2×C22⋊C4—38no—
295C2×C22⋊C4—68no—
296C2×C22⋊C4—68no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—68no—
299C2×C22⋊C4—68no—
300C2×C22⋊C4—124no—
301C2×C22⋊C4C22×D318yes—
302C2×C42—38no—
303C2×C42—34no—
304C2×C42—34no—
305C2×C42—68no—
306C2×C42—64no—
307C2×C42—64no—
308#95—34no—
309#95—34no—
310#95—38no—
311#95—38no—
312C23×C3—28no—
313C23×C3—28no—
314C23×C3—28no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—44no—
329C23×C3—44no—
330C23×C3—44no—
331C23×C3—44no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—48no—
335C23×C3—48no—
336C23×C3—48no—
337C23×C3—48no—
338C23×C3—48no—
339C23×C3C22×D414yes—
340C23×C3C22×D412yes—
341C23×C3C22×D412yes—
342C23×C3C2×#4112yes—
343C23×C3C2×#4112yes—
344C23×C3C2×#4112yes—
345C23×C3H(5,𝔽2)12yes—
346C23×C32−1+412yes—
347C23×C3C2511yescharacteristic
348C3×D4—432no—
349C3×D4—416no—
350C2×C3×C4—216no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—44no—
357C2×C3×C4—416no—
358C2×C3×C4#124116yescomplemented
359C2×C3×C4C22×D418yes—
360C2×C3×C4C22×D414yes—
361C2×C3×C4C2×#4114yes—
362C2×C3×C4C2×#4114yes—
363C2×Dic3—28no—
364C2×Dic3—28no—
365C2×Dic3—28no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—48no—
382C2×Dic3—48no—
383C2×Dic3—44no—
384C2×Dic3C22×D414yes—
385C2×Dic3C22×D414yes—
386C2×Dic3C2×#4114yes—
387Dic6—432no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—24no—
401C24D3×D414yes—
402C24C23×D314yes—
403C24C23×D312yes—
404C24C23×D311yescharacteristic
405C2×Q8—616no—
406C2×Q8—616no—
407C2×Q8—128no—
408C2×Q8—1232no—
409C2×D4—216no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—216no—
413C2×D4—28no—
414C2×D4—48no—
415C2×D4—432no—
416C2×D4—416no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—34no—
426C22×C4—34no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—416no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—64no—
445C22×C4—62no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—62no—
449C22×C4—64no—
450C22×C4#23714yes—
451C22×C4#23714yes—
452C22×C4C2×#6718yes—
453C22×C4C2×#6714yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D312yes—
457C4⋊C4—68no—
458C4⋊C4—616no—
459C4⋊C4—68no—
460C4⋊C4C2×#67116yes—
461C22⋊C4—216no—
462C22⋊C4—316no—
463C22⋊C4—616no—
464C22⋊C4—616no—
465C22⋊C4—68no—
466C22⋊C4—616no—
467C22⋊C4—616no—
468C22⋊C4—128no—
469C42—616no—
470C42—68no—
471C42—68no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—416no—
480C22×C3—44no—
481C22×C3—44no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—816no—
488C22×C3C2×#12414yes—
489C22×C3#53214yes—
490C22×C3#54214yes—
491C22×C3D4214yes—
492C22×C3#54714yes—
493C22×C3#54814yes—
494C22×C3#55514yes—
495C22×C3C23×D411yescharacteristic
496C22×C3C23×D411yescharacteristic
497C22×C3C22×#4111yescharacteristic
498C22×C3C22×#4111yescharacteristic
499C22×C3C22×#4111yescharacteristic
500C22×C3C2×H(5,𝔽2)11yescharacteristic
501C22×C3C2×2−1+411yescharacteristic
502C3×C4—28no—
503C3×C4—42no—
504C3×C4C2×#12418yescomplemented
505Dic3—24no—
506Dic3—24no—
507Dic3—24no—
508Dic3—44no—
509Dic3—44no—
510Dic3—42no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23C2×D3×D414yes—
539C23C2×D3×D412yes—
540C23C2×#23712yes—
541C23C2×#23712yes—
542C23#100412yes—
543C23D3×#4112yes—
544C23#101012yes—
545C23C22×#6712yes—
546C23C24×D311yescharacteristic
547Q8—1232no—
548D4—432no—
549D4—416no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—216no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—34no—
557C2×C4—34no—
558C2×C4—34no—
559C2×C4—44no—
560C2×C4—416no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—68no—
568C2×C4—68no—
569C2×C4—68no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—128no—
580C2×C4—128no—
581C2×C4—124no—
582C2×C4#936116yescomplemented
583C2×C4C2×#23714yes—
584C2×C4C2×#23714yes—
585C2×C4C22×#6718yes—
586C2×C4C22×#6714yes—
587C2×C3—44no—
588C2×C3—84no—
589C2×C3#249118yescomplemented
590C2×C3C22×#12411yescharacteristic
591C2×C3C2×#53211yescharacteristic
592C2×C3C2×#54211yescharacteristic
593C2×C3C2×D4211yescharacteristic
594C2×C3C2×#54711yescharacteristic
595C2×C3C2×#54811yescharacteristic
596C2×C3C2×#55511yescharacteristic
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—416no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—816no—
613C22#559614yes—
614C22#560314yes—
615C22#561114yes—
616C22#561614yes—
617C22C2×#93614yes—
618C22D4×#6714yes—
619C22#583114yes—
620C22C22×D3×D411yescharacteristic
621C22C22×#23711yescharacteristic
622C22C22×#23711yescharacteristic
623C22C2×#100411yescharacteristic
624C22C2×D3×#4111yescharacteristic
625C22C2×#101011yescharacteristic
626C22C23×#6711yescharacteristic
627C4—28no—
628C4—42no—
629C4—64no—
630C4—64no—
631C4—64no—
632C4—124no—
633C4—122no—
634C4—124no—
635C4C2×#93618yescomplemented
636C3C2×#249111yescharacteristic, complemented, Sylow
637C2—44no—
638C2—84no—
639C2#8266218yesdirect product factor
640C2C2×#559611yescharacteristic
641C2C2×#560311yescharacteristic
642C2C2×#561111yescharacteristic
643C2C2×#561611yescharacteristic
644C2C22×#93611yescharacteristic
645C2C2×D4×#6711yescharacteristic
646C2C2×#583111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
Fitting, prime radicalC22×C3×#121C2
socleC24×C3C24
hypercenterC22×#121D3
hypocenterC3C2×#2491
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. C22×#936C2semidirect
3. C22×#888C2semidirect
4. C2×#5222C2semidirect
5. C2×#5221C2semidirect
6. C2×#5221C2semidirect
7. C2×#5212C2semidirect
8. C2×C4×#217C2semidirect
9. #82662C2direct (N × H)
10. C2×Dic3×C4⋊C4C2semidirect
11. C2×#4947C2semidirect
12. C2×#4945C2semidirect
30. C2×#936C22semidirect
38. C2×#888C22semidirect
39. #5222C22semidirect
40. #5221C22semidirect
41. #5221C22semidirect
42. #5212C22semidirect
43. C4×#217C22semidirect
48. Dic3×C4⋊C4C22semidirect
49. #4947C22semidirect
50. #4945C22semidirect
101. C2×#217D4semidirect
241. #217C2×D4semidirect
358. C2×C3×C4#124semidirect
504. C3×C4C2×#124semidirect
582. C2×C4#936semidirect
589. C2×C3#2491semidirect
635. C4C2×#936semidirect
636. C3C2×#2491semidirect
639. C2#82662direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×#9362
C22×#8882
C2×#52222
C2×#52212
C2×#52122
C2×C4×#2172
#826622
C2×Dic3×C4⋊C42
C2×#49472
C2×#49452
C2×#24913