Finite Group Databasebeta

← C2×#83257 #10964293

Subgroups — C2×#83257 #10964293

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×C4⋊Q8C211yesmaximal, characteristic, complemented
2C22×#947C212yesmaximal, complemented
3C22×#895C211yesmaximal, characteristic
4C2×#5231C212yesmaximal, complemented
5#83257C2116yesmaximal, direct product factor
6C2×#4991C212yesmaximal, complemented
7C2×#4987C211yesmaximal, characteristic, complemented
8C2×#4988C211yesmaximal, characteristic, complemented
9C2×#4990C212yesmaximal, complemented
10C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
11C2×#4939C211yesmaximal, characteristic, complemented
12C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
13C2×#2491—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C23×D12C2211yescharacteristic
16C23×D12C2212yes—
17C23×C4×D3C2211yescharacteristic
18C2×C3×C4⋊Q8—28no—
19C2×C3×C4⋊Q8C2218yescomplemented
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C2×#947—216no—
25C2×#947C22116yescomplemented
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2211yescharacteristic
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×(C2×D3)⋊C4C2212yes—
31C22×(C2×D3)⋊C4C2212yes—
32C22×(C2×D3)⋊C4C2212yes—
33C22×(C2×D3)⋊C4C2211yescharacteristic
34C22×C4⋊Dic3C2211yescharacteristic
35C22×C4×Dic3C2211yescharacteristic
36C22×C4×Dic3C2212yes—
37C2×#895—28no—
38C2×#895C2218yes—
39#5231C22116yescomplemented
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2211yescharacteristic
43C2×C3⋊#95C2211yescharacteristic
44#4991C22116yescomplemented
45#4990C22116yescomplemented
46#4988C2218yescomplemented
47#4987C2218yescomplemented
48Dic3×C4⋊C4C2218yescomplemented
49#4939C2218yescomplemented
50C4×(C2×D3)⋊C4C2218yescomplemented
51C22×C4⋊Q8D311yescharacteristic, complemented
52C22×#124—32no—
53C22×#121—31no—
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—
63C24×D3C2312yes—
64C24×D3C2311yescharacteristic
65C22×C3×Q8—28no—
66C22×C3×Q8—28no—
67C22×C3×Q8—44no—
68C22×C3×Q8C2318yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C22×D12—28no—
79C22×D12—28no—
80C22×D12—24no—
81C22×D12—24no—
82C22×D12—24no—
83C22×D12—44no—
84C22×D12C2314yes—
85C22×D12C2318yes—
86C22×C4×D3—24no—
87C22×C4×D3—24no—
88C22×C4×D3—24no—
89C22×C4×D3C2314yes—
90C3×C4⋊Q8—232no—
91C2×C3×C4⋊C4—28no—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4D418yes—
94C2×C3×C4⋊C4C2314yes—
95C2×C3×C4⋊C4C2318yes—
96C2×C3×C4⋊C4C2314yes—
97C2×C3×C42—24no—
98C2×C3×C42C2314yes—
99#947—264no—
100C2×(C2×D3)⋊C4—28no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—28no—
103C2×(C2×D3)⋊C4—24no—
104C2×(C2×D3)⋊C4—28no—
105C2×(C2×D3)⋊C4—28no—
106C2×(C2×D3)⋊C4—44no—
107C2×(C2×D3)⋊C4D418yescomplemented
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×(C2×D3)⋊C4C2318yes—
111C2×(C2×D3)⋊C4C2318yes—
112C2×(C2×D3)⋊C4C2318yes—
113C2×(C2×D3)⋊C4C2318yes—
114C2×(C2×D3)⋊C4C2318yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×C4⋊Dic3—24no—
117C2×C4⋊Dic3C2314yes—
118C2×C4×Dic3—24no—
119C2×C4×Dic3—28no—
120C2×C4×Dic3C2314yes—
121C2×C4×Dic3C2318yes—
122#895—232no—
123C3⋊#95C2314yes—
124C3⋊#95C2314yes—
125C3⋊#95C2318yes—
126C3⋊#95C2318yes—
127C23×Q8C2×D312yes—
128C23×D4—31no—
129C23×D4—32no—
130C24×C4—31no—
131C2×C4⋊Q8—28no—
132C2×C4⋊Q8C2×D318yescomplemented
133C2×#124—316no—
134C2×#124—616no—
135C2×#121—38no—
136C2×#121—68no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4C2×D312yes—
139C22×C4⋊C4C2×D311yescharacteristic
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4—31no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—32no—
148C22×C22⋊C4—32no—
149C22×C42—32no—
150C22×C42—31no—
151C22×C42C2×D311yescharacteristic
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×C3C2411yescharacteristic
165C23×D3—22no—
166C23×D3—22no—
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—24no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—24no—
176C23×D3—24no—
177C23×D3—24no—
178C23×D3C2×D414yes—
179C23×D3C2414yes—
180C23×D3C2412yes—
181C2×C3×Q8—216no—
182C2×C3×Q8—216no—
183C2×C3×Q8—48no—
184C2×C3×Q8—432no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D418yes—
196C22×C3×C4C2×D414yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4114yes—
199C22×C3×C4#4114yes—
200C22×C3×C4C2414yes—
201C22×C3×C4C2414yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2412yes—
204C22×C3×C4C2412yes—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—24no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3C2×D414yes—
217C22×Dic3C2412yes—
218C22×Dic3C2412yes—
219C22×Dic3C2414yes—
220C22×Dic3C2412yes—
221C2×D12—216no—
222C2×D12—216no—
223C2×D12—28no—
224C2×D12—28no—
225C2×D12—28no—
226C2×D12—48no—
227C2×D12—432no—
228C2×D12—416no—
229C2×C4×D3—28no—
230C2×C4×D3—28no—
231C2×C4×D3—28no—
232C2×C4×D3—416no—
233C3×C4⋊C4—216no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4C2×D4116yes—
236C3×C42—28no—
237(C2×D3)⋊C4—216no—
238(C2×D3)⋊C4—216no—
239(C2×D3)⋊C4—216no—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—216no—
242(C2×D3)⋊C4—216no—
243(C2×D3)⋊C4—48no—
244(C2×D3)⋊C4C2×D4116yescomplemented
245C4⋊Dic3—28no—
246C4×Dic3—28no—
247C4×Dic3—216no—
248C25—31no—
249C25—32no—
250C22×Q8—28no—
251C22×Q8—28no—
252C22×Q8—44no—
253C22×Q8C22×D318yes—
254C22×D4—34no—
255C22×D4—38no—
256C22×D4—68no—
257C22×D4—64no—
258C22×D4—64no—
259C22×D4—68no—
260C22×D4—64no—
261C22×D4—124no—
262C23×C4—31no—
263C23×C4—32no—
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—34no—
267C23×C4—64no—
268C23×C4—64no—
269C23×C4—64no—
270C23×C4C22×D312yes—
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C23×C4C22×D312yes—
274C23×C4C22×D311yescharacteristic
275C4⋊Q8—232no—
276#124—664no—
277#121—632no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4#6718yes—
283C2×C4⋊C4C22×D314yes—
284C2×C4⋊C4C22×D318yes—
285C2×C4⋊C4C22×D314yes—
286C2×C22⋊C4—38no—
287C2×C22⋊C4—34no—
288C2×C22⋊C4—38no—
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—68no—
298C2×C22⋊C4—64no—
299C2×C22⋊C4—68no—
300C2×C22⋊C4—68no—
301C2×C22⋊C4—124no—
302C2×C42—24no—
303C2×C42—38no—
304C2×C42—34no—
305C2×C42—68no—
306C2×C42—64no—
307C2×C42C22×D314yes—
308#95—34no—
309#95—34no—
310#95—38no—
311#95—38no—
312C23×C3C22×D412yes—
313C23×C3C22×D412yes—
314C23×C3C2×#4112yes—
315C23×C3C2×#4112yes—
316C23×C3C2×#4112yes—
317C23×C3H(5,𝔽2)12yes—
318C23×C32−1+412yes—
319C23×C3C2511yescharacteristic
320C22×D3—28no—
321C22×D3—28no—
322C22×D3—28no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—24no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—44no—
337C22×D3—44no—
338C22×D3—44no—
339C22×D3—44no—
340C22×D3—48no—
341C22×D3—48no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—48no—
345C22×D3—48no—
346C22×D3—48no—
347C22×D3C22×D414yes—
348C3×Q8—432no—
349C2×C3×C4—216no—
350C2×C3×C4—28no—
351C2×C3×C4—28no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—48no—
362C2×C3×C4#124116yescomplemented
363C2×C3×C4C22×D414yes—
364C2×C3×C4C22×D418yes—
365C2×C3×C4C22×D414yes—
366C2×C3×C4C2×#4114yes—
367C2×C3×C4C2×#4114yes—
368C2×C3×C4C2×#4114yes—
369C2×Dic3—28no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—44no—
382C2×Dic3—48no—
383C2×Dic3—44no—
384C2×Dic3C22×D414yes—
385D12—432no—
386D12—416no—
387C4×D3—416no—
388C24—34no—
389C24—32no—
390C24—34no—
391C24—62no—
392C24—62no—
393C24—62no—
394C24—62no—
395C24—62no—
396C24—62no—
397C24—64no—
398C24—64no—
399C24—64no—
400C24—64no—
401C24—64no—
402C24—64no—
403C24—64no—
404C24C23×D311yescharacteristic
405C2×Q8—216no—
406C2×Q8—216no—
407C2×Q8—48no—
408C2×Q8—432no—
409C2×D4—616no—
410C2×D4—68no—
411C2×D4—68no—
412C2×D4—616no—
413C2×D4—68no—
414C2×D4—128no—
415C2×D4—1232no—
416C2×D4—1216no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—34no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—64no—
437C22×C4—62no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—68no—
443C22×C4—68no—
444C22×C4—68no—
445C22×C4—1216no—
446C22×C4D3×D414yes—
447C22×C4#23914yes—
448C22×C4#23914yes—
449C22×C4#23914yes—
450C22×C4C2×#6718yes—
451C22×C4C2×#6714yes—
452C22×C4C23×D314yes—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D312yes—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—68no—
460C4⋊C4C2×#67116yes—
461C22⋊C4—316no—
462C22⋊C4—616no—
463C22⋊C4—616no—
464C22⋊C4—616no—
465C22⋊C4—68no—
466C22⋊C4—616no—
467C22⋊C4—616no—
468C22⋊C4—128no—
469C42—28no—
470C42—616no—
471C42—68no—
472C22×C3C2×#12414yes—
473C22×C3#53214yes—
474C22×C3#54214yes—
475C22×C3D4214yes—
476C22×C3#54714yes—
477C22×C3#54814yes—
478C22×C3#55514yes—
479C22×C3C23×D411yescharacteristic
480C22×C3C23×D411yescharacteristic
481C22×C3C22×#4111yescharacteristic
482C22×C3C22×#4111yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C2×H(5,𝔽2)11yescharacteristic
485C22×C3C2×2−1+411yescharacteristic
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—416no—
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—816no—
502C3×C4—28no—
503C3×C4—24no—
504C3×C4—24no—
505C3×C4—44no—
506C3×C4C2×#12418yescomplemented
507Dic3—24no—
508Dic3—42no—
509Dic3—44no—
510Dic3—42no—
511C23—34no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—62no—
517C23—62no—
518C23—68no—
519C23—68no—
520C23—68no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23—124no—
529C23—124no—
530C23—124no—
531C23—124no—
532C23—128no—
533C23—128no—
534C23—128no—
535C23—128no—
536C23—128no—
537C23—128no—
538C23—128no—
539C23C2×D3×D412yes—
540C23#100412yes—
541C23C2×#23912yes—
542C23C2×#23912yes—
543C23C2×#23912yes—
544C23#100712yes—
545C23C22×#6712yes—
546C23C24×D311yescharacteristic
547Q8—432no—
548D4—1232no—
549D4—1216no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—216no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—34no—
563C2×C4—48no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—68no—
567C2×C4—64no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—128no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4—1216no—
580C2×C4#947116yescomplemented
581C2×C4C2×D3×D414yes—
582C2×C4C2×#23914yes—
583C2×C4C2×#23914yes—
584C2×C4C2×#23914yes—
585C2×C4C22×#6718yes—
586C2×C4C22×#6714yes—
587C2×C3#249118yescomplemented
588C2×C3C22×#12411yescharacteristic
589C2×C3C2×#53211yescharacteristic
590C2×C3C2×#54211yescharacteristic
591C2×C3C2×D4211yescharacteristic
592C2×C3C2×#54711yescharacteristic
593C2×C3C2×#54811yescharacteristic
594C2×C3C2×#55511yescharacteristic
595D3—44no—
596D3—84no—
597C22—64no—
598C22—64no—
599C22—64no—
600C22—1216no—
601C22—122no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—124no—
606C22—124no—
607C22—124no—
608C22—124no—
609C22—124no—
610C22—124no—
611C22—124no—
612C22—2416no—
613C22#572514yes—
614C22#572614yes—
615C22#573314yes—
616C22#573414yes—
617C22D4×#6714yes—
618C22C2×#94714yes—
619C22#581714yes—
620C22C22×D3×D411yescharacteristic
621C22C2×#100411yescharacteristic
622C22C22×#23911yescharacteristic
623C22C22×#23911yescharacteristic
624C22C22×#23911yescharacteristic
625C22C2×#100711yescharacteristic
626C22C23×#6711yescharacteristic
627C4—24no—
628C4—28no—
629C4—24no—
630C4—44no—
631C4—64no—
632C4—124no—
633C4—122no—
634C4—122no—
635C4C2×#94718yescomplemented
636C3C2×#249111yescharacteristic, complemented, Sylow
637C2—124no—
638C2—244no—
639C2#8325718yesdirect product factor
640C2C2×#572511yescharacteristic
641C2C2×#573411yescharacteristic
642C2C2×#573311yescharacteristic
643C2C2×#572611yescharacteristic
644C2C2×D4×#6711yescharacteristic
645C2C22×#94711yescharacteristic
646C2C2×#581711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×C4⋊Q8C2
socleC24×C3C24
prime radicalC22×#895C2
hypercenterC22×C4⋊Q8D3
hypocenterC3C2×#2491
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
2. C22×#947C2semidirect
4. C2×#5231C2semidirect
5. #83257C2direct (N × H)
6. C2×#4991C2semidirect
7. C2×#4987C2semidirect
8. C2×#4988C2semidirect
9. C2×#4990C2semidirect
10. C2×Dic3×C4⋊C4C2semidirect
11. C2×#4939C2semidirect
12. C2×C4×(C2×D3)⋊C4C2semidirect
19. C2×C3×C4⋊Q8C22semidirect
25. C2×#947C22semidirect
39. #5231C22semidirect
44. #4991C22semidirect
45. #4990C22semidirect
46. #4988C22semidirect
47. #4987C22semidirect
48. Dic3×C4⋊C4C22semidirect
49. #4939C22semidirect
50. C4×(C2×D3)⋊C4C22semidirect
51. C22×C4⋊Q8D3semidirect
107. C2×(C2×D3)⋊C4D4semidirect
132. C2×C4⋊Q8C2×D3semidirect
244. (C2×D3)⋊C4C2×D4semidirect
362. C2×C3×C4#124semidirect
506. C3×C4C2×#124semidirect
580. C2×C4#947semidirect
587. C2×C3#2491semidirect
635. C4C2×#947semidirect
636. C3C2×#2491semidirect
639. C2#83257direct (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⋊Q82
C22×#9472
C22×#8952
C2×#52312
#832572
C2×#49912
C2×#49872
C2×#49882
C2×#49902
C2×Dic3×C4⋊C42
C2×#49392
C2×C4×(C2×D3)⋊C42
C2×#24913