Finite Group Databasebeta

← C2×#76231 #10964239

Subgroups — C2×#76231 #10964239

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#894C212yesmaximal
2C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×C4×D12C211yesmaximal, characteristic
4C2×C3×#389C211yesmaximal, characteristic, complemented
5C2×#4987C211yesmaximal, characteristic, complemented
6C2×#4987C211yesmaximal, characteristic, complemented
7C2×#4987C211yesmaximal, characteristic, complemented
8C2×#4982C212yesmaximal, complemented
9C2×#4928C211yesmaximal, characteristic, complemented
10#76231C2116yesmaximal, direct product factor
11C2×#4675C212yesmaximal, complemented
12C2×#4664C212yesmaximal, complemented
13C2×#2486—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×C4×D3C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2211yescharacteristic
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C2×#894—216no—
33C2×#894C22116yes—
34C2×D3×C4⋊C4—28no—
35C2×D3×C4⋊C4C2218yescomplemented
36C2×C4×D12—28no—
37C2×C4×D12C2218yes—
38C3×#389C2218yescomplemented
39C2×C3×#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4987C2218yescomplemented
45#4987C2218yescomplemented
46#4987C2218yescomplemented
47#4982C22116yescomplemented
48#4928C2218yescomplemented
49#4675C22116yescomplemented
50#4664C22116yescomplemented
51C22×#123—32no—
52C22×C4×D4—31no—
53C23×C4⋊C4—31no—
54#2486—316no—
55C2×#398—32no—
56C2×#397—32no—
57C2×#389—31no—
58C2×#389D311yescharacteristic, complemented
59C2×#381—32no—
60C2×#380—31no—
61C2×#380—31no—
62C2×#380—31no—
63C24×D3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2312yes—
73C23×Dic3C2311yescharacteristic
74C22×D12—24no—
75C22×D12—24no—
76C22×D12D418yes—
77C22×D12C2314yes—
78C22×C4×D3—24no—
79C22×C4×D3—24no—
80C22×C4×D3—28no—
81C22×C4×D3—28no—
82C22×C4×D3—28no—
83C22×C4×D3—24no—
84C22×C4×D3—24no—
85C22×C4×D3—24no—
86C22×C4×D3D418yes—
87C22×C4×D3C2318yes—
88C22×C4×D3C2314yes—
89C22×C4×D3C2314yes—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—28no—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4C2318yes—
94C2×C3×C4⋊C4C2314yes—
95C2×C3×C4⋊C4C2314yes—
96C2×C3×C42—24no—
97C2×C3×C42C2314yes—
98C2×(C2×D3)⋊C4—24no—
99C2×(C2×D3)⋊C4—28no—
100C2×(C2×D3)⋊C4—28no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—24no—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2314yes—
106C2×(C2×D3)⋊C4C2318yes—
107C2×(C2×D3)⋊C4C2318yes—
108C2×C4⋊Dic3—24no—
109C2×C4⋊Dic3—24no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3C2314yes—
112C2×C4⋊Dic3C2314yes—
113C2×C4⋊Dic3C2314yes—
114C2×Dic3⋊C4—28no—
115C2×Dic3⋊C4—28no—
116C2×Dic3⋊C4C2318yes—
117C2×Dic3⋊C4C2318yes—
118#894—264no—
119D3×C4⋊C4—232no—
120C4×D12—232no—
121C3×#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C3⋊#95C2318yes—
125C3⋊#95C2318yes—
126C23×D4—31no—
127C24×C4—31no—
128C24×C4—32no—
129C24×C4—31no—
130C2×#123—316no—
131C2×#123—616no—
132C2×C4×D4—38no—
133C2×C4×D4—68no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—38no—
140C22×C4⋊C4—68no—
141C22×C4⋊C4C2×D312yes—
142C22×C4⋊C4C2×D311yescharacteristic
143C22×C4⋊C4C2×D311yescharacteristic
144C22×C22⋊C4—31no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—31no—
148C22×C22⋊C4—32no—
149C22×C42C2×D311yescharacteristic
150#398—316no—
151#397—316no—
152#389—38no—
153#389C2×D318yescomplemented
154#381—316no—
155#380—38no—
156#380—38no—
157#380—38no—
158C2×#95—32no—
159C2×#95—32no—
160C2×#95—32no—
161C2×#95—32no—
162C2×#95C2×D312yes—
163C24×C3C2411yescharacteristic
164C23×D3—22no—
165C23×D3—22no—
166C23×D3—22no—
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—22no—
172C23×D3—22no—
173C23×D3—22no—
174C23×D3C2×D414yes—
175C23×D3C2×D414yes—
176C23×D3C2×D414yes—
177C23×D3#4114yes—
178C23×D3C2412yes—
179C23×D3C2412yes—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4C2×D414yes—
192C22×C3×C4C2×D414yes—
193C22×C3×C4#4114yes—
194C22×C3×C4#4114yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2412yes—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3C2×D414yes—
212C22×Dic3C2412yes—
213C22×Dic3C2414yes—
214C22×Dic3C2414yes—
215C22×Dic3C2412yes—
216C2×D12—232no—
217C2×D12—28no—
218C2×D12—28no—
219C2×D12C2×D4116yes—
220C2×C4×D3—232no—
221C2×C4×D3—28no—
222C2×C4×D3—28no—
223C2×C4×D3—216no—
224C2×C4×D3—216no—
225C2×C4×D3—216no—
226C2×C4×D3—28no—
227C2×C4×D3—28no—
228C2×C4×D3—28no—
229C2×C4×D3—432no—
230C2×C4×D3—416no—
231C2×C4×D3C2×D4116yes—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—216no—
234C3×C4⋊C4—28no—
235C3×C42—28no—
236(C2×D3)⋊C4—28no—
237(C2×D3)⋊C4—216no—
238(C2×D3)⋊C4—216no—
239(C2×D3)⋊C4—216no—
240(C2×D3)⋊C4—28no—
241C4⋊Dic3—28no—
242C4⋊Dic3—28no—
243C4⋊Dic3—28no—
244Dic3⋊C4—216no—
245Dic3⋊C4—216no—
246C25—31no—
247C25—31no—
248C22×D4—38no—
249C22×D4—34no—
250C22×D4—64no—
251C22×D4—64no—
252C23×C4—31no—
253C23×C4—38no—
254C23×C4—34no—
255C23×C4—32no—
256C23×C4—32no—
257C23×C4—31no—
258C23×C4—38no—
259C23×C4—34no—
260C23×C4—68no—
261C23×C4—68no—
262C23×C4—68no—
263C23×C4—64no—
264C23×C4—64no—
265C23×C4—64no—
266C23×C4—64no—
267C23×C4—64no—
268C23×C4C22×D312yes—
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D312yes—
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273#123—664no—
274C4×D4—632no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—38no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—38no—
282C2×C4⋊C4—34no—
283C2×C4⋊C4—632no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—64no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4—68no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4C22×D314yes—
290C2×C4⋊C4C22×D318yes—
291C2×C4⋊C4C22×D314yes—
292C2×C22⋊C4—34no—
293C2×C22⋊C4—34no—
294C2×C22⋊C4—38no—
295C2×C22⋊C4—38no—
296C2×C22⋊C4—38no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—64no—
299C2×C22⋊C4—68no—
300C2×C22⋊C4—68no—
301C2×C22⋊C4—68no—
302C2×C42—24no—
303C2×C42C22×D314yes—
304#95—38no—
305#95—38no—
306#95—38no—
307#95—38no—
308#95C22×D318yes—
309C23×C3C22×D412yes—
310C23×C3C22×D412yes—
311C23×C3C2×#4112yes—
312C23×C3C2×#4112yes—
313C23×C3C2×#4112yes—
314C23×C32−1+412yes—
315C23×C32−1+412yes—
316C23×C3C2511yescharacteristic
317C22×D3—28no—
318C22×D3—28no—
319C22×D3—28no—
320C22×D3—28no—
321C22×D3—28no—
322C22×D3—28no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—28no—
328C22×D3—28no—
329C22×D3—28no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—44no—
337C22×D3—44no—
338C22×D3—44no—
339C22×D3—44no—
340C22×D3#121116yes—
341C22×D3C22×D414yes—
342C22×D3C22×D414yes—
343C22×D3C22×D414yes—
344C22×D3C2×#4114yes—
345C2×C3×C4—28no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—48no—
358C2×C3×C4—44no—
359C2×C3×C4—48no—
360C2×C3×C4#123116yescomplemented
361C2×C3×C4C22×D414yes—
362C2×C3×C4C22×D414yes—
363C2×C3×C4C2×#4114yes—
364C2×C3×C4C2×#4114yes—
365C2×Dic3—28no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—48no—
378C2×Dic3—44no—
379C2×Dic3—48no—
380C2×Dic3C22×D414yes—
381D12—232no—
382C4×D3—232no—
383C4×D3—432no—
384C4×D3—416no—
385C24—34no—
386C24—34no—
387C24—34no—
388C24—32no—
389C24—34no—
390C24—32no—
391C24—62no—
392C24—62no—
393C24—62no—
394C24—62no—
395C24—62no—
396C24—62no—
397C24—62no—
398C24—62no—
399C24—62no—
400C24—62no—
401C24C23×D311yescharacteristic
402C2×D4—316no—
403C2×D4—68no—
404C2×D4—68no—
405C2×D4—632no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—316no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—34no—
423C22×C4—616no—
424C22×C4—616no—
425C22×C4—616no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—632no—
429C22×C4—68no—
430C22×C4—68no—
431C22×C4—62no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—62no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—68no—
441C22×C4—68no—
442C22×C4—68no—
443C22×C4—1232no—
444C22×C4—1216no—
445C22×C4#23514yes—
446C22×C4D3×D414yes—
447C22×C4D3×D414yes—
448C22×C4#23914yes—
449C22×C4C23×D312yes—
450C22×C4C23×D314yes—
451C22×C4C23×D314yes—
452C22×C4C23×D312yes—
453C22×C4C23×D312yes—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—616no—
458C4⋊C4—68no—
459C4⋊C4—68no—
460C4⋊C4—616no—
461C4⋊C4—68no—
462C22⋊C4—68no—
463C22⋊C4—68no—
464C22⋊C4—616no—
465C22⋊C4—616no—
466C22⋊C4—616no—
467C42—28no—
468C22×C3C2×#12114yes—
469C22×C3C2×#12314yes—
470C22×C3#53614yes—
471C22×C3#54714yes—
472C22×C3#54714yes—
473C22×C3#55214yes—
474C22×C3#55214yes—
475C22×C3C23×D411yescharacteristic
476C22×C3C23×D411yescharacteristic
477C22×C3C22×#4111yescharacteristic
478C22×C3C22×#4111yescharacteristic
479C22×C3C22×#4111yescharacteristic
480C22×C3C2×2−1+411yescharacteristic
481C22×C3C2×2−1+411yescharacteristic
482C2×D3—24no—
483C2×D3—24no—
484C2×D3—24no—
485C2×D3—24no—
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—232no—
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—416no—
497C2×D3C2×#12118yes—
498C3×C4—24no—
499C3×C4—44no—
500C3×C4—42no—
501C3×C4—44no—
502C3×C4C2×#12318yescomplemented
503Dic3—24no—
504Dic3—44no—
505Dic3—42no—
506Dic3—44no—
507C23—34no—
508C23—34no—
509C23—34no—
510C23—316no—
511C23—34no—
512C23—68no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—62no—
517C23—68no—
518C23—68no—
519C23—68no—
520C23—68no—
521C23—68no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—68no—
529C23—68no—
530C23—68no—
531C23—124no—
532C23—124no—
533C23—124no—
534C23—124no—
535C23C2×#23512yes—
536C23C2×D3×D412yes—
537C23C2×D3×D412yes—
538C23C2×#23912yes—
539C23#100712yes—
540C23D3×#4112yes—
541C23#101012yes—
542C23C24×D311yescharacteristic
543D4—632no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—34no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—68no—
563C2×C4—632no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—1232no—
574C2×C4—128no—
575C2×C4—124no—
576C2×C4—128no—
577C2×C4—1216no—
578C2×C4#894116yescomplemented
579C2×C4C2×#23514yes—
580C2×C4C2×D3×D414yes—
581C2×C4C2×D3×D414yes—
582C2×C4C2×#23914yes—
583C2×C3#248618yescomplemented
584C2×C3C22×#12111yescharacteristic
585C2×C3C22×#12311yescharacteristic
586C2×C3C2×#53611yescharacteristic
587C2×C3C2×#54711yescharacteristic
588C2×C3C2×#54711yescharacteristic
589C2×C3C2×#55211yescharacteristic
590C2×C3C2×#55211yescharacteristic
591D3—28no—
592D3—44no—
593C22—38no—
594C22—64no—
595C22—64no—
596C22—64no—
597C22—64no—
598C22—632no—
599C22—64no—
600C22—64no—
601C22—64no—
602C22—64no—
603C22—64no—
604C22—122no—
605C22—122no—
606C22—1216no—
607C22—122no—
608C22—122no—
609C22C2×#89414yes—
610C22#555114yes—
611C22#558114yes—
612C22D3×#12114yes—
613C22#562814yes—
614C22#569414yes—
615C22#572614yes—
616C22C22×#23511yescharacteristic
617C22C22×D3×D411yescharacteristic
618C22C22×D3×D411yescharacteristic
619C22C22×#23911yescharacteristic
620C22C2×#100711yescharacteristic
621C22C2×D3×#4111yescharacteristic
622C22C2×#101011yescharacteristic
623C4—24no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—64no—
628C4—124no—
629C4—124no—
630C4—122no—
631C4C2×#89418yescomplemented
632C3C2×#248611yescharacteristic, complemented, Sylow
633C2—68no—
634C2—124no—
635C2#7623118yesdirect product factor
636C2C22×#89411yescharacteristic
637C2C2×#555111yescharacteristic
638C2C2×#558111yescharacteristic
639C2C2×D3×#12111yescharacteristic
640C2C2×#562811yescharacteristic
641C2C2×#569411yescharacteristic
642C2C2×#572611yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#389C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3C2×#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×D3×C4⋊C4C2semidirect
4. C2×C3×#389C2semidirect
5. C2×#4987C2semidirect
6. C2×#4987C2semidirect
7. C2×#4987C2semidirect
8. C2×#4982C2semidirect
9. C2×#4928C2semidirect
10. #76231C2direct (N × H)
11. C2×#4675C2semidirect
12. C2×#4664C2semidirect
35. C2×D3×C4⋊C4C22semidirect
38. C3×#389C22semidirect
44. #4987C22semidirect
45. #4987C22semidirect
46. #4987C22semidirect
47. #4982C22semidirect
48. #4928C22semidirect
49. #4675C22semidirect
50. #4664C22semidirect
58. C2×#389D3semidirect
153. #389C2×D3semidirect
360. C2×C3×C4#123semidirect
502. C3×C4C2×#123semidirect
578. C2×C4#894semidirect
583. C2×C3#2486semidirect
631. C4C2×#894semidirect
632. C3C2×#2486semidirect
635. C2#76231direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8942
C22×D3×C4⋊C42
C22×C4×D122
C2×C3×#3892
C2×#49872
C2×#49822
C2×#49282
#762312
C2×#46752
C2×#46642
C2×#24863