Finite Group Databasebeta

← C2×#76232 #10964360

Subgroups — C2×#76232 #10964360

Order 768.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×C4×D12C211yesmaximal, characteristic
3C2×C3×#389C211yesmaximal, characteristic, complemented
4C2×#4993C212yesmaximal, complemented
5C2×#4989C212yesmaximal, complemented
6C2×#4987C211yesmaximal, characteristic, complemented
7C2×#4987C211yesmaximal, characteristic, complemented
8C2×#4987C211yesmaximal, characteristic, complemented
9C2×#4928C211yesmaximal, characteristic, complemented
10#76232C2116yesmaximal, direct product factor
11C2×#4665C212yesmaximal
12C2×#4660C212yesmaximal, complemented
13C2×#2487—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C23×C4×D3C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C2×D3×C4⋊C4—28no—
33C2×D3×C4⋊C4C2218yescomplemented
34C2×C4×D12—28no—
35C2×C4×D12C2218yes—
36C3×#389C2218yescomplemented
37C2×C3×#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4993C22116yescomplemented
43#4989C22116yescomplemented
44#4987C2218yescomplemented
45#4987C2218yescomplemented
46#4987C2218yescomplemented
47#4928C2218yescomplemented
48#4665C22116yes—
49#4660C22116yescomplemented
50C22×C4×D4—31no—
51C23×C4⋊C4—31no—
52#2487—316no—
53C2×#400—32no—
54C2×#399—32no—
55C2×#389—31no—
56C2×#389D311yescharacteristic, complemented
57C2×#386—32no—
58C2×#385—32no—
59C2×#380—31no—
60C2×#380—31no—
61C2×#380—31no—
62C24×D3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C22×D12—24no—
74C22×D12—24no—
75C22×D12Q818yes—
76C22×D12C2314yes—
77C22×C4×D3—24no—
78C22×C4×D3—24no—
79C22×C4×D3—28no—
80C22×C4×D3—24no—
81C22×C4×D3—24no—
82C22×C4×D3—24no—
83C22×C4×D3—44no—
84C22×C4×D3—44no—
85C22×C4×D3D418yes—
86C22×C4×D3C2314yes—
87C22×C4×D3C2314yes—
88C22×C4×D3C2318yes—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—44no—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C4⋊C4C2314yes—
94C2×C3×C4⋊C4C2318yes—
95C2×C3×C42—24no—
96C2×C3×C42C2314yes—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—28no—
99C2×(C2×D3)⋊C4—24no—
100C2×(C2×D3)⋊C4—44no—
101C2×(C2×D3)⋊C4—44no—
102C2×(C2×D3)⋊C4C2318yes—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2314yes—
105C2×(C2×D3)⋊C4C2318yes—
106C2×(C2×D3)⋊C4C2318yes—
107C2×C4⋊Dic3—24no—
108C2×C4⋊Dic3—24no—
109C2×C4⋊Dic3—24no—
110C2×C4⋊Dic3C2314yes—
111C2×C4⋊Dic3C2314yes—
112C2×C4⋊Dic3C2314yes—
113C2×Dic3⋊C4—28no—
114C2×Dic3⋊C4—44no—
115C2×Dic3⋊C4C2318yes—
116C2×Dic3⋊C4C2318yes—
117D3×C4⋊C4—232no—
118C4×D12—232no—
119C3×#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C23×D4—31no—
125C24×C4—31no—
126C24×C4—31no—
127C24×C4—32no—
128C2×C4×D4—38no—
129C2×C4×D4—68no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—38no—
136C22×C4⋊C4—68no—
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D312yes—
140C22×C22⋊C4—31no—
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4—32no—
145C22×C42C2×D311yescharacteristic
146#400—316no—
147#399—316no—
148#389—38no—
149#389C2×D318yescomplemented
150#386—316no—
151#385—316no—
152#380—38no—
153#380—38no—
154#380—38no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95—32no—
159C2×#95C2×D312yes—
160C24×C3C2411yescharacteristic
161C23×D3—22no—
162C23×D3—22no—
163C23×D3—22no—
164C23×D3—22no—
165C23×D3—22no—
166C23×D3—22no—
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3C2×D414yes—
172C23×D3C2×Q814yes—
173C23×D3C2×Q814yes—
174C23×D3#4114yes—
175C23×D3C2412yes—
176C23×D3C2412yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×Q814yes—
190C22×C3×C4#4114yes—
191C22×C3×C4#4114yes—
192C22×C3×C4C2414yes—
193C22×C3×C4C2414yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2412yes—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—24no—
200C22×Dic3—22no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3C2×D414yes—
209C22×Dic3C2412yes—
210C22×Dic3C2414yes—
211C22×Dic3C2414yes—
212C22×Dic3C2412yes—
213C2×D12—232no—
214C2×D12—28no—
215C2×D12—28no—
216C2×D12C2×Q8116yes—
217C2×C4×D3—232no—
218C2×C4×D3—28no—
219C2×C4×D3—28no—
220C2×C4×D3—216no—
221C2×C4×D3—28no—
222C2×C4×D3—28no—
223C2×C4×D3—28no—
224C2×C4×D3—48no—
225C2×C4×D3—48no—
226C2×C4×D3—432no—
227C2×C4×D3—416no—
228C2×C4×D3C2×D4116yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—48no—
232C3×C42—28no—
233(C2×D3)⋊C4—28no—
234(C2×D3)⋊C4—216no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—48no—
237(C2×D3)⋊C4—48no—
238C4⋊Dic3—28no—
239C4⋊Dic3—28no—
240C4⋊Dic3—28no—
241Dic3⋊C4—216no—
242Dic3⋊C4—48no—
243C25—31no—
244C25—31no—
245C22×D4—38no—
246C22×D4—34no—
247C22×D4—64no—
248C22×D4—64no—
249C23×C4—31no—
250C23×C4—38no—
251C23×C4—34no—
252C23×C4—32no—
253C23×C4—32no—
254C23×C4—31no—
255C23×C4—34no—
256C23×C4—38no—
257C23×C4—68no—
258C23×C4—64no—
259C23×C4—64no—
260C23×C4—64no—
261C23×C4—64no—
262C23×C4—64no—
263C23×C4—124no—
264C23×C4—124no—
265C23×C4C22×D311yescharacteristic
266C23×C4C22×D312yes—
267C23×C4C22×D312yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C4×D4—632no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—34no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—34no—
277C2×C4⋊C4—38no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—632no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—124no—
285C2×C4⋊C4C22×D314yes—
286C2×C4⋊C4C22×D314yes—
287C2×C4⋊C4C22×D318yes—
288C2×C22⋊C4—34no—
289C2×C22⋊C4—34no—
290C2×C22⋊C4—38no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—38no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—68no—
296C2×C22⋊C4—124no—
297C2×C22⋊C4—124no—
298C2×C42—24no—
299C2×C42C22×D314yes—
300#95—38no—
301#95—38no—
302#95—38no—
303#95—38no—
304#95C22×D318yes—
305C23×C3C22×D412yes—
306C23×C3C22×Q812yes—
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3C2×#4112yes—
310C23×C3H(5,𝔽2)12yes—
311C23×C3H(5,𝔽2)12yes—
312C23×C3C2511yescharacteristic
313C22×D3—28no—
314C22×D3—28no—
315C22×D3—28no—
316C22×D3—28no—
317C22×D3—28no—
318C22×D3—28no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—28no—
324C22×D3—28no—
325C22×D3—28no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—44no—
333C22×D3—44no—
334C22×D3—44no—
335C22×D3—44no—
336C22×D3C22⋊Q8116yes—
337C22×D3C22×D414yes—
338C22×D3C22×Q814yes—
339C22×D3C22×Q814yes—
340C22×D3C2×#4114yes—
341C2×C3×C4—28no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—48no—
354C2×C3×C4—44no—
355C2×C3×C4—48no—
356C2×C3×C4#123116yes—
357C2×C3×C4C22×D414yes—
358C2×C3×C4C22×Q814yes—
359C2×C3×C4C2×#4114yes—
360C2×C3×C4C2×#4114yes—
361C2×Dic3—28no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—24no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—44no—
374C2×Dic3—48no—
375C2×Dic3—48no—
376C2×Dic3C22×D414yes—
377D12—232no—
378C4×D3—232no—
379C4×D3—432no—
380C4×D3—416no—
381C24—34no—
382C24—34no—
383C24—34no—
384C24—32no—
385C24—34no—
386C24—32no—
387C24—62no—
388C24—62no—
389C24—62no—
390C24—62no—
391C24—62no—
392C24—62no—
393C24—62no—
394C24—62no—
395C24—62no—
396C24—62no—
397C24C23×D311yescharacteristic
398C2×D4—316no—
399C2×D4—68no—
400C2×D4—68no—
401C2×D4—632no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—34no—
414C22×C4—32no—
415C22×C4—316no—
416C22×C4—32no—
417C22×C4—34no—
418C22×C4—34no—
419C22×C4—616no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—632no—
423C22×C4—68no—
424C22×C4—68no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—62no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—68no—
435C22×C4—68no—
436C22×C4—68no—
437C22×C4—128no—
438C22×C4—128no—
439C22×C4—1232no—
440C22×C4—1216no—
441C22×C4#23514yes—
442C22×C4D3×D414yes—
443C22×C4D3×Q814yes—
444C22×C4#23914yes—
445C22×C4C23×D314yes—
446C22×C4C23×D314yes—
447C22×C4C23×D312yes—
448C22×C4C23×D312yes—
449C22×C4C23×D312yes—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—48no—
453C4⋊C4—616no—
454C4⋊C4—68no—
455C4⋊C4—68no—
456C4⋊C4—68no—
457C4⋊C4—128no—
458C22⋊C4—68no—
459C22⋊C4—68no—
460C22⋊C4—616no—
461C22⋊C4—128no—
462C22⋊C4—128no—
463C42—28no—
464C22×C3C2×C22⋊Q814yes—
465C22×C3C2×#12314yes—
466C22×C3#53514yes—
467C22×C3#55314yes—
468C22×C3#55314yes—
469C22×C3#55414yes—
470C22×C3#55414yes—
471C22×C3C23×D411yescharacteristic
472C22×C3C23×Q811yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C22×#4111yescharacteristic
476C22×C3C2×H(5,𝔽2)11yescharacteristic
477C22×C3C2×H(5,𝔽2)11yescharacteristic
478C2×D3—24no—
479C2×D3—24no—
480C2×D3—24no—
481C2×D3—24no—
482C2×D3—24no—
483C2×D3—24no—
484C2×D3—232no—
485C2×D3—24no—
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—416no—
493C2×D3C2×C22⋊Q818yes—
494C3×C4—24no—
495C3×C4—44no—
496C3×C4—42no—
497C3×C4—44no—
498C3×C4C2×#12318yes—
499Dic3—24no—
500Dic3—42no—
501Dic3—44no—
502Dic3—44no—
503C23—34no—
504C23—34no—
505C23—34no—
506C23—316no—
507C23—34no—
508C23—68no—
509C23—62no—
510C23—62no—
511C23—62no—
512C23—62no—
513C23—68no—
514C23—68no—
515C23—68no—
516C23—68no—
517C23—68no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—68no—
525C23—68no—
526C23—68no—
527C23—124no—
528C23—124no—
529C23—124no—
530C23—124no—
531C23C2×#23512yes—
532C23C2×D3×D412yes—
533C23#100412yes—
534C23C2×D3×Q812yes—
535C23C2×#23912yes—
536C23D3×#4112yes—
537C23#100912yes—
538C23C24×D311yescharacteristic
539D4—632no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—34no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—68no—
559C2×C4—632no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—1232no—
570C2×C4—128no—
571C2×C4—124no—
572C2×C4—128no—
573C2×C4—1216no—
574C2×C4#894116yes—
575C2×C4C2×#23514yes—
576C2×C4C2×D3×D414yes—
577C2×C4C2×D3×Q814yes—
578C2×C4C2×#23914yes—
579C2×C3#248718yescomplemented
580C2×C3C22×C22⋊Q811yescharacteristic
581C2×C3C22×#12311yescharacteristic
582C2×C3C2×#53511yescharacteristic
583C2×C3C2×#55311yescharacteristic
584C2×C3C2×#55311yescharacteristic
585C2×C3C2×#55411yescharacteristic
586C2×C3C2×#55411yescharacteristic
587D3—28no—
588D3—44no—
589C22—38no—
590C22—64no—
591C22—64no—
592C22—64no—
593C22—64no—
594C22—64no—
595C22—632no—
596C22—64no—
597C22—64no—
598C22—64no—
599C22—64no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22—1216no—
604C22—122no—
605C22C2×#89414yes—
606C22#555814yes—
607C22#557914yes—
608C22#560714yes—
609C22D3×C22⋊Q814yes—
610C22#569514yes—
611C22#573014yes—
612C22C22×#23511yescharacteristic
613C22C22×D3×D411yescharacteristic
614C22C2×#100411yescharacteristic
615C22C22×D3×Q811yescharacteristic
616C22C22×#23911yescharacteristic
617C22C2×D3×#4111yescharacteristic
618C22C2×#100911yescharacteristic
619C4—24no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—64no—
624C4—124no—
625C4—124no—
626C4—122no—
627C4C2×#89418yes—
628C3C2×#248711yescharacteristic, complemented, Sylow
629C2—68no—
630C2—124no—
631C2#7623218yesdirect product factor
632C2C22×#89411yescharacteristic
633C2C2×#555811yescharacteristic
634C2C2×#557911yescharacteristic
635C2C2×#560711yescharacteristic
636C2C2×D3×C22⋊Q811yescharacteristic
637C2C2×#569511yescharacteristic
638C2C2×#573011yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#389C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3C2×#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×D3×C4⋊C4C2semidirect
3. C2×C3×#389C2semidirect
4. C2×#4993C2semidirect
5. C2×#4989C2semidirect
6. C2×#4987C2semidirect
7. C2×#4987C2semidirect
8. C2×#4987C2semidirect
9. C2×#4928C2semidirect
10. #76232C2direct (N × H)
12. C2×#4660C2semidirect
33. C2×D3×C4⋊C4C22semidirect
36. C3×#389C22semidirect
42. #4993C22semidirect
43. #4989C22semidirect
44. #4987C22semidirect
45. #4987C22semidirect
46. #4987C22semidirect
47. #4928C22semidirect
49. #4660C22semidirect
56. C2×#389D3semidirect
149. #389C2×D3semidirect
579. C2×C3#2487semidirect
628. C3C2×#2487semidirect
631. C2#76232direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C4⋊C42
C22×C4×D122
C2×C3×#3892
C2×#49932
C2×#49892
C2×#49872
C2×#49282
#762322
C2×#46652
C2×#46602
C2×#24873