Finite Group Databasebeta

← C2×#74655 #11001023

Subgroups — C2×#74655 #11001023

Order 768.

7502 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
1C24×C4×D3C211yesmaximal, characteristic
2C22×#896C212yesmaximal, complemented
3C22×#878C211yesmaximal, characteristic, complemented
4C2×C3×#385C211yesmaximal, characteristic, complemented
5C2×#5210C211yesmaximal, characteristic, complemented
6C2×#4989C212yesmaximal, complemented
7C2×#4955C211yesmaximal, characteristic
8C2×#4955C211yesmaximal, characteristic
9C2×#4942C211yesmaximal, characteristic, complemented
10#74655C2116yesmaximal, direct product factor
11C2×#4668C212yesmaximal, complemented
12C2×#4667C212yesmaximal, complemented
13C2×#2848—31nomaximal, Sylow
14C25×D3C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C23×C4×D3—28no—
18C23×C4×D3—216no—
19C23×C4×D3C2212yes—
20C23×C4×D3C2218yes—
21C23×C4×D3C2212yes—
22C23×Dic6C2211yescharacteristic, complemented
23C22×C3×C4⋊C4C2212yescomplemented
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2211yescharacteristic
26C22×#217C2211yescharacteristic
27C22×#217C2211yescharacteristic
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×(C2×D3)⋊C4C2212yes—
31C22×(C2×D3)⋊C4C2212yes—
32C22×C4⋊Dic3C2211yescharacteristic, complemented
33C22×Dic3⋊C4C2211yescharacteristic, complemented
34C22×Dic3⋊C4C2212yescomplemented
35C22×Dic3⋊C4C2211yescharacteristic, complemented
36C22×Dic3⋊C4C2212yescomplemented
37C2×#896—216no—
38C2×#896C22116yescomplemented
39C2×#878—28no—
40C2×#878C2218yescomplemented
41C3×#385C2218yescomplemented
42C2×C3×#95C2212yescomplemented
43#5210C2218yescomplemented
44C2×C3⋊#95C2212yescomplemented
45C2×C3⋊#95C2212yescomplemented
46#4989C22116yescomplemented
47#4955C2218yes—
48#4955C2218yes—
49#4942C2218yescomplemented
50#4668C22116yescomplemented
51#4667C22116yescomplemented
52C25×C4—31no—
53C22×C22⋊Q8—32no—
54C22×C22⋊Q8—31no—
55#2848—316no—
56C2×#385—32no—
57C2×#385—31no—
58C2×#385—32no—
59C2×#385D311yescharacteristic, complemented
60C2×#380—32no—
61C2×#380—31no—
62C2×#379—31no—
63C2×#379—31no—
64C25×C3C2311yescharacteristic
65C24×D3—24no—
66C24×D3—24no—
67C24×D3—28no—
68C24×D3—28no—
69C24×D3Q818yescomplemented
70C24×D3C2314yes—
71C24×D3C2312yes—
72C23×C3×C4—24no—
73C23×C3×C4—28no—
74C23×C3×C4C2311yescharacteristic
75C23×C3×C4C2314yes—
76C23×C3×C4C2312yes—
77C23×C3×C4C2311yescharacteristic
78C23×C3×C4C2312yes—
79C23×Dic3—28no—
80C23×Dic3D418yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2312yes—
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2314yes—
85C23×Dic3C2311yescharacteristic
86C23×Dic3C2311yescharacteristic
87C22×C4×D3—2128no—
88C22×C4×D3—28no—
89C22×C4×D3—28no—
90C22×C4×D3—232no—
91C22×C4×D3—28no—
92C22×C4×D3—28no—
93C22×C4×D3—28no—
94C22×C4×D3—264no—
95C22×C4×D3D4116yes—
96C22×C4×D3C2318yes—
97C22×C4×D3C2318yes—
98C22×Dic6—24no—
99C22×Dic6—28no—
100C22×Dic6C2314yescomplemented
101C2×C3×C4⋊C4—28no—
102C2×C3×C4⋊C4C2318yescomplemented
103C2×C3×C22⋊C4—24no—
104C2×C3×C22⋊C4—24no—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×#217—24no—
108C2×#217—24no—
109C2×#217C2314yes—
110C2×#217C2314yes—
111C2×(C2×D3)⋊C4—28no—
112C2×(C2×D3)⋊C4—28no—
113C2×(C2×D3)⋊C4—28no—
114C2×(C2×D3)⋊C4—28no—
115C2×(C2×D3)⋊C4C2318yes—
116C2×(C2×D3)⋊C4C2318yes—
117C2×(C2×D3)⋊C4C2318yes—
118C2×(C2×D3)⋊C4C2318yes—
119C2×C4⋊Dic3—24no—
120C2×C4⋊Dic3C2314yescomplemented
121C2×Dic3⋊C4—28no—
122C2×Dic3⋊C4—24no—
123C2×Dic3⋊C4—24no—
124C2×Dic3⋊C4—28no—
125C2×Dic3⋊C4C2314yescomplemented
126C2×Dic3⋊C4C2318yescomplemented
127C2×Dic3⋊C4C2318yescomplemented
128C2×Dic3⋊C4C2314yescomplemented
129#896—264no—
130#878—232no—
131C3×#95C2318yescomplemented
132C3⋊#95C2318yescomplemented
133C3⋊#95C2318yescomplemented
134C26—31no—
135C23×Q8—31no—
136C24×C4—31no—
137C24×C4—32no—
138C24×C4—38no—
139C24×C4—32no—
140C24×C4—68no—
141C24×C4—616no—
142C24×C4C2×D311yescharacteristic
143C2×C22⋊Q8—316no—
144C2×C22⋊Q8—38no—
145C2×C22⋊Q8—68no—
146C2×C22⋊Q8—616no—
147C22×C4⋊C4—31no—
148C22×C4⋊C4—32no—
149C22×C4⋊C4—31no—
150C22×C4⋊C4—32no—
151C22×C4⋊C4—31no—
152C22×C4⋊C4C2×D312yescomplemented
153C22×C22⋊C4—31no—
154C22×C22⋊C4—31no—
155C22×C22⋊C4—32no—
156C22×C22⋊C4—32no—
157C22×C22⋊C4—32no—
158C22×C22⋊C4—32no—
159C22×C22⋊C4C2×D311yescharacteristic
160C22×C22⋊C4C2×D311yescharacteristic
161#385—38no—
162#385—316no—
163#385—316no—
164#385C2×D318yescomplemented
165#380—38no—
166#380—316no—
167#379—38no—
168#379—38no—
169C2×#95—32no—
170C2×#95—32no—
171C2×#95C2×D312yescomplemented
172C24×C3—24no—
173C24×C3—24no—
174C24×C3C2×D414yes—
175C24×C3C2×Q814yes—
176C24×C3#4114yes—
177C24×C3C2412yes—
178C24×C3C2411yescharacteristic
179C23×D3—232no—
180C23×D3—264no—
181C23×D3—232no—
182C23×D3—232no—
183C23×D3—28no—
184C23×D3—24no—
185C23×D3—24no—
186C23×D3—28no—
187C23×D3—24no—
188C23×D3—24no—
189C23×D3—264no—
190C23×D3—216no—
191C23×D3—216no—
192C23×D3C2×D418yes—
193C23×D3C2×Q818yes—
194C23×D3C2×Q8116yescomplemented
195C23×D3#4118yes—
196C23×D3C2414yes—
197C22×C3×C4—232no—
198C22×C3×C4—28no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—216no—
202C22×C3×C4—22no—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—24no—
209C22×C3×C4C2×D418yes—
210C22×C3×C4C2414yes—
211C22×C3×C4C2412yes—
212C22×C3×C4C2412yes—
213C22×C3×C4C2414yes—
214C22×Dic3—24no—
215C22×Dic3—232no—
216C22×Dic3—216no—
217C22×Dic3—24no—
218C22×Dic3—22no—
219C22×Dic3—24no—
220C22×Dic3—22no—
221C22×Dic3—24no—
222C22×Dic3—24no—
223C22×Dic3—24no—
224C22×Dic3C2×D4116yes—
225C22×Dic3C2×D414yes—
226C22×Dic3C2×D414yes—
227C22×Dic3C2×D418yes—
228C22×Dic3C2414yes—
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C2×C4×D3—264no—
234C2×C4×D3—2256no—
235C2×C4×D3—216no—
236C2×C4×D3—216no—
237C2×C4×D3—264no—
238C2×C4×D3—216no—
239C2×C4×D3—216no—
240C2×C4×D3—216no—
241C2×C4×D3C2×D4132yes—
242C2×Dic6—28no—
243C2×Dic6—216no—
244C2×Dic6—416no—
245C3×C4⋊C4—216no—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—28no—
248#217—28no—
249#217—28no—
250(C2×D3)⋊C4—216no—
251(C2×D3)⋊C4—216no—
252(C2×D3)⋊C4—216no—
253(C2×D3)⋊C4—216no—
254C4⋊Dic3—28no—
255Dic3⋊C4—216no—
256Dic3⋊C4—28no—
257Dic3⋊C4—28no—
258Dic3⋊C4—216no—
259C25—32no—
260C25—38no—
261C25—34no—
262C25—68no—
263C25—68no—
264C25—64no—
265C25—64no—
266C25C22×D311yescharacteristic
267C22×Q8—34no—
268C22×Q8—64no—
269C22×Q8—68no—
270C23×C4—24no—
271C23×C4—28no—
272C23×C4—31no—
273C23×C4—38no—
274C23×C4—34no—
275C23×C4—31no—
276C23×C4—316no—
277C23×C4—38no—
278C23×C4—31no—
279C23×C4—32no—
280C23×C4—31no—
281C23×C4—38no—
282C23×C4—68no—
283C23×C4—68no—
284C23×C4—68no—
285C23×C4—632no—
286C23×C4—664no—
287C23×C4—68no—
288C23×C4—6128no—
289C23×C4—68no—
290C23×C4—68no—
291C23×C4C22×D314yes—
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C23×C4C22×D312yes—
295C23×C4C22×D312yes—
296C22⋊Q8—632no—
297C22⋊Q8—664no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—38no—
300C2×C4⋊C4—34no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—38no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—64no—
305C2×C4⋊C4—64no—
306C2×C4⋊C4—68no—
307C2×C4⋊C4—64no—
308C2×C4⋊C4—68no—
309C2×C4⋊C4C22×D318yescomplemented
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—34no—
314C2×C22⋊C4—38no—
315C2×C22⋊C4—38no—
316C2×C22⋊C4—38no—
317C2×C22⋊C4—38no—
318C2×C22⋊C4—68no—
319C2×C22⋊C4—68no—
320C2×C22⋊C4—68no—
321C2×C22⋊C4—68no—
322C2×C22⋊C4—64no—
323C2×C22⋊C4—64no—
324C2×C22⋊C4C22×D314yes—
325C2×C22⋊C4C22×D314yes—
326#95—38no—
327#95—38no—
328#95C22×D318yescomplemented
329C23×C3—28no—
330C23×C3—216no—
331C23×C3—28no—
332C23×C3—216no—
333C23×C3—24no—
334C23×C3—24no—
335C23×C3C22⋊Q8116yescomplemented
336C23×C3C22×D414yes—
337C23×C3C22×D414yes—
338C23×C3C22×D412yes—
339C23×C3C22×Q814yes—
340C23×C3C22×Q812yes—
341C23×C3C2×#4114yes—
342C23×C3C2×#4112yes—
343C23×C3C2×#4114yes—
344C23×C3C2511yescharacteristic
345C22×D3—232no—
346C22×D3—2256no—
347C22×D3—264no—
348C22×D3—232no—
349C22×D3—216no—
350C22×D3—216no—
351C22×D3—216no—
352C22×D3—232no—
353C22×D3—216no—
354C22×D3—216no—
355C22×D3—216no—
356C22×D3—24no—
357C22×D3—24no—
358C22×D3—24no—
359C22×D3—24no—
360C22×D3—264no—
361C22×D3C22⋊Q8132yescomplemented
362C22×D3C22×D418yes—
363C22×D3C22×Q818yes—
364C22×D3C2×#4118yes—
365C2×C3×C4—232no—
366C2×C3×C4—216no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—44no—
377C2×C3×C4—44no—
378C2×C3×C4—48no—
379C2×C3×C4C22×D418yes—
380C2×Dic3—28no—
381C2×Dic3—24no—
382C2×Dic3—232no—
383C2×Dic3—24no—
384C2×Dic3—22no—
385C2×Dic3—24no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—44no—
391C2×Dic3—44no—
392C2×Dic3—48no—
393C2×Dic3#120116yes—
394C2×Dic3C22×D414yes—
395C2×Dic3C22×D414yes—
396C2×Dic3C22×D418yes—
397C4×D3—264no—
398C4×D3—264no—
399Dic6—416no—
400C24—24no—
401C24—24no—
402C24—38no—
403C24—38no—
404C24—38no—
405C24—34no—
406C24—316no—
407C24—616no—
408C24—616no—
409C24—664no—
410C24—632no—
411C24—632no—
412C24—664no—
413C24—632no—
414C24—68no—
415C24—64no—
416C24—68no—
417C24—64no—
418C24—64no—
419C24—64no—
420C24#23514yes—
421C24D3×D414yes—
422C24D3×Q814yes—
423C24C23×D312yes—
424C24C23×D311yescharacteristic
425C2×Q8—68no—
426C2×Q8—616no—
427C2×Q8—1216no—
428C22×C4—232no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—216no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—34no—
441C22×C4—38no—
442C22×C4—32no—
443C22×C4—34no—
444C22×C4—316no—
445C22×C4—32no—
446C22×C4—332no—
447C22×C4—34no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—616no—
451C22×C4—616no—
452C22×C4—616no—
453C22×C4—664no—
454C22×C4—616no—
455C22×C4—6256no—
456C22×C4—664no—
457C22×C4—616no—
458C22×C4—632no—
459C22×C4—64no—
460C22×C4—616no—
461C22×C4—64no—
462C22×C4—62no—
463C22×C4—64no—
464C22×C4—62no—
465C22×C4—64no—
466C22×C4—64no—
467C22×C4—64no—
468C22×C4D3×D418yes—
469C22×C4C23×D314yes—
470C22×C4C23×D312yes—
471C22×C4C23×D312yes—
472C22×C4C23×D314yes—
473C4⋊C4—216no—
474C4⋊C4—68no—
475C4⋊C4—68no—
476C4⋊C4—616no—
477C4⋊C4—68no—
478C4⋊C4—616no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—616no—
482C22⋊C4—616no—
483C22⋊C4—616no—
484C22⋊C4—616no—
485C22⋊C4—68no—
486C22⋊C4—68no—
487C22×C3—232no—
488C22×C3—24no—
489C22×C3—28no—
490C22×C3—24no—
491C22×C3—28no—
492C22×C3C2×#12014yes—
493C22×C3C2×C22⋊Q814yes—
494C22×C3C2×C22⋊Q818yes—
495C22×C3C2×C22⋊Q818yescomplemented
496C22×C3#52514yes—
497C22×C3#52518yes—
498C22×C3C23×D412yes—
499C22×C3C23×D411yescharacteristic
500C22×C3C23×Q811yescharacteristic
501C22×C3C22×#4111yescharacteristic
502C22×C3C22×#4112yes—
503C2×D3—2128no—
504C2×D3—264no—
505C2×D3—28no—
506C2×D3—28no—
507C2×D3—28no—
508C2×D3—28no—
509C2×D3—28no—
510C2×D3—28no—
511C2×D3C2×C22⋊Q8116yescomplemented
512C3×C4—28no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—42no—
516Dic3—24no—
517Dic3—42no—
518Dic3—44no—
519Dic3—42no—
520Dic3C2×#12018yes—
521C23—28no—
522C23—24no—
523C23—24no—
524C23—216no—
525C23—28no—
526C23—216no—
527C23—38no—
528C23—38no—
529C23—38no—
530C23—332no—
531C23—664no—
532C23—632no—
533C23—664no—
534C23—6256no—
535C23—632no—
536C23—616no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23—64no—
541C23—616no—
542C23—632no—
543C23—616no—
544C23—616no—
545C23—616no—
546C23—616no—
547C23#896116yescomplemented
548C23C2×#23514yes—
549C23C2×#23512yes—
550C23C2×D3×D414yes—
551C23C2×D3×D412yes—
552C23C2×D3×D414yes—
553C23C2×D3×Q814yes—
554C23C2×D3×Q812yes—
555C23D3×#4114yes—
556C23C24×D311yescharacteristic
557Q8—1216no—
558C2×C4—232no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—216no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—38no—
570C2×C4—34no—
571C2×C4—316no—
572C2×C4—34no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—664no—
577C2×C4—664no—
578C2×C4—632no—
579C2×C4—64no—
580C2×C4—68no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—124no—
589C2×C4—128no—
590C2×C4—124no—
591C2×C4C2×D3×D418yes—
592C2×C3—28no—
593C2×C3#284818yescomplemented
594C2×C3C22×#12011yescharacteristic
595C2×C3C22×C22⋊Q811yescharacteristic
596C2×C3C22×C22⋊Q812yes—
597C2×C3C2×#52512yes—
598C2×C3C2×#52511yescharacteristic
599D3—216no—
600C22—28no—
601C22—24no—
602C22—232no—
603C22—28no—
604C22—24no—
605C22—316no—
606C22—6128no—
607C22—68no—
608C22—68no—
609C22—68no—
610C22—664no—
611C22—68no—
612C22—68no—
613C22—68no—
614C22C2×#89614yes—
615C22C2×#89618yescomplemented
616C22#554718yes—
617C22D3×#12014yes—
618C22D3×C22⋊Q818yes—
619C22#565614yes—
620C22C22×#23511yescharacteristic
621C22C22×D3×D412yes—
622C22C22×D3×D411yescharacteristic
623C22C22×D3×Q811yescharacteristic
624C22C2×D3×#4112yes—
625C4—28no—
626C4—38no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—64no—
631C4—124no—
632C4—122no—
633C4—122no—
634C3C2×#284811yescharacteristic, complemented, Sylow
635C2—28no—
636C2—616no—
637C2#7465518yesdirect product factor
638C2C22×#89611yescharacteristic
639C2C2×#554712yes—
640C2C2×D3×#12011yescharacteristic
641C2C2×D3×C22⋊Q812yes—
642C2C2×#565611yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#385C2
socleC24×C3C24
prime radicalC25×D3C22
hypercenterC2×#385D3
hypocenterC3C2×#2848
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
2. C22×#896C2semidirect
3. C22×#878C2semidirect
4. C2×C3×#385C2semidirect
5. C2×#5210C2semidirect
6. C2×#4989C2semidirect
9. C2×#4942C2semidirect
10. #74655C2direct (N × H)
11. C2×#4668C2semidirect
12. C2×#4667C2semidirect
22. C23×Dic6C22semidirect
23. C22×C3×C4⋊C4C22semidirect
32. C22×C4⋊Dic3C22semidirect
33. C22×Dic3⋊C4C22semidirect
34. C22×Dic3⋊C4C22semidirect
35. C22×Dic3⋊C4C22semidirect
36. C22×Dic3⋊C4C22semidirect
38. C2×#896C22semidirect
40. C2×#878C22semidirect
41. C3×#385C22semidirect
42. C2×C3×#95C22semidirect
43. #5210C22semidirect
44. C2×C3⋊#95C22semidirect
45. C2×C3⋊#95C22semidirect
46. #4989C22semidirect
49. #4942C22semidirect
50. #4668C22semidirect
51. #4667C22semidirect
59. C2×#385D3semidirect
69. C24×D3Q8semidirect
100. C22×Dic6C23semidirect
102. C2×C3×C4⋊C4C23semidirect
120. C2×C4⋊Dic3C23semidirect
125. C2×Dic3⋊C4C23semidirect
126. C2×Dic3⋊C4C23semidirect
127. C2×Dic3⋊C4C23semidirect
128. C2×Dic3⋊C4C23semidirect
131. C3×#95C23semidirect
132. C3⋊#95C23semidirect
133. C3⋊#95C23semidirect
152. C22×C4⋊C4C2×D3semidirect
164. #385C2×D3semidirect
171. C2×#95C2×D3semidirect
194. C23×D3C2×Q8semidirect
309. C2×C4⋊C4C22×D3semidirect
328. #95C22×D3semidirect
335. C23×C3C22⋊Q8semidirect
361. C22×D3C22⋊Q8semidirect
495. C22×C3C2×C22⋊Q8semidirect
511. C2×D3C2×C22⋊Q8semidirect
547. C23#896semidirect
593. C2×C3#2848semidirect
615. C22C2×#896semidirect
634. C3C2×#2848semidirect
637. C2#74655direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C4×D32
C22×#8962
C22×#8782
C2×C3×#3852
C2×#52102
C2×#49892
C2×#49552
C2×#49422
#746552
C2×#46682
C2×#46672
C2×#28483