Finite Group Databasebeta

← #10795131

Subgroups — #10795131

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5545C211yesmaximal, characteristic
2#73567C212yesmaximal, complemented
3C3×C4×#99C211yesmaximal, characteristic, complemented
4C4×#834C211yesmaximal, characteristic, complemented
5#70035C212yesmaximal
6#70028C212yesmaximal, complemented
7C4×#806C212yesmaximal, complemented
8#68969C212yesmaximal, complemented
9#68959C212yesmaximal, complemented
10#20724—31nomaximal, Sylow
11C22×D3×D4C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×D4×Dic3C2211yescharacteristic
14C2×C4×#67C411yescharacteristic, complemented
15C2×C4×#67C2211yescharacteristic
16C22×(C2×D3)⋊C4C411yescharacteristic, complemented
17C22×(C2×D3)⋊C4C2211yescharacteristic
18#5545—24no—
19#5545—24no—
20C2×#893C411yescharacteristic, complemented
21C2×#892C411yescharacteristic, complemented
22C2×#67⋊C4C411yescharacteristic, complemented
23C2×#67⋊C4C2211yescharacteristic
24C2×D3×C22⋊C4C411yescharacteristic, complemented
25C2×D3×C22⋊C4C2211yescharacteristic
26C2×C4×D12C411yescharacteristic, complemented
27C2×#872C411yescharacteristic, complemented
28C2×C3×#99C2212yes—
29C3×C4×C22⋊C4C2212yescomplemented
30C2×#834C2212yes—
31C4×#217C2212yescomplemented
32#4955C2214yes—
33C2×#806C2214yes—
34#4942C2214yescomplemented
35#4940—24no—
36#4669—24no—
37C3×#342C2212yescomplemented
38#4527C2214yescomplemented
39#4526C2212yescomplemented
40C2×#518—31no—
41#2312—32no—
42#1819—32no—
43#1799—32no—
44#1799—32no—
45#1781—32no—
46C4×#99—32no—
47C4×#99—31no—
48C4×#99D311yescharacteristic, complemented
49C24×D3C2×C411yescharacteristic
50C24×D3C2311yescharacteristic
51C22×C3×D4C2311yescharacteristic
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×#67C2×C411yescharacteristic
55C22×#67C2311yescharacteristic
56C23×Dic3C2×C411yescharacteristic
57C23×Dic3C2311yescharacteristic
58C2×D3×D4—24no—
59C2×D3×D4—24no—
60C2×D3×D4—24no—
61C22×D12C2×C411yescharacteristic
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3C2×C411yescharacteristic
65C22×C4×D3C2×C411yescharacteristic
66C22×C4×D3C2×C411yescharacteristic
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4C2×C411yescharacteristic
70C2×C3×C22⋊C4C2×C411yescharacteristic
71C2×C3×C22⋊C4C2314yes—
72C2×C3×C22⋊C4C2311yescharacteristic
73C2×C3×C42—22no—
74C2×C3×C42C2×C411yescharacteristic
75C2×#217C2×C411yescharacteristic
76C2×#217C2314yes—
77C2×#217C2311yescharacteristic
78D4×Dic3—24no—
79C4×#67—22no—
80C4×#67—22no—
81C4×#67—22no—
82C4×#67—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4C2×C411yescharacteristic
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×(C2×D3)⋊C4C2×C411yescharacteristic
90C2×(C2×D3)⋊C4C2×C411yescharacteristic
91C2×(C2×D3)⋊C4C2×C411yescharacteristic
92C2×(C2×D3)⋊C4C2×C411yescharacteristic
93C2×(C2×D3)⋊C4C2×C411yescharacteristic
94C2×(C2×D3)⋊C4C2×C411yescharacteristic
95C2×(C2×D3)⋊C4D414yes—
96C2×(C2×D3)⋊C4D414yes—
97C2×C4⋊Dic3C2×C411yescharacteristic
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4C2×C411yescharacteristic
100C2×Dic3⋊C4C2×C411yescharacteristic
101C2×C4×Dic3—22no—
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104C2×C4×Dic3C2×C411yescharacteristic
105#893—24no—
106#892—24no—
107#67⋊C4—24no—
108#67⋊C4—24no—
109D3×C22⋊C4—24no—
110D3×C22⋊C4—24no—
111C4×D12—22no—
112C4×D12—22no—
113#872—22no—
114#872—22no—
115C3×#99—24no—
116C3×#95—22no—
117#834—24no—
118C3⋊#95—22no—
119C3⋊#95—24no—
120#806—28no—
121C23×D4—31no—
122#518—64no—
123#518—64no—
124C2×C4×D4—31no—
125C2×C4×D4—31no—
126C2×C4×D4—31no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×C4×D4—31no—
130C2×C4×D4—31no—
131C2×C4×D4C2×D311yescharacteristic
132C2×#117—31no—
133C2×#117—31no—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—31no—
136C22×C22⋊C4—31no—
137C22×C22⋊C4—31no—
138C2×#99—32no—
139C2×#99—34no—
140C2×#99C2×D312yes—
141#388—64no—
142#388—64no—
143#380—34no—
144#379—34no—
145C4×C22⋊C4—32no—
146C4×C22⋊C4C2×D312yescomplemented
147#342—32no—
148#342—34no—
149#342C2×D312yescomplemented
150C24×C3C22×C411yescharacteristic
151C24×C3C2411yescharacteristic
152C23×D3—22no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—81no—
158C23×D3—81no—
159C23×D3—81no—
160C23×D3—81no—
161C23×D3C22×C411yescharacteristic
162C23×D3C22×C411yescharacteristic
163C23×D3C22×C411yescharacteristic
164C23×D3C2×D414yes—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4#4114yes—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—24no—
171C22×C3×C4—81no—
172C22×C3×C4—81no—
173C22×C3×C4C22×C411yescharacteristic
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C2×#67—22no—
181C2×#67—22no—
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—22no—
185C2×#67#4114yes—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—81no—
189C22×Dic3—81no—
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C411yescharacteristic
192C22×Dic3C22×C411yescharacteristic
193C22×Dic3C22×C411yescharacteristic
194C22×Dic3C22×C411yescharacteristic
195C22×Dic3C2×D414yes—
196D3×D4—232no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—22no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—81no—
208C2×C4×D3—81no—
209C2×C4×D3—81no—
210C2×C4×D3—81no—
211C3×C4⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42—82no—
219#217—22no—
220#217—22no—
221#217—24no—
222#217—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—21no—
226(C2×D3)⋊C4—21no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—84no—
229(C2×D3)⋊C4—84no—
230(C2×D3)⋊C4C22⋊C412yes—
231(C2×D3)⋊C4C22⋊C412yes—
232(C2×D3)⋊C4C22⋊C412yes—
233(C2×D3)⋊C4C22⋊C412yes—
234(C2×D3)⋊C4C22⋊C414yes—
235(C2×D3)⋊C4C22⋊C414yes—
236C4⋊Dic3—22no—
237Dic3⋊C4—21no—
238Dic3⋊C4—21no—
239Dic3⋊C4—22no—
240Dic3⋊C4—84no—
241C4×Dic3—22no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—82no—
245C25—31no—
246C25—31no—
247C22×D4—31no—
248C22×D4—31no—
249C22×D4—31no—
250C22×D4—64no—
251C22×D4—64no—
252C22×D4—64no—
253C22×D4C22×D311yescharacteristic
254C23×C4—31no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—31no—
261C23×C4C4×D311yescharacteristic
262C23×C4C22×D311yescharacteristic
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—62no—
266C4×D4—62no—
267C4×D4—64no—
268C4×D4—64no—
269C4×D4—64no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—64no—
273C4×D4—62no—
274C4×D4—62no—
275#117—62no—
276#117—62no—
277#117—64no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4C4×D311yescharacteristic
283C2×C22⋊C4—31no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—34no—
288C2×C22⋊C4—34no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—34no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—64no—
301C2×C22⋊C4—64no—
302C2×C22⋊C4C4×D311yescharacteristic
303C2×C22⋊C4C22×D314yes—
304C2×C22⋊C4C22×D311yescharacteristic
305C2×C42—22no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—62no—
310C2×C42C4×D311yescharacteristic
311#99—24no—
312#99—68no—
313#99—64no—
314#95—22no—
315#95—64no—
316#95—62no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3#11712yes—
324C23×C3C23×C411yescharacteristic
325C23×C3C22×D412yes—
326C23×C3C2×#4112yes—
327C23×C3C2×#4112yes—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—28no—
335C22×D3—28no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—44no—
341C22×D3—44no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—82no—
345C22×D3—82no—
346C22×D3—84no—
347C22×D3—84no—
348C22×D3—84no—
349C22×D3—84no—
350C22×D3C2×C22⋊C412yes—
351C22×D3C2×C22⋊C412yes—
352C22×D3#11712yes—
353C3×D4—28no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—44no—
365C2×C3×C4—44no—
366C2×C3×C4—44no—
367C2×C3×C4—81no—
368C2×C3×C4—81no—
369C2×C3×C4C2×C22⋊C412yes—
370C2×C3×C4C2×C22⋊C411yescharacteristic
371C2×C3×C4C2×C22⋊C411yescharacteristic
372C2×C3×C4#11712yes—
373#67—28no—
374#67—28no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—44no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—44no—
385C2×Dic3—44no—
386C2×Dic3—44no—
387C2×Dic3—82no—
388C2×Dic3C2×C22⋊C412yes—
389C2×Dic3C2×C22⋊C412yes—
390C2×Dic3#11712yes—
391D12—28no—
392C4×D3—28no—
393C4×D3—82no—
394C4×D3—82no—
395C4×D3—82no—
396C4×D3—82no—
397C24—31no—
398C24—31no—
399C24—31no—
400C24—34no—
401C24—62no—
402C24—62no—
403C24—62no—
404C24—62no—
405C24—62no—
406C24—241no—
407C24—241no—
408C24—241no—
409C24—241no—
410C24C2×C4×D311yescharacteristic
411C24C23×D311yescharacteristic
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—34no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—632no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4#23514yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—34no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—64no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4—241no—
448C22×C4—241no—
449C22×C4—241no—
450C22×C4—241no—
451C22×C4C2×C4×D311yescharacteristic
452C22×C4C2×C4×D311yescharacteristic
453C22×C4C2×C4×D311yescharacteristic
454C22×C4C2×C4×D311yescharacteristic
455C22×C4C2×C4×D311yescharacteristic
456C22×C4D3×D412yes—
457C22×C4D3×D412yes—
458C4⋊C4—22no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—244no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—32no—
469C22⋊C4—32no—
470C22⋊C4—34no—
471C22⋊C4—32no—
472C22⋊C4—32no—
473C22⋊C4—34no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—61no—
480C22⋊C4—61no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—244no—
484C22⋊C4—244no—
485C42—21no—
486C42—21no—
487C42—62no—
488C42—62no—
489C42—62no—
490C42—82no—
491C42—242no—
492C22×C3—22no—
493C22×C3—21no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3C22×C22⋊C411yescharacteristic
504C22×C3C2×#11711yescharacteristic
505C22×C3C2×#11711yescharacteristic
506C22×C3#52514yes—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—42no—
522C2×D3—88no—
523C2×D3—88no—
524C3×C4—22no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—22no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—84no—
531Dic3—22no—
532Dic3—22no—
533Dic3—22no—
534Dic3—41no—
535Dic3—41no—
536Dic3—84no—
537C23—21no—
538C23—21no—
539C23—32no—
540C23—32no—
541C23—32no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—68no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—68no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—124no—
559C23—124no—
560C23—121no—
561C23—121no—
562C23—242no—
563C23—242no—
564C23—244no—
565C23—244no—
566C23—244no—
567C23—244no—
568C23#87212yes—
569C23C22×C4×D311yescharacteristic
570C23C2×#23512yes—
571C23C2×D3×D412yes—
572C23D3×#4112yes—
573D4—28no—
574D4—68no—
575D4—68no—
576D4—68no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—68no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—124no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—124no—
606C2×C4—124no—
607C2×C4—124no—
608C2×C4—242no—
609C2×C4—242no—
610C2×C4—242no—
611C2×C4—242no—
612C2×C4—242no—
613C2×C4#87212yes—
614C2×C4D3×C22⋊C411yescharacteristic
615C2×C4D3×C22⋊C411yescharacteristic
616C2×C4D3×C22⋊C412yes—
617C2×C3—24no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—42no—
621C2×C3#228011yescharacteristic
622C2×C3#288311yescharacteristic
623C2×C3#288311yescharacteristic
624D3—42no—
625D3—42no—
626D3—42no—
627C22—22no—
628C22—21no—
629C22—22no—
630C22—21no—
631C22—22no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—62no—
639C22—62no—
640C22—62no—
641C22—62no—
642C22—62no—
643C22—62no—
644C22—62no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—121no—
652C22—121no—
653C22—248no—
654C22—248no—
655C22C2×#87211yescharacteristic
656C22C2×D3×C22⋊C411yescharacteristic
657C22D3×#11711yescharacteristic
658C22#554714yes—
659C4—22no—
660C4—21no—
661C4—21no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—62no—
666C4—62no—
667C4—62no—
668C4—84no—
669C4—121no—
670C4—121no—
671C4—244no—
672C3#2072411yescharacteristic, complemented, Sylow
673C2—24no—
674C2—21no—
675C2—21no—
676C2—42no—
677C2—122no—
678C2—122no—
679C2—122no—
680C2#7356111yescharacteristic
681C2#7868811yescharacteristic
682C2#7869411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×C4×#99C2
socleC22×C3C22×C22⋊C4
prime radicalC22×D3×D4C22
cosocleC24×C3C24
hypercenterC4×#99D3
hypocenterC3#20724
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #73567C2semidirect
3. C3×C4×#99C2semidirect
4. C4×#834C2semidirect
6. #70028C2semidirect
7. C4×#806C2semidirect
8. #68969C2semidirect
9. #68959C2semidirect
14. C2×C4×#67C4semidirect
16. C22×(C2×D3)⋊C4C4semidirect
20. C2×#893C4semidirect
21. C2×#892C4semidirect
22. C2×#67⋊C4C4semidirect
24. C2×D3×C22⋊C4C4semidirect
26. C2×C4×D12C4semidirect
27. C2×#872C4semidirect
29. C3×C4×C22⋊C4C22semidirect
31. C4×#217C22semidirect
34. #4942C22semidirect
37. C3×#342C22semidirect
38. #4527C22semidirect
39. #4526C22semidirect
48. C4×#99D3semidirect
146. C4×C22⋊C4C2×D3semidirect
149. #342C2×D3semidirect
672. C3#20724semidirect

Maximal subgroups

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

subgroupindex
C2×#55452
#735672
C3×C4×#992
C4×#8342
#700352
#700282
C4×#8062
#689692
#689592
#207243