Finite Group Databasebeta

← #10796982

Subgroups — #10796982

Order 768.

1994 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2C22×#926C211yesmaximal, characteristic, complemented
3C2×#5240C212yesmaximal, complemented
4#84027C211yesmaximal, characteristic, complemented
5#71705C214yesmaximal, complemented
6#71670C214yesmaximal, complemented
7#71592C212yesmaximal, complemented
8#20983—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C24×C3×C4C2211yescharacteristic
11C3×#525—22no—
12C3×#525—22no—
13C3×#525—21no—
14C3×#525—21no—
15C3×#525—21no—
16C3×#525—21no—
17C2×C3×#123C2212yes—
18C2×C3×C22⋊Q8C412yescomplemented
19C2×C3×#121C412yescomplemented
20C2×C3×#120C2212yes—
21C2×C3×C4×D4C414yescomplemented
22C2×C3×#117C2211yescharacteristic
23C22×#217C2212yes—
24C2×#926—21no—
25C2×#926—21no—
26C2×#926—28no—
27C2×#926C2212yes—
28C2×#926C2212yes—
29C2×#926C2212yes—
30C22×C4⋊Dic3—21no—
31C22×C4×Dic3—21no—
32#5247C2212yes—
33#5240—28no—
34C2×#834C2218yes—
35#5212—22no—
36#5211C2212yes—
37C4×#217—22no—
38#4526C2218yes—
39C2×#525D311yescharacteristic
40C22×#117—31no—
41C2×#410—32no—
42#2280—31no—
43#1899—34no—
44#1898—34no—
45#1740—32no—
46C25×C3C2311yescharacteristic
47C2×C3×#41—24no—
48C2×C3×#41—21no—
49C2×C3×#41—21no—
50C2×C3×#41—21no—
51C2×C3×#41—21no—
52C2×C3×#41—24no—
53C22×C3×Q8C2×C411yescharacteristic, complemented
54C22×C3×D4C2×C414yes—
55C22×C3×D4C2×C411yescharacteristic, complemented
56C22×C3×D4C2312yes—
57C23×C3×C4—24no—
58C23×C3×C4—24no—
59C23×C3×C4C2×C412yes—
60C23×C3×C4C2×C412yes—
61C23×C3×C4C2×C412yescomplemented
62C23×C3×C4D412yes—
63C23×C3×C4D412yes—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3—22no—
71C3×#123—24no—
72C3×#123—22no—
73C3×#123—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#120—22no—
83C3×#120—24no—
84C3×#120—22no—
85C3×C4×D4—24no—
86C3×C4×D4—24no—
87C3×C4×D4—24no—
88C3×C4×D4—24no—
89C3×#117—21no—
90C3×#117—21no—
91C3×#117—21no—
92C3×#117—21no—
93C2×C3×C4⋊C4C2×C412yescomplemented
94C2×C3×C4⋊C4C2×C414yescomplemented
95C2×C3×C22⋊C4C2×C414yes—
96C2×C3×C22⋊C4C2×C414yescomplemented
97C2×C3×C22⋊C4C2312yes—
98C2×C3×C42C2×C412yescomplemented
99C2×#217—22no—
100C2×#217—28no—
101C2×#217—24no—
102C2×#217C2314yes—
103C2×#217C2314yes—
104#926—28no—
105#926—28no—
106#926—24no—
107#926—28no—
108#926—24no—
109#926—24no—
110#926D418yescomplemented
111C2×C4⋊Dic3—28no—
112C2×C4⋊Dic3—22no—
113C2×C4⋊Dic3—22no—
114C2×C4⋊Dic3—22no—
115C2×C4×Dic3—28no—
116C2×C4×Dic3—22no—
117C2×C4×Dic3—22no—
118C2×C4×Dic3—22no—
119#834—216no—
120C3⋊#95—24no—
121C22×#41C2×D311yescharacteristic
122C24×C4C2×D311yescharacteristic
123#525—21no—
124#525—21no—
125#525—21no—
126#525—22no—
127#525—22no—
128#525—21no—
129C2×#123C2×D312yes—
130C2×C22⋊Q8Dic312yescomplemented
131C2×#121Dic312yescomplemented
132C2×#120C2×D312yes—
133C2×C4×D4Dic314yescomplemented
134C2×#117—32no—
135C2×#117—32no—
136C2×#117—32no—
137C2×#117—68no—
138C2×#117—61no—
139C2×#117—61no—
140C2×#117C2×D311yescharacteristic
141C22×C4⋊C4—61no—
142C22×C22⋊C4—32no—
143C22×C42—61no—
144#410—68no—
145C2×#99—38no—
146#381—32no—
147#380—62no—
148#379—32no—
149C4×C22⋊C4—62no—
150#342—38no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—44no—
154C24×C3C22×C412yes—
155C24×C3C22×C412yes—
156C24×C3C2×D411yescharacteristic
157C24×C3C2×D412yes—
158C24×C3C2×D411yescharacteristic
159C24×C3C2411yescharacteristic
160C3×#41—42no—
161C3×#41—42no—
162C3×#41—44no—
163C3×#41—44no—
164C3×#41—42no—
165C3×#41—42no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—24no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—24no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—24no—
178C2×C3×D4—48no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—28no—
182C22×C3×C4—28no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—28no—
188C22×C3×C4—28no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—28no—
196C22×C3×C4C22⋊C412yes—
197C22×C3×C4C22⋊C412yes—
198C22×C3×C4C22⋊C412yes—
199C22×C3×C4C22⋊C412yes—
200C22×C3×C4C22⋊C412yes—
201C22×C3×C4C22⋊C412yes—
202C22×C3×C4C22⋊C412yes—
203C22×C3×C4C22⋊C412yes—
204C22×C3×C4C22×C412yes—
205C22×C3×C4C22×C412yes—
206C22×C3×C4C22×C414yes—
207C22×C3×C4C22×C411yescharacteristic
208C22×C3×C4C22×C411yescharacteristic
209C22×C3×C4C2×D414yes—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4114yes—
218C22×Dic3—28no—
219C22×Dic3—24no—
220C22×Dic3—24no—
221C22×Dic3—48no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C42—22no—
233C3×C42—22no—
234#217—28no—
235#217—24no—
236#217—24no—
237#217—24no—
238#217—48no—
239#217C2×D418yes—
240C4⋊Dic3—28no—
241C4⋊Dic3—28no—
242C4⋊Dic3—44no—
243C4×Dic3—28no—
244C4×Dic3—28no—
245C4×Dic3—44no—
246C25C22×D311yescharacteristic
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—21no—
250C2×#41—21no—
251C2×#41—21no—
252C2×#41—21no—
253C22×Q8C2×Dic311yescharacteristic, complemented
254C22×D4C2×Dic314yes—
255C22×D4C2×Dic311yescharacteristic, complemented
256C22×D4C22×D312yes—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—62no—
260C23×C4C2×Dic312yes—
261C23×C4C2×Dic312yes—
262C23×C4C2×Dic312yescomplemented
263C23×C4#6711yescharacteristic
264C23×C4#6711yescharacteristic
265C23×C4#6711yescharacteristic
266C23×C4#6712yes—
267C23×C4#6712yes—
268C23×C4#6711yescharacteristic
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D311yescharacteristic
271#123—24no—
272#123—22no—
273#123—22no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281#121—22no—
282#120—22no—
283#120—24no—
284#120—22no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—38no—
294#117—68no—
295#117—68no—
296#117—68no—
297#117—64no—
298#117—64no—
299#117—64no—
300C2×C4⋊C4—68no—
301C2×C4⋊C4—62no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4C2×Dic312yescomplemented
305C2×C4⋊C4C2×Dic314yescomplemented
306C2×C22⋊C4—34no—
307C2×C22⋊C4—34no—
308C2×C22⋊C4—68no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—64no—
311C2×C22⋊C4C2×Dic314yes—
312C2×C22⋊C4C2×Dic314yescomplemented
313C2×C22⋊C4C22×D312yes—
314C2×C42—68no—
315C2×C42—62no—
316C2×C42—62no—
317C2×C42—62no—
318C2×C42C2×Dic312yescomplemented
319#99—616no—
320#95—64no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—21no—
326C23×C3—24no—
327C23×C3—21no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—22no—
331C23×C3—24no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—48no—
337C23×C3—48no—
338C23×C3—44no—
339C23×C3C2×C22⋊C412yes—
340C23×C3C2×C22⋊C412yes—
341C23×C3C23×C411yescharacteristic
342C23×C3C22×D412yes—
343C23×C3C22×D411yescharacteristic
344C23×C3C22×D411yescharacteristic
345C23×C3C2×#4112yes—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×D4—48no—
349C3×D4—42no—
350C3×D4—44no—
351C3×D4—42no—
352C3×D4—48no—
353C3×D4—42no—
354C3×D4—42no—
355C2×C3×C4—24no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—24no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—44no—
373C2×C3×C4—44no—
374C2×C3×C4—44no—
375C2×C3×C4C2×C22⋊C412yes—
376C2×C3×C4C2×C22⋊C412yes—
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C4×D414yes—
382C2×C3×C4C4×D414yes—
383C2×C3×C4#12012yes—
384C2×C3×C4#12012yes—
385C2×C3×C4#12114yes—
386C2×C3×C4#12114yes—
387C2×C3×C4#12312yes—
388C2×C3×C4#12312yes—
389C2×Dic3—28no—
390C2×Dic3—416no—
391C2×Dic3—48no—
392C2×Dic3—48no—
393C2×Dic3—44no—
394C24—22no—
395C24—22no—
396C24—44no—
397C24C22×Dic312yes—
398C24C22×Dic312yes—
399C24C2×#6711yescharacteristic
400C24C2×#6712yes—
401C24C2×#6711yescharacteristic
402C24C23×D311yescharacteristic
403#41—42no—
404#41—42no—
405#41—42no—
406#41—42no—
407#41—44no—
408#41—44no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—48no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—68no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—128no—
443C22×C4#21712yes—
444C22×C4#21712yes—
445C22×C4#21712yes—
446C22×C4#21712yes—
447C22×C4#21712yes—
448C22×C4#21712yes—
449C22×C4#21712yes—
450C22×C4#21712yes—
451C22×C4D3×D414yes—
452C22×C4#23714yes—
453C22×C4C22×Dic312yes—
454C22×C4C22×Dic312yes—
455C22×C4C22×Dic314yes—
456C22×C4C22×Dic311yescharacteristic
457C22×C4C22×Dic311yescharacteristic
458C22×C4C2×#6712yes—
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—68no—
470C4⋊C4—68no—
471C4⋊C4—124no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—38no—
479C22⋊C4—68no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—128no—
484C42—22no—
485C42—22no—
486C42—68no—
487C42—68no—
488C42—124no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—48no—
502C22×C3—48no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—44no—
506C22×C3—42no—
507C22×C3—84no—
508C22×C3#41014yescomplemented
509C22×C3C22×C22⋊C411yescharacteristic
510C22×C3C2×C4×D412yes—
511C22×C3C2×#12011yescharacteristic
512C22×C3C2×#12112yes—
513C22×C3C2×#12311yescharacteristic
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—44no—
517C3×C4—42no—
518C3×C4#38612yes—
519C3×C4#38612yes—
520C3×C4#38612yes—
521C3×C4#38612yes—
522Dic3—48no—
523Dic3—84no—
524C23—24no—
525C23—21no—
526C23—21no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—21no—
531C23—21no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—48no—
540C23—48no—
541C23—44no—
542C23C2×#21712yes—
543C23C2×#21712yes—
544C23C2×D3×D412yes—
545C23C2×#23712yes—
546C23C23×Dic311yescharacteristic
547C23C22×#6711yescharacteristic
548C23C22×#6711yescharacteristic
549Q8—42no—
550Q8—42no—
551D4—48no—
552D4—48no—
553D4—42no—
554D4—42no—
555D4—44no—
556D4—42no—
557D4—42no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—68no—
579C2×C4—1216no—
580C2×C4—128no—
581C2×C4—128no—
582C2×C4—124no—
583C2×C4D4×Dic314yes—
584C2×C4D4×Dic314yes—
585C2×C4#93512yes—
586C2×C4#93512yes—
587C2×C4#93712yes—
588C2×C4#93712yes—
589C2×C4#93914yes—
590C2×C4#93914yes—
591C2×C4C2×#21712yes—
592C2×C4C2×#21712yes—
593C2×C4C2×#21711yescharacteristic
594C2×C4C2×#21711yescharacteristic
595C2×C4C2×#21711yescharacteristic
596C2×C4C2×#21711yescharacteristic
597C2×C3—21no—
598C2×C3—22no—
599C2×C3—21no—
600C2×C3—24no—
601C2×C3—22no—
602C2×C3—42no—
603C2×C3—82no—
604C2×C3C2×#38611yescharacteristic
605C2×C3C2×#41012yes—
606C22—24no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—48no—
619C22—48no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—84no—
625C22#524014yescomplemented
626C22C2×D4×Dic312yes—
627C22C2×#93511yescharacteristic
628C22C2×#93711yescharacteristic
629C22C2×#93912yes—
630C22C22×#21711yescharacteristic
631C4—22no—
632C4—22no—
633C4—42no—
634C4—44no—
635C4—128no—
636C4—244no—
637C4#522012yes—
638C4#522012yes—
639C4#522012yes—
640C4#522012yes—
641C3#2098311yescharacteristic, complemented, Sylow
642C2—21no—
643C2—22no—
644C2—24no—
645C2—21no—
646C2—22no—
647C2—42no—
648C2—82no—
649C2C2×#522011yescharacteristic
650C2C2×#524012yes—
subgroupG/H
centerC2×C4C2×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
Fitting, prime radicalC2×C3×#525C2
socleC22×C3C22×C22⋊C4
cosocleC24×C3C24
hypercenterC2×#525D3
hypocenterC3#20983
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. C22×#926C2semidirect
3. C2×#5240C2semidirect
4. #84027C2semidirect
5. #71705C2semidirect
6. #71670C2semidirect
7. #71592C2semidirect
18. C2×C3×C22⋊Q8C4semidirect
19. C2×C3×#121C4semidirect
21. C2×C3×C4×D4C4semidirect
53. C22×C3×Q8C2×C4semidirect
55. C22×C3×D4C2×C4semidirect
61. C23×C3×C4C2×C4semidirect
93. C2×C3×C4⋊C4C2×C4semidirect
94. C2×C3×C4⋊C4C2×C4semidirect
96. C2×C3×C22⋊C4C2×C4semidirect
98. C2×C3×C42C2×C4semidirect
110. #926D4semidirect
130. C2×C22⋊Q8Dic3semidirect
131. C2×#121Dic3semidirect
133. C2×C4×D4Dic3semidirect
253. C22×Q8C2×Dic3semidirect
255. C22×D4C2×Dic3semidirect
262. C23×C4C2×Dic3semidirect
304. C2×C4⋊C4C2×Dic3semidirect
305. C2×C4⋊C4C2×Dic3semidirect
312. C2×C22⋊C4C2×Dic3semidirect
318. C2×C42C2×Dic3semidirect
508. C22×C3#410semidirect
625. C22#5240semidirect
641. C3#20983semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5252
C22×#9262
C2×#52402
#840272
#717052
#716702
#715922
#209833