Finite Group Databasebeta

← #10798373

Subgroups — #10798373

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic
2#75969C211yesmaximal, characteristic, complemented
3#73751C211yesmaximal, characteristic, complemented
4C3×#1918C211yesmaximal, characteristic, complemented
5#70609C211yesmaximal, characteristic, complemented
6#69626C211yesmaximal, characteristic, complemented
7#69463C211yesmaximal, characteristic, complemented
8#68897C211yesmaximal, characteristic
9#68895C211yesmaximal, characteristic, complemented
10#68792C211yesmaximal, characteristic, complemented
11#68784C211yesmaximal, characteristic, complemented
12#68782C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic, complemented
14#68673C211yesmaximal, characteristic
15#68598C211yesmaximal, characteristic, complemented
16#21134—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#4990—21no—
30#4988C2211yescharacteristic
31#4987—21no—
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4939C2211yescharacteristic
43C4×(C2×D3)⋊C4—21no—
44C4×(C2×D3)⋊C4—21no—
45#4932C2211yescharacteristic
46#4872C418yescomplemented
47#4670—21no—
48#4666—21no—
49#4664—21no—
50D3×#95—21no—
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×D8—31no—
62#2319—31no—
63#2275—31no—
64#1926—31no—
65#1926—31no—
66#1918—31no—
67#1918D311yescharacteristic, complemented
68#1879—31no—
69#1879—31no—
70#1864—31no—
71#1864—31no—
72C4×#106—31no—
73#1762—31no—
74#1762—31no—
75#1746—31no—
76C22×D12C2311yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×D24—22no—
110C2×D24—22no—
111C2×D24C2×C414yes—
112#893—22no—
113#893—22no—
114#893—22no—
115#893—22no—
116#893C2×C414yes—
117#893C2×C414yescomplemented
118C4×D12—22no—
119C4×D12—22no—
120C3×#107C2×C414yescomplemented
121C3×#95—21no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—22no—
132#821—22no—
133#815—22no—
134C8×Dic3C2×C414yescomplemented
135#809—22no—
136#809—22no—
137#809C2×C414yes—
138#809C2×C414yescomplemented
139#808—22no—
140#808—22no—
141C22×D8—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—31no—
146C4×D8—38no—
147C2×#107C2×D311yescharacteristic
148C2×#106—31no—
149C2×#106—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#97—31no—
157C2×C4×C8—31no—
158#390—31no—
159#390—31no—
160#389—31no—
161#389C2×D311yescharacteristic
162#388—61no—
163#388—61no—
164#381—61no—
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4C2×D311yescharacteristic
170C4×C22⋊C4—61no—
171C4×C22⋊C4—61no—
172C4×C22⋊C4—61no—
173#376—31no—
174C2×#95—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C23×D3—21no—
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12C22×C412yes—
209C2×D12C22×C412yes—
210C2×D12#4112yes—
211C2×D12#4112yes—
212C2×C4×D3—24no—
213C2×C4×D3—24no—
214C2×C4×D3—24no—
215C2×C4×D3—24no—
216C2×C4×D3—81no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×C4×D3—81no—
222C2×C4×D3—81no—
223C2×C4×D3—81no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8C22×C412yes—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4C22×C412yes—
237C3×C4⋊C4C22×C412yes—
238C3×C4⋊C4#4112yes—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—81no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—81no—
247(C2×D3)⋊C4—81no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—81no—
254C4⋊Dic3#4112yes—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—81no—
258C4×Dic3—81no—
259C4×Dic3—81no—
260C4×Dic3C22×C412yes—
261C4×Dic3#4112yes—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C22×C412yes—
268D24—28no—
269C22×D4—31no—
270C22×D4—31no—
271C23×C4—61no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C2×D8—34no—
276C2×D8—62no—
277C2×D8—62no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C4×D4—34no—
283C4×D4—34no—
284C4×D4—62no—
285C4×D4—62no—
286C4×D4—62no—
287C4×D4—62no—
288C4×D4—62no—
289C4×D4—62no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—21no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107C4×D314yescomplemented
314#106—62no—
315#106—62no—
316#102—34no—
317#102—34no—
318#102—62no—
319#102—62no—
320#102—62no—
321#102—62no—
322#97—62no—
323C4×C8—34no—
324#95—21no—
325#95—21no—
326#95—31no—
327#95—31no—
328#95—31no—
329#95—31no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3C22×D411yescharacteristic
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4C23×C411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3C4×D412yes—
390C2×Dic3C4×D412yes—
391D12—22no—
392D12—22no—
393D12—22no—
394D12—22no—
395D12—22no—
396D12—41no—
397D12—41no—
398D12#11714yes—
399C4×D3—84no—
400C4×D3—84no—
401C4×D3—84no—
402C4×D3—84no—
403C3×C8—22no—
404C3×C8—41no—
405C3⋊C8—22no—
406C3⋊C8—41no—
407C24—61no—
408C24—61no—
409C2×D4—32no—
410C2×D4—32no—
411C2×D4—32no—
412C2×D4—32no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—241no—
436C22×C4—241no—
437C22×C4—241no—
438C22×C4—241no—
439C22×C4—241no—
440C22×C4—241no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446D8—68no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—32no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8C2×C4×D312yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—32no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—81no—
469C4⋊C4—241no—
470C4⋊C4C2×C4×D312yes—
471C4⋊C4C2×C4×D312yes—
472C4⋊C4#23912yes—
473C22⋊C4—64no—
474C22⋊C4—64no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C22⋊C4—241no—
478C22⋊C4—241no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—32no—
484C42—32no—
485C42—61no—
486C42—61no—
487C42—81no—
488C42—241no—
489C42—241no—
490C42—241no—
491C22×C3C2×C4×D411yescharacteristic
492C22×C3#52511yescharacteristic
493C22×C3#52511yescharacteristic
494C22×C3#57611yescharacteristic
495C22×C3#57611yescharacteristic
496C22×C3#57711yescharacteristic
497C22×C3#57711yescharacteristic
498C2×D3—24no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—41no—
513C3×C4—42no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—81no—
518C3×C4C2×#11711yescharacteristic
519C3×C4C4×#4111yescharacteristic
520C3×C4#52911yescharacteristic
521C3×C4#52911yescharacteristic
522Dic3—22no—
523Dic3—22no—
524Dic3—41no—
525Dic3—42no—
526Dic3—41no—
527Dic3—81no—
528C23—62no—
529C23—62no—
530C23—62no—
531C23—62no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23C2×D3×D411yescharacteristic
543D4—34no—
544D4—62no—
545D4—62no—
546D4—62no—
547D4—62no—
548D4—62no—
549D4—121no—
550D4—121no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—244no—
588C2×C4—244no—
589C2×C4—244no—
590C2×C4—244no—
591C2×C4C22×C4×D311yescharacteristic
592C2×C4C2×#23911yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C8—22no—
598C8—41no—
599C8—62no—
600C8—121no—
601C2×C3#229011yescharacteristic
602C2×C3#294811yescharacteristic
603C2×C3#295211yescharacteristic
604C2×C3#311811yescharacteristic
605C2×C3#311811yescharacteristic
606C2×C3#311911yescharacteristic
607C2×C3#311911yescharacteristic
608D3—44no—
609D3—82no—
610C22—64no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22C4×D3×D411yescharacteristic
625C22#562711yescharacteristic
626C22#565611yescharacteristic
627C22#575711yescharacteristic
628C22#576911yescharacteristic
629C22#577811yescharacteristic
630C22#577811yescharacteristic
631C4—42no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—62no—
637C4—62no—
638C4—81no—
639C4—121no—
640C4—121no—
641C4—122no—
642C4—241no—
643C4D3×#11711yescharacteristic
644C4C4×#23911yescharacteristic
645C4#569111yescharacteristic
646C4#570611yescharacteristic
647C3#2113411yescharacteristic, complemented, Sylow
648C2—124no—
649C2—242no—
650C2#7493711yescharacteristic
651C2#7958811yescharacteristic
652C2#7983211yescharacteristic
653C2#8095211yescharacteristic
654C2#8120811yescharacteristic
655C2#8127211yescharacteristic
656C2#8149611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1918C2
socleC23×C3C22×D4
prime radicalC22×D24C22
cosocleC22×C3×C4C24
hypercenter#1918D3
hypocenterC3#21134
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #75969C2semidirect
3. #73751C2semidirect
4. C3×#1918C2semidirect
5. #70609C2semidirect
6. #69626C2semidirect
7. #69463C2semidirect
9. #68895C2semidirect
10. #68792C2semidirect
11. #68784C2semidirect
12. #68782C2semidirect
13. #68782C2semidirect
15. #68598C2semidirect
46. #4872C4semidirect
67. #1918D3semidirect
117. #893C2×C4semidirect
120. C3×#107C2×C4semidirect
134. C8×Dic3C2×C4semidirect
138. #809C2×C4semidirect
313. #107C4×D3semidirect
647. C3#21134semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
#759692
#737512
C3×#19182
#706092
#696262
#694632
#688972
#688952
#687922
#687842
#687822
#686732
#685982
#211343