Finite Group Databasebeta

← #10812349

Subgroups — #10812349

Order 768.

1006 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4804C211yesmaximal, characteristic
2#76613C211yesmaximal, characteristic
3#75979C211yesmaximal, characteristic, complemented
4C3×#1933C211yesmaximal, characteristic, complemented
5#70267C211yesmaximal, characteristic
6#70253C211yesmaximal, characteristic, complemented
7#70241C211yesmaximal, characteristic, complemented
8C8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
9C4×#821C211yesmaximal, characteristic, complemented
10#69666C211yesmaximal, characteristic, complemented
11#69626C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic, complemented
13#68868C211yesmaximal, characteristic, complemented
14#68813C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#22572—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4990—21no—
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4984C2211yescharacteristic
38#4977—21no—
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4804C418yescomplemented
47#4670—21no—
48#4665—21no—
49#4665—21no—
50#4664—21no—
51D3×#95—21no—
52#4648—21no—
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×#416—31no—
62#2346—31no—
63#2339—31no—
64#1933—31no—
65#1933D311yescharacteristic, complemented
66#1927—31no—
67#1927—31no—
68#1864—31no—
69#1864—31no—
70#1825—31no—
71#1807—31no—
72#1788—31no—
73C4×#102—31no—
74C4×#102—31no—
75C8×C22⋊C4—31no—
76C22×C3×Q8C2311yescharacteristic
77C2×#239—22no—
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—41no—
81C2×#239—41no—
82C2×#239C2×C414yes—
83C22×D12—21no—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×C4×D3—21no—
87C22×C4×D3—21no—
88C22×C4×D3C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D3×C8—41no—
114C2×D3×C8—41no—
115C2×D3×C8C2×C414yescomplemented
116#893—22no—
117#893—41no—
118#892—22no—
119#892C2×C414yescomplemented
120C4×D12—22no—
121C4×D12—41no—
122C3×#103—41no—
123C3×#103C2×C414yescomplemented
124C3×C4×C8—41no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—41no—
129#835C2×C414yescomplemented
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#821—22no—
140#821C2×C414yes—
141(C2×D3)⋊C8—22no—
142C8×Dic3—41no—
143#809—22no—
144#809C2×C414yes—
145#808—41no—
146#808—41no—
147C22×#41—31no—
148C23×C8—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×#117—31no—
152C2×#106—31no—
153C2×#106—31no—
154#416—38no—
155C2×#103—31no—
156C2×#103—31no—
157C2×#103C2×D311yescharacteristic
158C2×#103C2×D311yescharacteristic
159C2×#102—31no—
160C2×#102—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×C22⋊C8—31no—
164C2×C4×C8—31no—
165C2×C4×C8C2×D311yescharacteristic
166#391—31no—
167#391C2×D311yescharacteristic
168#389—31no—
169#389C2×D311yescharacteristic
170#388—61no—
171#388—61no—
172#386—61no—
173#386—61no—
174#382—61no—
175#381—61no—
176#380—31no—
177#380—31no—
178C4×C4⋊C4—31no—
179C4×C4⋊C4—31no—
180C4×C22⋊C4—31no—
181#376—61no—
182C2×#95—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191C23×D3—21no—
192C23×D3C2×D411yescharacteristic
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8C22×C412yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212#239—28no—
213#239—44no—
214#239—44no—
215#239—44no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—24no—
220C2×D12—42no—
221C2×D12—42no—
222C2×D12C22×C412yes—
223C2×D12#4112yes—
224C2×D12#4112yes—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—24no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×C4×D3C22×C412yes—
239C2×C4×D3#4112yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C22×C412yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—81no—
252C3×C4⋊C4C22×C412yes—
253C3×C42—21no—
254C3×C42—21no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—81no—
268C4⋊Dic3C22×C412yes—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—24no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C22×C412yes—
280D3×C8—44no—
281C2×#41—34no—
282C2×#41—62no—
283C2×#41—62no—
284C2×#41—62no—
285C2×#41—121no—
286C2×#41—121no—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C22×D4—61no—
290C23×C4—31no—
291C23×C4—61no—
292C23×C4—61no—
293C23×C4—61no—
294C22×C8—31no—
295C22×C8—34no—
296C22×C8—31no—
297C22×C8—121no—
298C22×C8—121no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—121no—
304C4×D4—121no—
305#117—34no—
306#117—62no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—31no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—121no—
328#106—121no—
329#103—34no—
330#103—41no—
331#103—121no—
332#103C4×D314yescomplemented
333#102—34no—
334#102—34no—
335#102—62no—
336#102—62no—
337C22⋊C8—62no—
338C4×C8—41no—
339C4×C8—121no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3C22×D411yescharacteristic
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—22no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3C4×D412yes—
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—41no—
370C3×Q8—41no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#13512yes—
389C2×C3×C4C23×C411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3C4×D412yes—
412C2×Dic3#13512yes—
413D12—22no—
414D12—22no—
415D12—22no—
416D12—48no—
417D12#11714yes—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—48no—
423C4×D3—84no—
424C4×D3—84no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C24—31no—
430C24—61no—
431#41—68no—
432#41—124no—
433#41—124no—
434#41—124no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D312yes—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—32no—
444C2×D4—64no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—122no—
449C2×D4—122no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—31no—
459C22×C4—64no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—32no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—124no—
490C2×C8C2×C4×D312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—81no—
503C4⋊C4—241no—
504C4⋊C4C2×C4×D312yes—
505C22⋊C4—62no—
506C22⋊C4—64no—
507C22⋊C4—62no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—241no—
511C22⋊C4—241no—
512C42—21no—
513C42—21no—
514C42—64no—
515C42—61no—
516C42—61no—
517C42—122no—
518C42—122no—
519C22×C3C2×C4×D411yescharacteristic
520C22×C3#54611yescharacteristic
521C22×C3#54611yescharacteristic
522C22×C3C2×#13511yescharacteristic
523C22×C3C2×#13511yescharacteristic
524C22×C3#57711yescharacteristic
525C22×C3#57711yescharacteristic
526C2×D3—22no—
527C2×D3—24no—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—41no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—41no—
538C2×D3—42no—
539C2×D3—42no—
540C3×C4—22no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—41no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4C2×#11711yescharacteristic
547C3×C4#51811yescharacteristic
548C3×C4#55111yescharacteristic
549C3×C4#55111yescharacteristic
550Dic3—22no—
551Dic3—22no—
552Dic3—42no—
553Dic3—41no—
554Dic3—81no—
555Dic3—81no—
556C23—32no—
557C23—62no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—62no—
564C23—62no—
565C23—61no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23C2×D3×D411yescharacteristic
571Q8—22no—
572Q8—22no—
573Q8—41no—
574Q8—41no—
575D4—34no—
576D4—62no—
577D4—62no—
578D4—62no—
579D4—128no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—32no—
588C2×C4—32no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—128no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—244no—
621C2×C4—244no—
622C2×C4#91912yes—
623C2×C4C22×C4×D311yescharacteristic
624C2×C4#100411yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4D3×#4111yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4#100911yescharacteristic
629C8—22no—
630C8—41no—
631C8—62no—
632C8—121no—
633C2×C3#234211yescharacteristic
634C2×C3C4×#13511yescharacteristic
635C2×C3#295211yescharacteristic
636C2×C3#329611yescharacteristic
637C2×C3#329611yescharacteristic
638C2×C3#330311yescharacteristic
639C2×C3#330311yescharacteristic
640D3—42no—
641D3—44no—
642C22—62no—
643C22—62no—
644C22—64no—
645C22—62no—
646C22—122no—
647C22—121no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—121no—
652C22—121no—
653C22—122no—
654C22—122no—
655C22—121no—
656C22C4×D3×D411yescharacteristic
657C22#561011yescharacteristic
658C22#567611yescharacteristic
659C22#576211yescharacteristic
660C22C2×#91911yescharacteristic
661C22D3×#13511yescharacteristic
662C22#577011yescharacteristic
663C4—22no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—62no—
668C4—62no—
669C4—81no—
670C4—81no—
671C4—122no—
672C4—121no—
673C4—241no—
674C4—241no—
675C4D3×#11711yescharacteristic
676C4#554511yescharacteristic
677C4#565911yescharacteristic
678C4#570511yescharacteristic
679C3#2257211yescharacteristic, complemented, Sylow
680C2—122no—
681C2—124no—
682C2#7583311yescharacteristic
683C2#7965011yescharacteristic
684C2C4×#91911yescharacteristic
685C2#8068211yescharacteristic
686C2#8077511yescharacteristic
687C2#8161511yescharacteristic
688C2#8165211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC22×#239C22
cosocleC22×C3×C4C24
hypercenter#1933D3
hypocenterC3#22572
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
3. #75979C2semidirect
4. C3×#1933C2semidirect
6. #70253C2semidirect
7. #70241C2semidirect
8. C8×(C2×D3)⋊C4C2semidirect
9. C4×#821C2semidirect
10. #69666C2semidirect
11. #69626C2semidirect
12. #68895C2semidirect
13. #68868C2semidirect
14. #68813C2semidirect
15. #68780C2semidirect
46. #4804C4semidirect
65. #1933D3semidirect
115. C2×D3×C8C2×C4semidirect
119. #892C2×C4semidirect
123. C3×#103C2×C4semidirect
129. #835C2×C4semidirect
332. #103C4×D3semidirect
679. C3#22572semidirect

Maximal subgroups

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

subgroupindex
C2×#48042
#766132
#759792
C3×#19332
#702672
#702532
#702412
C8×(C2×D3)⋊C42
C4×#8212
#696662
#696262
#688952
#688682
#688132
#687802
#225723