Finite Group Databasebeta

← #10742733

Subgroups — #10742733

Order 768.

2462 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#896C212yesmaximal, complemented
2C2×#4989C212yesmaximal, complemented
3C2×#4665C212yesmaximal
4C3×#1444C211yesmaximal, characteristic, complemented
5#71570C212yesmaximal, complemented
6#69428C212yesmaximal, complemented
7#69426C211yesmaximal, characteristic
8#69424C211yesmaximal, characteristic, complemented
9#69409C212yesmaximal, complemented
10#15951—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C23×C4×D3C2212yes—
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C2×#896—28no—
24C2×#896—28no—
25C2×#896—28no—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35#4989—28no—
36#4665—28no—
37C22×C22⋊Q8—32no—
38C2×#386—32no—
39C2×#385—32no—
40#1446—31no—
41#1444D311yescharacteristic, complemented
42#1441—32no—
43#1439—32no—
44#1439—32no—
45#1438—31no—
46C24×D3C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C22×D12—24no—
56C22×D12—24no—
57C22×D12—41no—
58C22×D12—41no—
59C22×D12Q814yescomplemented
60C22×C4×D3—24no—
61C22×C4×D3—24no—
62C22×C4×D3—24no—
63C22×C4×D3—24no—
64C22×C4×D3—24no—
65C22×C4×D3—42no—
66C22×C4×D3—42no—
67C22×Dic6—24no—
68C22×Dic6—24no—
69C22×Dic6—41no—
70C22×Dic6—41no—
71C22×Dic6D414yescomplemented
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—41no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—42no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4D418yes—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4D418yescomplemented
98#896—264no—
99C3×#95—24no—
100C3×#95—22no—
101C3×#95—41no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—41no—
106C3⋊#95—42no—
107C3⋊#95—41no—
108C23×Q8—31no—
109C23×D4—31no—
110C24×C4—32no—
111C2×C22⋊Q8—68no—
112C2×C22⋊Q8—68no—
113C2×C22⋊Q8—68no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4C2×D312yes—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4—32no—
122C22×C22⋊C4—32no—
123#386—68no—
124#385—68no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D312yes—
133C2×#95C2×D311yescharacteristic
134C24×C3C2411yescharacteristic
135C23×D3—22no—
136C23×D3—22no—
137C23×D3—22no—
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3C2×D414yes—
148C23×D3C2×Q814yes—
149C23×D3#4114yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D414yes—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4#4112yes—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3C2×D414yes—
197C22×Dic3C2×D414yes—
198C22×Dic3C2×D414yes—
199C22×Dic3C2×D414yes—
200C2×D12—216no—
201C2×D12—44no—
202C2×D12—44no—
203C2×D12—48no—
204C2×D12—44no—
205C2×C4×D3—216no—
206C2×C4×D3—48no—
207C2×C4×D3—48no—
208C2×C4×D3—48no—
209C2×C4×D3—48no—
210C2×C4×D3—48no—
211C2×C4×D3—48no—
212C2×Dic6—216no—
213C2×Dic6—44no—
214C2×Dic6—44no—
215C2×Dic6—44no—
216C2×Dic6—48no—
217C3×C4⋊C4—216no—
218C3×C4⋊C4—44no—
219(C2×D3)⋊C4—216no—
220(C2×D3)⋊C4—216no—
221(C2×D3)⋊C4—216no—
222(C2×D3)⋊C4—48no—
223(C2×D3)⋊C4—48no—
224Dic3⋊C4—216no—
225Dic3⋊C4—216no—
226C25—32no—
227C22×Q8—34no—
228C22×Q8—64no—
229C22×Q8—64no—
230C22×Q8—121no—
231C22×Q8—121no—
232C22×D4—34no—
233C22×D4—64no—
234C22×D4—64no—
235C22×D4—121no—
236C22×D4—121no—
237C23×C4—32no—
238C23×C4—32no—
239C23×C4—32no—
240C23×C4—64no—
241C23×C4—64no—
242C23×C4—64no—
243C23×C4—64no—
244C23×C4—64no—
245C23×C4—122no—
246C23×C4—122no—
247C23×C4C22×D311yescharacteristic
248C23×C4C22×D312yes—
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C22⋊Q8—664no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—38no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—64no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4—122no—
279#95—22no—
280#95—24no—
281#95—41no—
282#95—64no—
283#95—64no—
284#95—64no—
285#95—122no—
286#95—121no—
287#95—121no—
288C23×C3C22×D411yescharacteristic
289C23×C3C22×D412yes—
290C23×C3C22×D412yes—
291C23×C3C22×Q811yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4112yes—
294C23×C3H(5,𝔽2)11yescharacteristic
295C23×C3H(5,𝔽2)11yescharacteristic
296C23×C3H(5,𝔽2)11yescharacteristic
297C23×C32−1+411yescharacteristic
298C23×C32−1+412yes—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—24no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3C22⋊Q818yescomplemented
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3#12018yes—
384D12—416no—
385C4×D3—432no—
386Dic6—416no—
387C24—34no—
388C24—34no—
389C24—34no—
390C24—62no—
391C24—62no—
392C24—62no—
393C24—62no—
394C24—62no—
395C24—62no—
396C24—62no—
397C24—62no—
398C24—62no—
399C24—62no—
400C24—62no—
401C24—62no—
402C24C23×D311yescharacteristic
403C2×Q8—616no—
404C2×Q8—124no—
405C2×Q8—124no—
406C2×Q8—124no—
407C2×Q8—128no—
408C2×D4—616no—
409C2×D4—124no—
410C2×D4—128no—
411C2×D4—124no—
412C2×D4—124no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—34no—
440C22×C4—34no—
441C22×C4—616no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—128no—
460C22×C4—128no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4—128no—
464C22×C4—128no—
465C22×C4#23512yes—
466C22×C4D3×D414yes—
467C22×C4D3×D412yes—
468C22×C4D3×D414yes—
469C22×C4D3×Q812yes—
470C4⋊C4—216no—
471C4⋊C4—44no—
472C4⋊C4—616no—
473C4⋊C4—616no—
474C22⋊C4—616no—
475C22⋊C4—616no—
476C22⋊C4—616no—
477C22⋊C4—128no—
478C22⋊C4—128no—
479C22×C3C2×#12012yes—
480C22×C3C2×C22⋊Q812yes—
481C22×C3#52512yes—
482C22×C3#53411yescharacteristic
483C22×C3#53612yes—
484C22×C3#53712yes—
485C22×C3#53912yes—
486C22×C3#53911yescharacteristic
487C22×C3#54311yescharacteristic
488C22×C3D4211yescharacteristic
489C22×C3D4211yescharacteristic
490C22×C3D4212yes—
491C22×C3#54612yes—
492C22×C3#54712yes—
493C22×C3#54812yes—
494C22×C3#54812yes—
495C22×C3D4×Q811yescharacteristic
496C22×C3D4×Q812yes—
497C22×C3#55011yescharacteristic
498C22×C3#55111yescharacteristic
499C22×C3#55411yescharacteristic
500C22×C3#56211yescharacteristic
501C22×C3#56311yescharacteristic
502C2×D3—28no—
503C2×D3—28no—
504C2×D3—28no—
505C2×D3—44no—
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—44no—
512C2×D3—44no—
513C2×D3—44no—
514C2×D3—44no—
515C2×D3—44no—
516C2×D3—44no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—44no—
520C3×C4—42no—
521C3×C4—42no—
522Dic3—28no—
523Dic3—44no—
524Dic3—82no—
525C23—38no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—64no—
530C23—64no—
531C23—64no—
532C23—64no—
533C23—64no—
534C23—64no—
535C23—64no—
536C23—64no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23—64no—
541C23—64no—
542C23—64no—
543C23—64no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23—122no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—122no—
556C23—122no—
557C23—122no—
558C23—122no—
559C23—122no—
560C23C2×#23511yescharacteristic
561C23C2×D3×D411yescharacteristic
562C23C2×D3×D412yes—
563C23C2×D3×D412yes—
564C23#100411yescharacteristic
565C23C2×D3×Q811yescharacteristic
566C23#100711yescharacteristic
567C23#100712yes—
568C23D3×#4112yes—
569C23#100911yescharacteristic
570C23#100911yescharacteristic
571Q8—1216no—
572D4—1216no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—38no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—1232no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C3#240311yescharacteristic
625C2×C3#240712yes—
626C2×C3#240912yes—
627C2×C3#243711yescharacteristic
628C2×C3#244912yes—
629C2×C3#246311yescharacteristic
630C2×C3#268011yescharacteristic
631C2×C3#268112yes—
632C2×C3#268311yescharacteristic
633C2×C3#272811yescharacteristic
634C2×C3#273211yescharacteristic
635D3—48no—
636C22—68no—
637C22—68no—
638C22—68no—
639C22—124no—
640C22—124no—
641C22—124no—
642C22—124no—
643C22—124no—
644C22—124no—
645C22—124no—
646C22—124no—
647C22—124no—
648C22—124no—
649C22—124no—
650C22—124no—
651C22#550712yes—
652C22#551511yescharacteristic
653C22#554712yes—
654C22#555311yescharacteristic
655C22#557911yescharacteristic
656C22D3×#12012yes—
657C22#559011yescharacteristic
658C22#560511yescharacteristic
659C22#560912yes—
660C22D3×C22⋊Q812yes—
661C22#562912yes—
662C22#563312yes—
663C22#563312yes—
664C22#563411yescharacteristic
665C22#565412yes—
666C22#565911yescharacteristic
667C22#569212yes—
668C22#570011yescharacteristic
669C22#571711yescharacteristic
670C22#572412yes—
671C22#572711yescharacteristic
672C22#572712yes—
673C22#573511yescharacteristic
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—68no—
680C4—124no—
681C4—242no—
682C3#1595111yescharacteristic, complemented, Sylow
683C2—128no—
684C2#7437111yescharacteristic
685C2#7440412yes—
686C2#7461311yescharacteristic
687C2#7613411yescharacteristic
688C2#7622512yes—
689C2#7675311yescharacteristic
690C2#7717212yes—
691C2#7726311yescharacteristic
692C2#7787011yescharacteristic
693C2#7797012yes—
694C2#7810211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1444C2
prime radicalC23×D12C22
hypercenter#1444D3
hypocenterC3#15951
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
1. C22×#896C2semidirect
2. C2×#4989C2semidirect
4. C3×#1444C2semidirect
5. #71570C2semidirect
6. #69428C2semidirect
8. #69424C2semidirect
9. #69409C2semidirect
41. #1444D3semidirect
59. C22×D12Q8semidirect
71. C22×Dic6D4semidirect
97. C2×Dic3⋊C4D4semidirect
333. C22×D3C22⋊Q8semidirect
682. C3#15951semidirect

Maximal subgroups

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

subgroupindex
C22×#8962
C2×#49892
C2×#46652
C3×#14442
#715702
#694282
#694262
#694242
#694092
#159513