Finite Group Databasebeta

← #10742732

Subgroups — #10742732

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×#946C212yesmaximal, complemented
2C2×#4665C212yesmaximal
3C2×#4668C212yesmaximal, complemented
4C3×#1446C211yesmaximal, characteristic, complemented
5#69428C212yesmaximal, complemented
6#69427C212yesmaximal, complemented
7#69426C211yesmaximal, characteristic
8#69420C211yesmaximal, characteristic, complemented
9#69410C212yesmaximal, complemented
10#15951—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C23×C4×D3C2212yes—
14C2×#946—28no—
15C2×#946—28no—
16C2×#946—28no—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×C4⋊Dic3C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35#4668—28no—
36#4665—28no—
37C22×C22⋊Q8—32no—
38C2×#386—32no—
39C2×#385—32no—
40#1446D311yescharacteristic, complemented
41#1444—31no—
42#1441—32no—
43#1439—32no—
44#1439—32no—
45#1438—31no—
46C24×D3C2312yes—
47C22×C3×Q8—24no—
48C22×C3×Q8—24no—
49C22×C3×Q8—41no—
50C22×C3×Q8—41no—
51C22×C3×Q8D414yescomplemented
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C22×D12—24no—
61C22×D12—24no—
62C22×D12—41no—
63C22×D12—41no—
64C22×D12Q814yescomplemented
65C22×C4×D3—24no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3—42no—
71C22×C4×D3—42no—
72#946—264no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
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—42no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4D418yes—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—41no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4D418yescomplemented
99C3×#95—24no—
100C3×#95—24no—
101C3×#95—41no—
102C3×#95—41no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—41no—
107C3⋊#95—42no—
108C23×Q8C2×D311yescharacteristic
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⋊C4—32no—
117C22×C4⋊C4—31no—
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—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95C2×D312yes—
131C2×#95C2×D312yes—
132C2×#95C2×D311yescharacteristic
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—
150C2×C3×Q8—216no—
151C2×C3×Q8—44no—
152C2×C3×Q8—44no—
153C2×C3×Q8—48no—
154C2×C3×Q8—44no—
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—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×Q812yes—
183C22×C3×C4#4112yes—
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×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3C2×D414yes—
203C22×Dic3C2×D414yes—
204C22×Dic3C2×D414yes—
205C2×D12—216no—
206C2×D12—44no—
207C2×D12—44no—
208C2×D12—48no—
209C2×D12—44no—
210C2×C4×D3—216no—
211C2×C4×D3—48no—
212C2×C4×D3—48no—
213C2×C4×D3—48no—
214C2×C4×D3—48no—
215C2×C4×D3—48no—
216C2×C4×D3—48no—
217(C2×D3)⋊C4—216no—
218(C2×D3)⋊C4—216no—
219(C2×D3)⋊C4—216no—
220(C2×D3)⋊C4—48no—
221(C2×D3)⋊C4—48no—
222C4⋊Dic3—216no—
223C4⋊Dic3—44no—
224Dic3⋊C4—216no—
225Dic3⋊C4—216no—
226C25—32no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—41no—
230C22×Q8—41no—
231C22×Q8D1214yescomplemented
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×D312yes—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D312yes—
252C22⋊Q8—664no—
253C2×C4⋊C4—38no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—121no—
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—24no—
280#95—24no—
281#95—41no—
282#95—41no—
283#95—64no—
284#95—64no—
285#95—62no—
286#95—121no—
287#95—122no—
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+412yes—
298C23×C32−1+411yescharacteristic
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
334C3×Q8—416no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4#12018yes—
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—24no—
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—42no—
384C2×Dic3—42no—
385D12—416no—
386C4×D3—432no—
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—216no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—48no—
407C2×Q8—44no—
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—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—616no—
440C22×C4—62no—
441C22×C4—62no—
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—128no—
459C22×C4—128no—
460C22×C4—128no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4—128no—
464C22×C4C2×D1212yes—
465C22×C4C2×D1214yes—
466C22×C4D3×D414yes—
467C22×C4D3×D414yes—
468C22×C4D3×Q812yes—
469C22×C4#23912yes—
470C4⋊C4—616no—
471C4⋊C4—616no—
472C4⋊C4—616no—
473C4⋊C4—124no—
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×C3D4212yes—
490C22×C3D4211yescharacteristic
491C22×C3#54612yes—
492C22×C3#54712yes—
493C22×C3#54812yes—
494C22×C3#54812yes—
495C22×C3D4×Q812yes—
496C22×C3D4×Q811yescharacteristic
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—28no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—82no—
522Dic3—44no—
523Dic3—44no—
524Dic3—44no—
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—
560C23C22×D1211yescharacteristic
561C23C2×D3×D412yes—
562C23C2×D3×D412yes—
563C23#100411yescharacteristic
564C23#100411yescharacteristic
565C23C2×D3×Q811yescharacteristic
566C23C2×#23911yescharacteristic
567C23#100712yes—
568C23D3×#4112yes—
569C23#100911yescharacteristic
570C23#101011yescharacteristic
571Q8—416no—
572D4—1216no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—1232no—
611C2×C4—122no—
612C2×C4—122no—
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#88218yes—
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—
651C22C2×#88212yes—
652C22#549311yescharacteristic
653C22Q8×D1211yescharacteristic
654C22#557612yes—
655C22#557711yescharacteristic
656C22#559011yescharacteristic
657C22#559012yes—
658C22#559011yescharacteristic
659C22D3×C22⋊Q812yes—
660C22#562712yes—
661C22#562812yes—
662C22#562912yes—
663C22#563411yescharacteristic
664C22#563812yes—
665C22#564511yescharacteristic
666C22#567612yes—
667C22#567712yes—
668C22#567911yescharacteristic
669C22#568411yescharacteristic
670C22#572612yes—
671C22#572712yes—
672C22#573011yescharacteristic
673C22#573511yescharacteristic
674C4—28no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—124no—
680C4—124no—
681C4—124no—
682C3#1595111yescharacteristic, complemented, Sylow
683C2—128no—
684C2#7385211yescharacteristic
685C2#7661812yes—
686C2#7661912yes—
687C2#7665411yescharacteristic
688C2#7665611yescharacteristic
689C2#7678711yescharacteristic
690C2#7686511yescharacteristic
691C2#7687512yes—
692C2#7725212yes—
693C2#7726211yescharacteristic
694C2#7726311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1446C2
prime radicalC23×D12C22
hypercenter#1446D3
hypocenterC3#15951
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×#946C2semidirect
3. C2×#4668C2semidirect
4. C3×#1446C2semidirect
5. #69428C2semidirect
6. #69427C2semidirect
8. #69420C2semidirect
9. #69410C2semidirect
40. #1446D3semidirect
51. C22×C3×Q8D4semidirect
64. C22×D12Q8semidirect
98. C2×Dic3⋊C4D4semidirect
231. C22×Q8D12semidirect
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×#9462
C2×#46652
C2×#46682
C3×#14462
#694282
#694272
#694262
#694202
#694102
#159513