Finite Group Databasebeta

← #10751711

Subgroups — #10751711

Order 768.

1706 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C2×#4990C211yesmaximal, characteristic
3C2×#4673C212yesmaximal
4C3×#1441C211yesmaximal, characteristic, complemented
5#71570C211yesmaximal, characteristic
6#71565C211yesmaximal, characteristic, complemented
7#69428C212yesmaximal
8#69427C211yesmaximal, characteristic
9#69424C212yesmaximal, complemented
10#69414C212yesmaximal, complemented
11#69401C211yesmaximal, characteristic, complemented
12#16602—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C2×#888—28no—
25C2×#888—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4990—24no—
39#4673—28no—
40C22×C4⋊Q8—31no—
41C2×#393—32no—
42C2×#388—31no—
43#1446—32no—
44#1442—32no—
45#1441—31no—
46#1441D311yescharacteristic, complemented
47#1439—31no—
48#1439—32no—
49#1439—31no—
50#1434—31no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—42no—
65C22×Dic6—42no—
66C22×Dic6—42no—
67C22×Dic6—42no—
68C22×Dic6D418yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—42no—
84C2×(C2×D3)⋊C4—42no—
85C2×(C2×D3)⋊C4—42no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4Q814yescomplemented
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3D414yes—
95#888—232no—
96C3×#95—24no—
97C3×#95—22no—
98C3×#95—42no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C23×Q8—32no—
109C2×C4⋊Q8—64no—
110C2×C4⋊Q8—68no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C22⋊C4—31no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C42—31no—
121#393—68no—
122#388—64no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95C2×D312yes—
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D312yes—
135C24×C3C2411yescharacteristic
136C23×D3—21no—
137C23×D3—22no—
138C23×D3—21no—
139C23×D3—21no—
140C23×D3—21no—
141C23×D3—21no—
142C23×D3—22no—
143C23×D3—21no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3C2×Q812yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×Q812yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3C2×D414yes—
202C22×Dic3C2×D414yes—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×Q812yes—
206C22×Dic3#4112yes—
207C2×Dic6—216no—
208C2×Dic6—48no—
209C2×Dic6—48no—
210C2×Dic6—48no—
211C2×Dic6—48no—
212C2×Dic6—48no—
213C2×Dic6—48no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—44no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—48no—
218(C2×D3)⋊C4—48no—
219(C2×D3)⋊C4—44no—
220(C2×D3)⋊C4—44no—
221C4⋊Dic3—28no—
222Dic3⋊C4—216no—
223C4×Dic3—28no—
224C25—31no—
225C22×Q8—38no—
226C22×Q8—64no—
227C22×Q8—64no—
228C22×Q8—122no—
229C22×Q8—122no—
230C22×Q8—122no—
231C22×Q8—122no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D312yes—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C4⋊Q8—632no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—122no—
264C2×C22⋊C4—122no—
265C2×C22⋊C4—122no—
266C2×C22⋊C4—122no—
267C2×C42—34no—
268C2×C42—64no—
269#95—24no—
270#95—22no—
271#95—42no—
272#95—64no—
273#95—62no—
274#95—62no—
275#95—64no—
276#95—64no—
277#95—62no—
278#95—122no—
279#95—122no—
280#95—122no—
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D412yes—
283C23×C3C22×Q811yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3H(5,𝔽2)11yescharacteristic
288C23×C3H(5,𝔽2)12yes—
289C23×C3H(5,𝔽2)11yescharacteristic
290C23×C32−1+412yes—
291C23×C32−1+412yes—
292C22×D3—22no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—42no—
314C22×D3—42no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4#12414yes—
346C2×Dic3—22no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3#12014yes—
374C2×Dic3C22⋊Q814yes—
375Dic6—432no—
376C24—32no—
377C24—61no—
378C24—62no—
379C24—61no—
380C24—61no—
381C24—61no—
382C24—62no—
383C24—61no—
384C24—61no—
385C24—62no—
386C24—62no—
387C24C23×D311yescharacteristic
388C2×Q8—616no—
389C2×Q8—128no—
390C2×Q8—128no—
391C2×Q8—128no—
392C2×Q8—128no—
393C2×Q8—128no—
394C2×Q8—128no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—34no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—34no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4#23512yes—
451C22×C4D3×D414yes—
452C22×C4D3×D412yes—
453C22×C4#23712yes—
454C22×C4D3×Q812yes—
455C4⋊C4—28no—
456C4⋊C4—44no—
457C4⋊C4—616no—
458C4⋊C4—68no—
459C22⋊C4—68no—
460C22⋊C4—128no—
461C22⋊C4—124no—
462C22⋊C4—128no—
463C22⋊C4—124no—
464C42—68no—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×C22⋊Q811yescharacteristic
467C22×C3C2×#12411yescharacteristic
468C22×C3#53412yes—
469C22×C3#53412yes—
470C22×C3#53511yescharacteristic
471C22×C3#53511yescharacteristic
472C22×C3#53611yescharacteristic
473C22×C3#53611yescharacteristic
474C22×C3#54212yes—
475C22×C3#54212yes—
476C22×C3#54212yes—
477C22×C3#54412yes—
478C22×C3#54812yes—
479C22×C3#54812yes—
480C22×C3#54812yes—
481C22×C3D4×Q812yes—
482C22×C3#55411yescharacteristic
483C22×C3#55411yescharacteristic
484C22×C3#55511yescharacteristic
485C22×C3#55511yescharacteristic
486C22×C3#56312yes—
487C22×C3#56312yes—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—24no—
500C3×C4—42no—
501C3×C4—44no—
502C3×C4—81no—
503C3×C4—82no—
504Dic3—24no—
505Dic3—24no—
506Dic3—44no—
507Dic3—81no—
508Dic3—82no—
509C23—62no—
510C23—64no—
511C23—64no—
512C23—62no—
513C23—62no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—121no—
518C23—122no—
519C23—121no—
520C23—122no—
521C23—122no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—122no—
526C23—122no—
527C23—121no—
528C23—122no—
529C23—122no—
530C23—121no—
531C23—121no—
532C23C2×#23511yescharacteristic
533C23C2×D3×D412yes—
534C23C2×D3×D411yescharacteristic
535C23C2×#23711yescharacteristic
536C23#100411yescharacteristic
537C23C2×D3×Q811yescharacteristic
538C23#100712yes—
539C23D3×#4111yescharacteristic
540C23#100911yescharacteristic
541C23#100912yes—
542C23#101012yes—
543Q8—1232no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—34no—
555C2×C4—34no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—62no—
582C2×C4—64no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4#88514yes—
604C2×C3#240911yescharacteristic
605C2×C3#243611yescharacteristic
606C2×C3#256311yescharacteristic
607C2×C3#261311yescharacteristic
608C2×C3#262811yescharacteristic
609C2×C3#264011yescharacteristic
610C2×C3#272912yes—
611C2×C3#272912yes—
612C2×C3#273112yes—
613C2×C3#273112yes—
614C2×C3#282111yescharacteristic
615D3—82no—
616C22—124no—
617C22—122no—
618C22—122no—
619C22—124no—
620C22—122no—
621C22—242no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22—242no—
626C22—241no—
627C22C2×#88511yescharacteristic
628C22#551111yescharacteristic
629C22#553211yescharacteristic
630C22#555212yes—
631C22#558212yes—
632C22D3×#12011yescharacteristic
633C22#559812yes—
634C22#560512yes—
635C22#560711yescharacteristic
636C22#561611yescharacteristic
637C22D3×C22⋊Q811yescharacteristic
638C22#563312yes—
639C22#563511yescharacteristic
640C22#564312yes—
641C22#564612yes—
642C22#564811yescharacteristic
643C22#565312yes—
644C22#567411yescharacteristic
645C22#568912yes—
646C22#569912yes—
647C22#571012yes—
648C22#572411yescharacteristic
649C22#572712yes—
650C4—24no—
651C4—44no—
652C4—42no—
653C4—64no—
654C4—64no—
655C4—82no—
656C4—81no—
657C4—124no—
658C4—241no—
659C4—242no—
660C3#1660211yescharacteristic, complemented, Sylow
661C2—242no—
662C2#7394211yescharacteristic
663C2#7470111yescharacteristic
664C2#7517812yes—
665C2#7613312yes—
666C2#7664311yescharacteristic
667C2#7712211yescharacteristic
668C2#7717211yescharacteristic
669C2#7737112yes—
670C2#7792511yescharacteristic
671C2#7798512yes—
672C2#7820911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1441C2
prime radicalC24×D3C23
hypercenter#1441D3
hypocenterC3#16602
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
4. C3×#1441C2semidirect
6. #71565C2semidirect
9. #69424C2semidirect
10. #69414C2semidirect
11. #69401C2semidirect
46. #1441D3semidirect
87. C2×(C2×D3)⋊C4Q8semidirect
660. C3#16602semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49902
C2×#46732
C3×#14412
#715702
#715652
#694282
#694272
#694242
#694142
#694012
#166023