Finite Group Databasebeta

← #10732193

Subgroups — #10732193

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic, complemented
2C2×#4979C211yesmaximal, characteristic, complemented
3C2×#4990C211yesmaximal, characteristic
4C2×#4673C211yesmaximal, characteristic
5C2×#4669C212yesmaximal
6C3×#1441C211yesmaximal, characteristic, complemented
7#71570C211yesmaximal, characteristic
8#69428C212yesmaximal
9#69409C212yesmaximal, complemented
10#69403C211yesmaximal, characteristic, complemented
11#69401C212yesmaximal, complemented
12#15176—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#887—28no—
26C2×#887—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38#4990—24no—
39#4979—24no—
40#4673—24no—
41#4669—28no—
42C22×C4×Q8—31no—
43C2×#395—31no—
44C2×#393—31no—
45C2×#388—32no—
46C2×#388—31no—
47#1441—32no—
48#1441D311yescharacteristic, complemented
49#1439—32no—
50#1439—31no—
51#1434—32no—
52#1434—31no—
53C24×D3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6D414yes—
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—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3D418yes—
98C2×C4×Dic3D414yes—
99#887—232no—
100C3×#95—24no—
101C3×#95—22no—
102C3×#95—24no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C23×Q8—31no—
112C2×C4×Q8—64no—
113C2×C4×Q8—68no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C22⋊C4—32no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4—31no—
121C22×C22⋊C4—32no—
122C22×C22⋊C4—31no—
123C22×C42—32no—
124C22×C42—31no—
125#395—64no—
126#393—64no—
127#388—68no—
128#388—64no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—31no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C2×#95C2×D312yes—
140C24×C3C2411yescharacteristic
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—21no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—21no—
149C23×D3—22no—
150C23×D3—21no—
151C23×D3#4112yes—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
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—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D414yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4#4112yes—
182C22×C3×C4#4112yes—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D414yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×D414yes—
210C22×Dic3#4114yes—
211C22×Dic3#4114yes—
212C2×Dic6—28no—
213C2×Dic6—216no—
214C2×Dic6—28no—
215C2×Dic6—28no—
216C2×Dic6—216no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—44no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—216no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—44no—
223(C2×D3)⋊C4—48no—
224Dic3⋊C4—216no—
225Dic3⋊C4—48no—
226C4×Dic3—216no—
227C4×Dic3—28no—
228C25—31no—
229C22×Q8—34no—
230C22×Q8—64no—
231C22×Q8—64no—
232C22×Q8—62no—
233C22×Q8—62no—
234C23×C4—32no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D312yes—
241C23×C4C22×D312yes—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C4×Q8—632no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—38no—
271C2×C42—34no—
272C2×C42—64no—
273C2×C42—64no—
274C2×C42—64no—
275#95—24no—
276#95—22no—
277#95—24no—
278#95—64no—
279#95—64no—
280#95—62no—
281#95—64no—
282#95—62no—
283#95—64no—
284#95—64no—
285#95—62no—
286C23×C3C22×D411yescharacteristic
287C23×C3C22×D412yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4112yes—
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4112yes—
294C23×C32−1+411yescharacteristic
295C23×C32−1+412yes—
296C23×C32−1+411yescharacteristic
297C22×D3—24no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—24no—
301C22×D3—22no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—41no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—41no—
313C22×D3—42no—
314C22×D3—41no—
315C22×D3—42no—
316C22×D3—41no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4#12114yes—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3#12014yes—
379C2×Dic3#12118yes—
380Dic6—232no—
381C24—32no—
382C24—61no—
383C24—61no—
384C24—62no—
385C24—61no—
386C24—61no—
387C24—61no—
388C24—62no—
389C24—62no—
390C24—62no—
391C24—61no—
392C24C23×D311yescharacteristic
393C2×Q8—616no—
394C2×Q8—68no—
395C2×Q8—68no—
396C2×Q8—616no—
397C2×Q8—68no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—34no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4#23512yes—
454C22×C4D3×D414yes—
455C22×C4D3×D412yes—
456C22×C4D3×D412yes—
457C22×C4#23712yes—
458C4⋊C4—28no—
459C4⋊C4—44no—
460C4⋊C4—616no—
461C4⋊C4—128no—
462C22⋊C4—616no—
463C22⋊C4—68no—
464C22⋊C4—124no—
465C22⋊C4—128no—
466C22⋊C4—124no—
467C42—68no—
468C42—616no—
469C22×C3C2×#12011yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3C2×#12112yes—
472C22×C3#52511yescharacteristic
473C22×C3#52512yes—
474C22×C3#52512yes—
475C22×C3#52511yescharacteristic
476C22×C3#53211yescharacteristic
477C22×C3#53212yes—
478C22×C3#53611yescharacteristic
479C22×C3#53612yes—
480C22×C3#53612yes—
481C22×C3#53611yescharacteristic
482C22×C3#54811yescharacteristic
483C22×C3#54812yes—
484C22×C3#54811yescharacteristic
485C22×C3#55212yes—
486C22×C3#55212yes—
487C22×C3#55211yescharacteristic
488C22×C3#55211yescharacteristic
489C22×C3#55612yes—
490C22×C3#55612yes—
491C22×C3#55612yes—
492C2×D3—44no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—44no—
497C2×D3—81no—
498C2×D3—82no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—82no—
502C2×D3—81no—
503C3×C4—24no—
504C3×C4—44no—
505C3×C4—42no—
506C3×C4—81no—
507C3×C4—82no—
508Dic3—28no—
509Dic3—24no—
510Dic3—42no—
511Dic3—44no—
512Dic3—81no—
513C23—62no—
514C23—64no—
515C23—64no—
516C23—62no—
517C23—62no—
518C23—122no—
519C23—122no—
520C23—121no—
521C23—121no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—122no—
529C23—122no—
530C23—121no—
531C23—122no—
532C23—121no—
533C23—122no—
534C23—122no—
535C23—121no—
536C23C2×#23511yescharacteristic
537C23C2×D3×D411yescharacteristic
538C23C2×D3×D411yescharacteristic
539C23C2×D3×D412yes—
540C23C2×#23711yescharacteristic
541C23#100712yes—
542C23#100711yescharacteristic
543C23D3×#4111yescharacteristic
544C23D3×#4112yes—
545C23D3×#4112yes—
546C23#101011yescharacteristic
547Q8—632no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—34no—
559C2×C4—38no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—62no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—122no—
607C2×C4#88414yes—
608C2×C3#238811yescharacteristic
609C2×C3#253611yescharacteristic
610C2×C3#253612yes—
611C2×C3#254512yes—
612C2×C3#254511yescharacteristic
613C2×C3#259912yes—
614C2×C3#259911yescharacteristic
615C2×C3#260412yes—
616C2×C3#260411yescharacteristic
617C2×C3#261411yescharacteristic
618C2×C3#284711yescharacteristic
619D3—82no—
620C22—122no—
621C22—124no—
622C22—122no—
623C22—124no—
624C22—122no—
625C22—242no—
626C22—242no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22C2×#88411yescharacteristic
632C22#551311yescharacteristic
633C22#552911yescharacteristic
634C22#554712yes—
635C22#556812yes—
636C22D3×#12011yescharacteristic
637C22#559612yes—
638C22#559811yescharacteristic
639C22D3×#12112yes—
640C22#560611yescharacteristic
641C22#562812yes—
642C22#562811yescharacteristic
643C22#563312yes—
644C22#563311yescharacteristic
645C22#564112yes—
646C22#565612yes—
647C22#565611yescharacteristic
648C22#566811yescharacteristic
649C22#569312yes—
650C22#569312yes—
651C22#570212yes—
652C22#572412yes—
653C22#572411yescharacteristic
654C4—24no—
655C4—44no—
656C4—42no—
657C4—68no—
658C4—64no—
659C4—82no—
660C4—81no—
661C4—122no—
662C4—124no—
663C4—241no—
664C3#1517611yescharacteristic, complemented, Sylow
665C2—242no—
666C2#7393411yescharacteristic
667C2#7493511yescharacteristic
668C2#7537112yes—
669C2#7609712yes—
670C2#7653411yescharacteristic
671C2#7688911yescharacteristic
672C2#7725611yescharacteristic
673C2#7731512yes—
674C2#7778412yes—
675C2#7781011yescharacteristic
676C2#7820411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1441C2
prime radicalC24×D3C23
hypercenter#1441D3
hypocenterC3#15176
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
1. C22×#887C2semidirect
2. C2×#4979C2semidirect
6. C3×#1441C2semidirect
9. #69409C2semidirect
10. #69403C2semidirect
11. #69401C2semidirect
48. #1441D3semidirect
664. C3#15176semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×#49792
C2×#49902
C2×#46732
C2×#46692
C3×#14412
#715702
#694282
#694092
#694032
#694012
#151763