Finite Group Databasebeta

← #10732192

Subgroups — #10732192

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×C4×Dic6C211yesmaximal, characteristic, complemented
2C2×#4940C211yesmaximal, characteristic
3C2×#4673C211yesmaximal, characteristic
4C2×#4656C211yesmaximal, characteristic, complemented
5C2×#4670C212yesmaximal
6C3×#1434C211yesmaximal, characteristic, complemented
7#69428C212yesmaximal
8#69427C211yesmaximal, characteristic
9#69411C212yesmaximal, complemented
10#69410C211yesmaximal, characteristic, complemented
11#69402C212yesmaximal, complemented
12#15176—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2212yes—
25C2×C4×Dic6—24no—
26C2×C4×Dic6—28no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4940—24no—
39#4673—24no—
40#4670—28no—
41#4656—24no—
42C22×C4×Q8—31no—
43C2×#395—31no—
44C2×#393—31no—
45C2×#388—31no—
46C2×#388—32no—
47#1441—31no—
48#1441—32no—
49#1439—31no—
50#1439—32no—
51#1434—32no—
52#1434D311yescharacteristic, complemented
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×C42—24no—
70C2×C3×C42D414yes—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3—24no—
98C2×C4×Dic3D418yes—
99C4×Dic6—232no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—22no—
103C3×#95—24no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C23×Q8—31no—
112C2×C4×Q8—68no—
113C2×C4×Q8—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4—31no—
122C22×C22⋊C4—31no—
123C22×C42—32no—
124C22×C42C2×D311yescharacteristic
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—32no—
134C2×#95—32no—
135C2×#95—31no—
136C2×#95C2×D311yescharacteristic
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C2×#95C2×D312yes—
140C24×C3C2411yescharacteristic
141C23×D3—21no—
142C23×D3—22no—
143C23×D3—21no—
144C23×D3—22no—
145C23×D3—21no—
146C23×D3—21no—
147C23×D3—22no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—22no—
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—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4#4114yes—
180C22×C3×C4#4112yes—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3C2×D414yes—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×D414yes—
210C22×Dic3#4112yes—
211C22×Dic3#4114yes—
212C2×Dic6—28no—
213C2×Dic6—216no—
214C2×Dic6—28no—
215C2×Dic6—28no—
216C2×Dic6—216no—
217C3×C42—28no—
218(C2×D3)⋊C4—216no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—44no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—48no—
223C4⋊Dic3—28no—
224C4⋊Dic3—48no—
225Dic3⋊C4—216no—
226Dic3⋊C4—44no—
227C4×Dic3—216no—
228C25—31no—
229C22×Q8—34no—
230C22×Q8—64no—
231C22×Q8—64no—
232C22×Q8—62no—
233C22×Q8—62no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D312yes—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C4×Q8—632no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—24no—
271C2×C42—38no—
272C2×C42—64no—
273C2×C42—64no—
274C2×C42D1214yes—
275#95—24no—
276#95—24no—
277#95—22no—
278#95—22no—
279#95—64no—
280#95—64no—
281#95—64no—
282#95—64no—
283#95—62no—
284#95—62no—
285#95—64no—
286C23×C3C22×D411yescharacteristic
287C23×C3C22×D412yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C2×#4112yes—
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4112yes—
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C32−1+411yescharacteristic
295C23×C32−1+411yescharacteristic
296C23×C32−1+412yes—
297C22×D3—22no—
298C22×D3—24no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—24no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—42no—
315C22×D3—41no—
316C22×D3—42no—
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—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
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—42no—
346C2×C3×C4—42no—
347C2×C3×C4#12014yes—
348C2×C3×C4#12114yes—
349C2×Dic3—24no—
350C2×Dic3—24no—
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—22no—
360C2×Dic3—22no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—42no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3#12118yes—
380Dic6—232no—
381C24—32no—
382C24—61no—
383C24—62no—
384C24—61no—
385C24—62no—
386C24—61no—
387C24—62no—
388C24—61no—
389C24—61no—
390C24—61no—
391C24—62no—
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—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—34no—
421C22×C4—34no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—32no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4C2×D1212yes—
452C22×C4C2×D1212yes—
453C22×C4C2×D1212yes—
454C22×C4#23514yes—
455C22×C4D3×D414yes—
456C22×C4D3×D412yes—
457C22×C4#23912yes—
458C4⋊C4—616no—
459C4⋊C4—68no—
460C4⋊C4—128no—
461C4⋊C4—124no—
462C22⋊C4—68no—
463C22⋊C4—616no—
464C22⋊C4—128no—
465C22⋊C4—124no—
466C22⋊C4—124no—
467C42—28no—
468C42—616no—
469C22×C3C2×#12011yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3C2×#12112yes—
472C22×C3#52511yescharacteristic
473C22×C3#52511yescharacteristic
474C22×C3#52512yes—
475C22×C3#52512yes—
476C22×C3#53211yescharacteristic
477C22×C3#53212yes—
478C22×C3#53612yes—
479C22×C3#53612yes—
480C22×C3#53611yescharacteristic
481C22×C3#53611yescharacteristic
482C22×C3#54811yescharacteristic
483C22×C3#54811yescharacteristic
484C22×C3#54812yes—
485C22×C3#55211yescharacteristic
486C22×C3#55211yescharacteristic
487C22×C3#55212yes—
488C22×C3#55212yes—
489C22×C3#55612yes—
490C22×C3#55612yes—
491C22×C3#55612yes—
492C2×D3—42no—
493C2×D3—44no—
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—24no—
505C3×C4—42no—
506C3×C4—44no—
507C3×C4—82no—
508Dic3—28no—
509Dic3—44no—
510Dic3—42no—
511Dic3—81no—
512Dic3—81no—
513C23—62no—
514C23—62no—
515C23—64no—
516C23—62no—
517C23—64no—
518C23—122no—
519C23—122no—
520C23—121no—
521C23—121no—
522C23—122no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—122no—
533C23—122no—
534C23—122no—
535C23—122no—
536C23C22×D1211yescharacteristic
537C23C2×#23512yes—
538C23C2×D3×D412yes—
539C23C2×D3×D411yescharacteristic
540C23C2×#23911yescharacteristic
541C23D3×#4111yescharacteristic
542C23D3×#4111yescharacteristic
543C23D3×#4112yes—
544C23#101011yescharacteristic
545C23#101011yescharacteristic
546C23#101012yes—
547Q8—632no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—38no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—122no—
591C2×C4—121no—
592C2×C4—122no—
593C2×C4—121no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4#88214yes—
607C2×C4#89514yes—
608C2×C3#238811yescharacteristic
609C2×C3#253611yescharacteristic
610C2×C3#253612yes—
611C2×C3#254512yes—
612C2×C3#254511yescharacteristic
613C2×C3#259911yescharacteristic
614C2×C3#259912yes—
615C2×C3#260412yes—
616C2×C3#260411yescharacteristic
617C2×C3#261411yescharacteristic
618C2×C3#284711yescharacteristic
619D3—82no—
620C22—122no—
621C22—122no—
622C22—124no—
623C22—122no—
624C22—124no—
625C22—241no—
626C22—242no—
627C22—242no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22#547711yescharacteristic
632C22C2×#88211yescharacteristic
633C22C2×#89511yescharacteristic
634C22#552411yescharacteristic
635C22#552411yescharacteristic
636C22#552611yescharacteristic
637C22#552611yescharacteristic
638C22#553112yes—
639C22#553312yes—
640C22#554712yes—
641C22#554812yes—
642C22#555212yes—
643C22#558412yes—
644C22D3×#12112yes—
645C22#562712yes—
646C22#564812yes—
647C22#564812yes—
648C22#567711yescharacteristic
649C22#567711yescharacteristic
650C22#569411yescharacteristic
651C22#569411yescharacteristic
652C22#570212yes—
653C22#570812yes—
654C4—24no—
655C4—24no—
656C4—44no—
657C4—42no—
658C4—68no—
659C4—82no—
660C4—122no—
661C4—124no—
662C4—241no—
663C4—241no—
664C3#1517611yescharacteristic, complemented, Sylow
665C2—242no—
666C2#7340311yescharacteristic
667C2#7358011yescharacteristic
668C2#7371011yescharacteristic
669C2#7404811yescharacteristic
670C2#7413511yescharacteristic
671C2#7424011yescharacteristic
672C2#7430811yescharacteristic
673C2#7512512yes—
674C2#7512612yes—
675C2#7559212yes—
676C2#7559312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1434C2
prime radicalC24×D3C23
hypercenter#1434D3
hypocenterC3#15176
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic6C2semidirect
4. C2×#4656C2semidirect
6. C3×#1434C2semidirect
9. #69411C2semidirect
10. #69410C2semidirect
11. #69402C2semidirect
52. #1434D3semidirect
664. C3#15176semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic62
C2×#49402
C2×#46732
C2×#46562
C2×#46702
C3×#14342
#694282
#694272
#694112
#694102
#694022
#151763