Finite Group Databasebeta

← #10728711

Subgroups — #10728711

Order 768.

1742 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×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×#5231C211yesmaximal, characteristic
3C2×#4674C211yesmaximal, characteristic
4C2×#4670C211yesmaximal, characteristic
5C3×#1446C211yesmaximal, characteristic, complemented
6#69427C212yesmaximal
7#69422C211yesmaximal, characteristic, complemented
8#69412C212yesmaximal, complemented
9#69408C212yesmaximal
10#69407C212yesmaximal, complemented
11#69402C211yesmaximal, characteristic, complemented
12#14882—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×Q8×Dic3—24no—
15C2×Q8×Dic3—24no—
16C2×Q8×Dic3—24no—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2212yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31#5231—24no—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4674—24no—
41#4670—24no—
42C22×C4×Q8—31no—
43C2×#397—31no—
44C2×#393—31no—
45C2×#388—31no—
46#1446D311yescharacteristic, complemented
47#1445—31no—
48#1442—32no—
49#1439—32no—
50#1437—32no—
51#1436—32no—
52#1434—31no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—24no—
57C22×C3×Q8—24no—
58C22×C3×Q8D414yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69Q8×Dic3—232no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—42no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3D414yes—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3D414yes—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—24no—
101C3×#95—24no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C23×Q8C2×D311yescharacteristic
111C2×C4×Q8—64no—
112C2×C4×Q8—64no—
113C2×C4×Q8—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4—31no—
121C22×C22⋊C4—32no—
122C22×C42—31no—
123C22×C42—32no—
124#397—64no—
125#393—64no—
126#388—64no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D312yes—
137C2×#95C2×D312yes—
138C2×#95C2×D311yescharacteristic
139C24×C3C2411yescharacteristic
140C23×D3—21no—
141C23×D3—21no—
142C23×D3—21no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—21no—
146C23×D3—22no—
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—22no—
150C23×D3#4112yes—
151C2×C3×Q8—28no—
152C2×C3×Q8—28no—
153C2×C3×Q8—216no—
154C2×C3×Q8—28no—
155C2×C3×Q8—216no—
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—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
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×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4114yes—
183C22×C3×C4#4114yes—
184C22×C3×C4#4112yes—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D412yes—
215C22×Dic3#4114yes—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—48no—
218(C2×D3)⋊C4—44no—
219(C2×D3)⋊C4—48no—
220(C2×D3)⋊C4—44no—
221C4⋊Dic3—216no—
222C4⋊Dic3—28no—
223C4⋊Dic3—44no—
224C4×Dic3—216no—
225C4×Dic3—28no—
226C25—31no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—24no—
230C22×Q8—24no—
231C22×Q8D1214yes—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D312yes—
242C4×Q8—632no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—64no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—122no—
265C2×C42—34no—
266C2×C42—64no—
267C2×C42—64no—
268C2×C42—64no—
269C2×C42—62no—
270C2×C42—62no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—64no—
276#95—62no—
277#95—64no—
278#95—62no—
279#95—64no—
280#95—64no—
281#95—122no—
282#95—122no—
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D411yescharacteristic
285C23×C3C22×D411yescharacteristic
286C23×C3C2×#4112yes—
287C23×C3C2×#4112yes—
288C23×C3C2×#4111yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4112yes—
291C23×C3H(5,𝔽2)11yescharacteristic
292C23×C32−1+412yes—
293C23×C32−1+411yescharacteristic
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—24no—
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—41no—
305C22×D3—42no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—41no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C3×Q8—232no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
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#12014yes—
346C2×C3×C4#12318yes—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—41no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3#12414yes—
378C24—32no—
379C24—61no—
380C24—61no—
381C24—61no—
382C24—62no—
383C24—62no—
384C24—61no—
385C24—62no—
386C24—61no—
387C24—62no—
388C24—61no—
389C24C23×D311yescharacteristic
390C2×Q8—28no—
391C2×Q8—216no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—216no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—32no—
418C22×C4—34no—
419C22×C4—32no—
420C22×C4—34no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
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×C4C2×D1212yes—
449C22×C4C2×D1214yes—
450C22×C4D3×D412yes—
451C22×C4D3×D412yes—
452C22×C4#23714yes—
453C22×C4#23714yes—
454C22×C4#23912yes—
455C4⋊C4—616no—
456C4⋊C4—68no—
457C4⋊C4—124no—
458C22⋊C4—68no—
459C22⋊C4—124no—
460C22⋊C4—128no—
461C22⋊C4—124no—
462C22⋊C4—128no—
463C42—616no—
464C42—68no—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×#12312yes—
467C22×C3#52511yescharacteristic
468C22×C3#52511yescharacteristic
469C22×C3#52512yes—
470C22×C3#52511yescharacteristic
471C22×C3C2×#12411yescharacteristic
472C22×C3#52912yes—
473C22×C3#52912yes—
474C22×C3#53611yescharacteristic
475C22×C3#53711yescharacteristic
476C22×C3#54112yes—
477C22×C3#54212yes—
478C22×C3#54611yescharacteristic
479C22×C3#54711yescharacteristic
480C22×C3#54812yes—
481C22×C3#54811yescharacteristic
482C22×C3#54812yes—
483C22×C3#55011yescharacteristic
484C22×C3#55212yes—
485C22×C3#55612yes—
486C22×C3#55612yes—
487C22×C3#55912yes—
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—28no—
501C3×C4—42no—
502C3×C4—81no—
503C3×C4—82no—
504Dic3—24no—
505Dic3—42no—
506Dic3—44no—
507Dic3—81no—
508Dic3—82no—
509C23—64no—
510C23—62no—
511C23—64no—
512C23—62no—
513C23—62no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—121no—
518C23—122no—
519C23—121no—
520C23—121no—
521C23—122no—
522C23—122no—
523C23—121no—
524C23—122no—
525C23—122no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—122no—
530C23—122no—
531C23—121no—
532C23C22×D1211yescharacteristic
533C23C2×D3×D411yescharacteristic
534C23C2×D3×D411yescharacteristic
535C23C2×#23712yes—
536C23C2×#23712yes—
537C23C2×#23911yescharacteristic
538C23#100712yes—
539C23#100711yescharacteristic
540C23D3×#4112yes—
541C23D3×#4111yescharacteristic
542C23#100911yescharacteristic
543Q8—232no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—34no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—64no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—122no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—122no—
596C2×C4—121no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4#88214yes—
603C2×C4#88618yes—
604C2×C3#239811yescharacteristic
605C2×C3#240711yescharacteristic
606C2×C3#242112yes—
607C2×C3#243611yescharacteristic
608C2×C3#244812yes—
609C2×C3#245812yes—
610C2×C3#246912yes—
611C2×C3#259211yescharacteristic
612C2×C3#259911yescharacteristic
613C2×C3#263711yescharacteristic
614C2×C3#284911yescharacteristic
615D3—82no—
616C22—122no—
617C22—124no—
618C22—122no—
619C22—122no—
620C22—124no—
621C22—241no—
622C22—241no—
623C22—242no—
624C22—241no—
625C22—241no—
626C22—242no—
627C22C2×#88211yescharacteristic
628C22C2×#88612yes—
629C22#557611yescharacteristic
630C22#557612yes—
631C22#557711yescharacteristic
632C22#558611yescharacteristic
633C22#558612yes—
634C22#558611yescharacteristic
635C22#562412yes—
636C22#562711yescharacteristic
637C22#562911yescharacteristic
638C22#563312yes—
639C22#563912yes—
640C22#564112yes—
641C22#564312yes—
642C22D3×#12411yescharacteristic
643C22#567611yescharacteristic
644C22#568012yes—
645C22#568112yes—
646C22#572411yescharacteristic
647C22#572611yescharacteristic
648C22#573112yes—
649C22#573212yes—
650C4—24no—
651C4—28no—
652C4—42no—
653C4—64no—
654C4—81no—
655C4—82no—
656C4—124no—
657C4—122no—
658C4—241no—
659C4—242no—
660C3#1488211yescharacteristic, complemented, Sylow
661C2—242no—
662C2#7384711yescharacteristic
663C2#7650011yescharacteristic
664C2#7650712yes—
665C2#7661811yescharacteristic
666C2#7662212yes—
667C2#7677911yescharacteristic
668C2#7678512yes—
669C2#7680411yescharacteristic
670C2#7718112yes—
671C2#7719611yescharacteristic
672C2#7720511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1446C2
prime radicalC24×D3C23
hypercenter#1446D3
hypocenterC3#14882
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic3C2semidirect
5. C3×#1446C2semidirect
7. #69422C2semidirect
8. #69412C2semidirect
10. #69407C2semidirect
11. #69402C2semidirect
46. #1446D3semidirect
660. C3#14882semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×#52312
C2×#46742
C2×#46702
C3×#14462
#694272
#694222
#694122
#694082
#694072
#694022
#148823