Finite Group Databasebeta

← #10751708

Subgroups — #10751708

Order 768.

1706 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic, complemented
2C2×#5231C211yesmaximal, characteristic
3C2×#4673C211yesmaximal, characteristic
4C2×#4670C211yesmaximal, characteristic
5C3×#1446C211yesmaximal, characteristic, complemented
6#69428C212yesmaximal
7#69428C212yesmaximal
8#69424C211yesmaximal, characteristic, complemented
9#69413C212yesmaximal, complemented
10#69411C212yesmaximal, complemented
11#69402C211yesmaximal, characteristic, complemented
12#16602—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×#944—24no—
16C2×#944—24no—
17C2×#944—24no—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31#5231—24no—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4673—24no—
41#4670—24no—
42C22×C4⋊Q8—31no—
43C2×#393—31no—
44C2×#393—31no—
45C2×#388—31no—
46#1446—31no—
47#1446D311yescharacteristic, complemented
48#1442—32no—
49#1441—32no—
50#1439—32no—
51#1439—32no—
52#1434—31no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C22×C3×Q8—42no—
58C22×C3×Q8D414yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—42no—
72C22×Dic6—42no—
73C22×Dic6D414yes—
74#944—232no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
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—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—42no—
85C2×(C2×D3)⋊C4—42no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4—42no—
88C2×(C2×D3)⋊C4Q814yescomplemented
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3D414yes—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—42no—
103C3×#95—42no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C23×Q8—31no—
113C23×Q8C2×D311yescharacteristic
114C2×C4⋊Q8—64no—
115C2×C4⋊Q8—64no—
116C2×C4⋊Q8—64no—
117C22×C4⋊C4—32no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4—31no—
120C22×C22⋊C4—31no—
121C22×C22⋊C4—31no—
122C22×C22⋊C4—31no—
123C22×C22⋊C4—32no—
124C22×C22⋊C4—32no—
125C22×C42—31no—
126#393—64no—
127#393—64no—
128#388—64no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D311yescharacteristic
139C2×#95C2×D312yes—
140C2×#95C2×D312yes—
141C24×C3C2411yescharacteristic
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—21no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—22no—
149C23×D3—22no—
150C23×D3—21no—
151C23×D3—21no—
152C23×D3C2×Q812yes—
153C2×C3×Q8—28no—
154C2×C3×Q8—48no—
155C2×C3×Q8—48no—
156C2×C3×Q8—44no—
157C2×C3×Q8—44no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
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—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×Q812yes—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
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×Dic3C2×Q812yes—
216C22×Dic3#4112yes—
217C22×Dic3#4114yes—
218C2×Dic6—28no—
219C2×Dic6—48no—
220C2×Dic6—48no—
221C2×Dic6—44no—
222C2×Dic6—44no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—48no—
225(C2×D3)⋊C4—48no—
226(C2×D3)⋊C4—44no—
227(C2×D3)⋊C4—44no—
228C4⋊Dic3—44no—
229Dic3⋊C4—216no—
230Dic3⋊C4—216no—
231C4×Dic3—28no—
232C25—31no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—34no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8—62no—
239C22×Q8—62no—
240C22×Q8—122no—
241C22×Q8—122no—
242C22×Q8D1214yes—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—32no—
246C23×C4—31no—
247C23×C4—32no—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D312yes—
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D311yescharacteristic
253C4⋊Q8—632no—
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—62no—
261C2×C4⋊C4—64no—
262C2×C22⋊C4—34no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—122no—
273C2×C22⋊C4—122no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4—122no—
276C2×C42—34no—
277C2×C42—62no—
278C2×C42—62no—
279#95—22no—
280#95—22no—
281#95—42no—
282#95—42no—
283#95—64no—
284#95—64no—
285#95—62no—
286#95—64no—
287#95—64no—
288#95—62no—
289#95—122no—
290#95—122no—
291C23×C3C22×D411yescharacteristic
292C23×C3C22×D411yescharacteristic
293C23×C3C22×D411yescharacteristic
294C23×C3C22×Q811yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4112yes—
297C23×C3H(5,𝔽2)12yes—
298C23×C3H(5,𝔽2)12yes—
299C23×C32−1+411yescharacteristic
300C23×C32−1+412yes—
301C23×C32−1+411yescharacteristic
302C22×D3—22no—
303C22×D3—24no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—24no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C3×Q8—416no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
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—24no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4#12014yes—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—42no—
375C2×Dic3—41no—
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×Dic3C22⋊Q814yes—
385C2×Dic3#12414yes—
386Dic6—416no—
387C24—32no—
388C24—61no—
389C24—61no—
390C24—61no—
391C24—62no—
392C24—62no—
393C24—61no—
394C24—62no—
395C24—62no—
396C24—61no—
397C24—61no—
398C24C23×D311yescharacteristic
399C2×Q8—28no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—68no—
405C2×Q8—128no—
406C2×Q8—128no—
407C2×Q8—124no—
408C2×Q8—124no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4C2×D1214yes—
465C22×C4C2×D1212yes—
466C22×C4D3×D412yes—
467C22×C4D3×D412yes—
468C22×C4D3×Q812yes—
469C4⋊C4—616no—
470C4⋊C4—616no—
471C4⋊C4—124no—
472C22⋊C4—68no—
473C22⋊C4—128no—
474C22⋊C4—124no—
475C22⋊C4—128no—
476C22⋊C4—124no—
477C42—68no—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3C2×#12411yescharacteristic
481C22×C3#53412yes—
482C22×C3#53412yes—
483C22×C3#53511yescharacteristic
484C22×C3#53511yescharacteristic
485C22×C3#53611yescharacteristic
486C22×C3#53611yescharacteristic
487C22×C3#54212yes—
488C22×C3#54212yes—
489C22×C3#54212yes—
490C22×C3#54412yes—
491C22×C3#54812yes—
492C22×C3#54811yescharacteristic
493C22×C3#54811yescharacteristic
494C22×C3#54812yes—
495C22×C3D4×Q811yescharacteristic
496C22×C3D4×Q811yescharacteristic
497C22×C3#55412yes—
498C22×C3#55512yes—
499C22×C3#56312yes—
500C22×C3#56312yes—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—44no—
505C2×D3—44no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—82no—
511C2×D3—82no—
512C3×C4—24no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—81no—
516C3×C4—82no—
517Dic3—24no—
518Dic3—24no—
519Dic3—44no—
520Dic3—81no—
521Dic3—82no—
522C23—64no—
523C23—62no—
524C23—64no—
525C23—62no—
526C23—62no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—122no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—122no—
537C23—121no—
538C23—122no—
539C23—121no—
540C23—122no—
541C23—122no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23C22×D1211yescharacteristic
546C23C2×D3×D411yescharacteristic
547C23C2×D3×D411yescharacteristic
548C23#100412yes—
549C23#100412yes—
550C23C2×D3×Q811yescharacteristic
551C23#100712yes—
552C23#100711yescharacteristic
553C23D3×#4112yes—
554C23D3×#4111yescharacteristic
555C23#101011yescharacteristic
556Q8—416no—
557Q8—1216no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—34no—
568C2×C4—34no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—122no—
612C2×C4—121no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4#88214yes—
618C2×C3#240911yescharacteristic
619C2×C3#243611yescharacteristic
620C2×C3#256311yescharacteristic
621C2×C3#261311yescharacteristic
622C2×C3#262811yescharacteristic
623C2×C3#264011yescharacteristic
624C2×C3#272912yes—
625C2×C3#272912yes—
626C2×C3#273112yes—
627C2×C3#273112yes—
628C2×C3#282111yescharacteristic
629D3—82no—
630C22—122no—
631C22—124no—
632C22—122no—
633C22—124no—
634C22—122no—
635C22—241no—
636C22—242no—
637C22—242no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22C2×#88211yescharacteristic
642C22#549312yes—
643C22Q8×D1211yescharacteristic
644C22#557611yescharacteristic
645C22#557612yes—
646C22#558912yes—
647C22#559211yescharacteristic
648C22#559211yescharacteristic
649C22D3×C22⋊Q811yescharacteristic
650C22#562811yescharacteristic
651C22#563312yes—
652C22#563612yes—
653C22#564012yes—
654C22#564012yes—
655C22#564612yes—
656C22#566912yes—
657C22#567012yes—
658C22D3×#12411yescharacteristic
659C22#567711yescharacteristic
660C22#572411yescharacteristic
661C22#572711yescharacteristic
662C22#572912yes—
663C22#573512yes—
664C4—24no—
665C4—44no—
666C4—42no—
667C4—64no—
668C4—64no—
669C4—81no—
670C4—82no—
671C4—124no—
672C4—242no—
673C4—241no—
674C3#1660211yescharacteristic, complemented, Sylow
675C2—242no—
676C2#7385511yescharacteristic
677C2#7650011yescharacteristic
678C2#7657212yes—
679C2#7661911yescharacteristic
680C2#7665512yes—
681C2#7679911yescharacteristic
682C2#7681511yescharacteristic
683C2#7683412yes—
684C2#7720211yescharacteristic
685C2#7722312yes—
686C2#7727611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1446C2
prime radicalC24×D3C23
hypercenter#1446D3
hypocenterC3#16602
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×#944C2semidirect
5. C3×#1446C2semidirect
8. #69424C2semidirect
9. #69413C2semidirect
10. #69411C2semidirect
11. #69402C2semidirect
47. #1446D3semidirect
88. C2×(C2×D3)⋊C4Q8semidirect
674. C3#16602semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×#52312
C2×#46732
C2×#46702
C3×#14462
#694282
#694242
#694132
#694112
#694022
#166023