Finite Group Databasebeta

← #10751700

Subgroups — #10751700

Order 768.

1750 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×#897C211yesmaximal, characteristic
2C2×#4985C211yesmaximal, characteristic, complemented
3C2×#4668C211yesmaximal, characteristic
4C2×#4656C211yesmaximal, characteristic, complemented
5C3×#1445C211yesmaximal, characteristic, complemented
6#69429C212yesmaximal
7#69427C212yesmaximal
8(C2×D3)⋊#95C211yesmaximal, characteristic
9#69424C211yesmaximal, characteristic, complemented
10#69419C212yesmaximal, complemented
11#69412C212yesmaximal, complemented
12#16601—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C2×#897—24no—
25C2×#897—24no—
26C2×#897—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4985—24no—
40#4668—24no—
41#4656—24no—
42C22×C22⋊Q8—31no—
43C2×#400—31no—
44C2×#395—31no—
45C2×#385—31no—
46#1446—31no—
47#1445D311yescharacteristic, complemented
48#1444—32no—
49#1442—32no—
50#1440—32no—
51#1439—32no—
52C22⋊#95—31no—
53C24×D3C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—42no—
67C22×C4×D3—42no—
68C22×C4×D3D414yes—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—42no—
72C22×Dic6—42no—
73C22×Dic6D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D414yes—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—42no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4Q814yescomplemented
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—42no—
94C2×C4⋊Dic3—42no—
95C2×C4⋊Dic3—42no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99#897—232no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—24no—
103C3×#95—24no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—22no—
109C3⋊#95—42no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C23×Q8—31no—
113C24×C4—31no—
114C2×C22⋊Q8—64no—
115C2×C22⋊Q8—64no—
116C2×C22⋊Q8—64no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C22⋊C4—31no—
123C22×C22⋊C4—32no—
124C22×C22⋊C4—32no—
125C22×C22⋊C4—31no—
126#400—64no—
127#395—64no—
128#385—64no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C2×#95C2×D312yes—
140C2×#95C2×D311yescharacteristic
141C24×C3C2411yescharacteristic
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—21no—
148C23×D3—22no—
149C23×D3—21no—
150C23×D3C2×D412yes—
151C23×D3C2×Q812yes—
152C23×D3#4112yes—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
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—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×Q812yes—
181C22×C3×C4#4114yes—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3C2×D412yes—
211C22×Dic3C2×D414yes—
212C22×Dic3C2×Q812yes—
213C2×C4×D3—28no—
214C2×C4×D3—44no—
215C2×C4×D3—44no—
216C2×C4×D3—48no—
217C2×C4×D3—48no—
218C2×Dic6—28no—
219C2×Dic6—48no—
220C2×Dic6—44no—
221C2×Dic6—44no—
222C2×Dic6—48no—
223C3×C4⋊C4—28no—
224(C2×D3)⋊C4—216no—
225(C2×D3)⋊C4—28no—
226(C2×D3)⋊C4—48no—
227(C2×D3)⋊C4—44no—
228C4⋊Dic3—216no—
229C4⋊Dic3—48no—
230C4⋊Dic3—44no—
231Dic3⋊C4—44no—
232C25—31no—
233C22×Q8—34no—
234C22×Q8—62no—
235C22×Q8—62no—
236C22×Q8—122no—
237C22×Q8—122no—
238C23×C4—32no—
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—34no—
242C23×C4—31no—
243C23×C4—32no—
244C23×C4—62no—
245C23×C4—62no—
246C23×C4—122no—
247C23×C4—122no—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D312yes—
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D311yescharacteristic
253C22⋊Q8—632no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—122no—
265C2×C4⋊C4—122no—
266C2×C4⋊C4—122no—
267C2×C4⋊C4D1214yes—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4—122no—
279#95—22no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—62no—
284#95—64no—
285#95—64no—
286#95—62no—
287#95—64no—
288#95—122no—
289#95—122no—
290#95—122no—
291C23×C3C22×D411yescharacteristic
292C23×C3C22×D411yescharacteristic
293C23×C3C22×D411yescharacteristic
294C23×C3C22×Q811yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3C2×#4111yescharacteristic
297C23×C3H(5,𝔽2)12yes—
298C23×C3H(5,𝔽2)12yes—
299C23×C32−1+411yescharacteristic
300C23×C32−1+411yescharacteristic
301C23×C32−1+412yes—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—22no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3C22⋊Q814yes—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
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—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4#12014yes—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—41no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C4×D3—416no—
386Dic6—416no—
387C24—32no—
388C24—32no—
389C24—32no—
390C24—61no—
391C24—61no—
392C24—62no—
393C24—61no—
394C24—62no—
395C24—62no—
396C24—62no—
397C24—61no—
398C24C23×D311yescharacteristic
399C2×Q8—68no—
400C2×Q8—128no—
401C2×Q8—124no—
402C2×Q8—124no—
403C2×Q8—128no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
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—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—68no—
447C22×C4—62no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—124no—
460C22×C4—124no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4C2×D1214yes—
464C22×C4C2×D1212yes—
465C22×C4D3×D412yes—
466C22×C4D3×D412yes—
467C22×C4#23714yes—
468C22×C4D3×Q812yes—
469C4⋊C4—28no—
470C4⋊C4—616no—
471C4⋊C4—124no—
472C4⋊C4—128no—
473C4⋊C4—124no—
474C22⋊C4—68no—
475C22⋊C4—616no—
476C22⋊C4—128no—
477C22⋊C4—124no—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3#53211yescharacteristic
481C22×C3#53411yescharacteristic
482C22×C3#53411yescharacteristic
483C22×C3#53412yes—
484C22×C3#53512yes—
485C22×C3#53611yescharacteristic
486C22×C3#53611yescharacteristic
487C22×C3#53912yes—
488C22×C3#53912yes—
489C22×C3#54212yes—
490C22×C3#54412yes—
491C22×C3#54712yes—
492C22×C3#54811yescharacteristic
493C22×C3#54811yescharacteristic
494C22×C3#54812yes—
495C22×C3D4×Q811yescharacteristic
496C22×C3D4×Q811yescharacteristic
497C22×C3#55112yes—
498C22×C3#55412yes—
499C22×C3#56312yes—
500C22×C3#56312yes—
501C2×D3—24no—
502C2×D3—24no—
503C2×D3—24no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—44no—
512C3×C4—24no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—44no—
516C3×C4—81no—
517Dic3—44no—
518Dic3—42no—
519Dic3—42no—
520Dic3—81no—
521Dic3—82no—
522C23—34no—
523C23—64no—
524C23—62no—
525C23—62no—
526C23—64no—
527C23—62no—
528C23—62no—
529C23—64no—
530C23—64no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23C22×D1211yescharacteristic
546C23C2×D3×D411yescharacteristic
547C23C2×D3×D411yescharacteristic
548C23C2×#23712yes—
549C23C2×D3×Q811yescharacteristic
550C23#100711yescharacteristic
551C23D3×#4111yescharacteristic
552C23#100912yes—
553C23#100912yes—
554C23#101011yescharacteristic
555C23#101012yes—
556Q8—1216no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—1216no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
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#243211yescharacteristic
620C2×C3#256411yescharacteristic
621C2×C3#261311yescharacteristic
622C2×C3#262711yescharacteristic
623C2×C3#266311yescharacteristic
624C2×C3#272012yes—
625C2×C3#272312yes—
626C2×C3#272812yes—
627C2×C3#272912yes—
628C2×C3#281711yescharacteristic
629D3—44no—
630C22—64no—
631C22—64no—
632C22—64no—
633C22—124no—
634C22—122no—
635C22—122no—
636C22—124no—
637C22—124no—
638C22—124no—
639C22—122no—
640C22—122no—
641C22C2×#88211yescharacteristic
642C22#552512yes—
643C22#555512yes—
644C22Q8×D1211yescharacteristic
645C22#557611yescharacteristic
646C22#559812yes—
647C22D3×C22⋊Q811yescharacteristic
648C22#562811yescharacteristic
649C22#563512yes—
650C22#564312yes—
651C22#565311yescharacteristic
652C22#565311yescharacteristic
653C22#565312yes—
654C22#565812yes—
655C22#566512yes—
656C22#567511yescharacteristic
657C22#567711yescharacteristic
658C22#568912yes—
659C22#569912yes—
660C22#570712yes—
661C22#571212yes—
662C22#571611yescharacteristic
663C22#572711yescharacteristic
664C4—24no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—81no—
669C4—122no—
670C4—122no—
671C4—124no—
672C4—241no—
673C4—242no—
674C3#1660111yescharacteristic, complemented, Sylow
675C2—124no—
676C2#7413411yescharacteristic
677C2#7516912yes—
678C2#7517412yes—
679C2#7644511yescharacteristic
680C2#7661911yescharacteristic
681C2#7770312yes—
682C2#7798512yes—
683C2#7805811yescharacteristic
684C2#7810811yescharacteristic
685C2#7831611yescharacteristic
686C2#7836011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1445C2
prime radicalC24×D3C23
hypercenter#1445D3
hypocenterC3#16601
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#4985C2semidirect
4. C2×#4656C2semidirect
5. C3×#1445C2semidirect
9. #69424C2semidirect
10. #69419C2semidirect
11. #69412C2semidirect
47. #1445D3semidirect
87. C2×(C2×D3)⋊C4Q8semidirect
674. C3#16601semidirect

Maximal subgroups

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

subgroupindex
C22×#8972
C2×#49852
C2×#46682
C2×#46562
C3×#14452
#694292
#694272
(C2×D3)⋊#952
#694242
#694192
#694122
#166013