Finite Group Databasebeta

← #10731737

Subgroups — #10731737

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 700 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×#4667C212yesmaximal
5C2×#4666C211yesmaximal, characteristic
6C3×#1446C211yesmaximal, characteristic, complemented
7#69428C212yesmaximal
8#69424C211yesmaximal, characteristic, complemented
9#69414C212yesmaximal, complemented
10Dic3⋊#95C211yesmaximal, characteristic, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#15125—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C2×#944—28no—
17C2×#944—24no—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31#5231—24no—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4673—24no—
40#4667—28no—
41#4666—24no—
42C22×C4⋊Q8—31no—
43C2×#393—31no—
44C2×#393—31no—
45C2×#380—32no—
46C2×C4×C22⋊C4—31no—
47#1446—31no—
48#1446D311yescharacteristic, complemented
49#1442—32no—
50#1439—32no—
51C4⋊#95—32no—
52C4⋊#95—31no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×C4×D3—22no—
70C22×C4×D3—24no—
71C22×C4×D3—24no—
72C22×C4×D3—22no—
73C22×C4×D3Q814yescomplemented
74C22×Dic6—22no—
75C22×Dic6—24no—
76C22×Dic6—22no—
77C22×Dic6—22no—
78C22×Dic6—42no—
79#944—232no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—42no—
89C2×(C2×D3)⋊C4—42no—
90C2×(C2×D3)⋊C4D414yes—
91C2×(C2×D3)⋊C4Q814yescomplemented
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4D418yes—
101C2×C4×Dic3—24no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3D414yes—
105C3×#95—22no—
106C3×#95—24no—
107C3×#95—22no—
108C3×#95—42no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C23×Q8—31no—
117C23×Q8C2×D311yescharacteristic
118C24×C4—31no—
119C2×C4⋊Q8—64no—
120C2×C4⋊Q8—68no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—32no—
124C22×C22⋊C4—31no—
125C22×C22⋊C4—31no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C42—31no—
129C22×C42—31no—
130#393—64no—
131#393—64no—
132#380—68no—
133C4×C22⋊C4—64no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—31no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95C2×D311yescharacteristic
142C2×#95C2×D312yes—
143C2×#95C2×D311yescharacteristic
144C2×#95C2×D312yes—
145C24×C3C2411yescharacteristic
146C23×D3—21no—
147C23×D3—22no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—22no—
154C23×D3C2×D412yes—
155C23×D3C2×Q812yes—
156C23×D3C2×Q812yes—
157C2×C3×Q8—28no—
158C2×C3×Q8—48no—
159C2×C3×Q8—48no—
160C2×C3×Q8—44no—
161C2×C3×Q8—44no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×Q812yes—
189C22×C3×C4C2×Q812yes—
190C22×C3×C4#4112yes—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
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×D412yes—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D414yes—
215C22×Dic3C2×Q812yes—
216C22×Dic3C2×Q812yes—
217C22×Dic3#4112yes—
218C22×Dic3#4114yes—
219C22×Dic3#4112yes—
220C2×C4×D3—28no—
221C2×C4×D3—216no—
222C2×C4×D3—44no—
223C2×C4×D3—44no—
224C2×C4×D3—48no—
225C2×Dic6—28no—
226C2×Dic6—48no—
227C2×Dic6—48no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—28no—
232(C2×D3)⋊C4—48no—
233(C2×D3)⋊C4—48no—
234Dic3⋊C4—216no—
235Dic3⋊C4—216no—
236Dic3⋊C4—48no—
237C4×Dic3—28no—
238C4×Dic3—44no—
239C25—31no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—64no—
246C22×Q8—62no—
247C22×Q8—62no—
248C22×Q8—62no—
249C22×Q8—122no—
250C23×C4—31no—
251C23×C4—31no—
252C23×C4—32no—
253C23×C4—31no—
254C23×C4—34no—
255C23×C4—32no—
256C23×C4—62no—
257C23×C4—64no—
258C23×C4—64no—
259C23×C4—62no—
260C23×C4C22×D312yes—
261C23×C4C22×D312yes—
262C23×C4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
264C23×C4C22×D311yescharacteristic
265C4⋊Q8—632no—
266C2×C4⋊C4—38no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—34no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—122no—
286C2×C22⋊C4—122no—
287C2×C42—34no—
288C2×C42—64no—
289C2×C42—62no—
290C2×C42—64no—
291#95—22no—
292#95—24no—
293#95—22no—
294#95—42no—
295#95—64no—
296#95—62no—
297#95—64no—
298#95—64no—
299#95—64no—
300#95—62no—
301#95—122no—
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D412yes—
304C23×C3C22×D411yescharacteristic
305C23×C3C22×Q811yescharacteristic
306C23×C3C22×Q811yescharacteristic
307C23×C3C2×#4111yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3C2×#4111yescharacteristic
311C23×C3H(5,𝔽2)12yes—
312C23×C32−1+412yes—
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—24no—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—24no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—42no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3C4⋊Q814yescomplemented
336C3×Q8—416no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4#12114yes—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—42no—
376C2×Dic3—41no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3#12014yes—
390C2×Dic3C22⋊Q814yes—
391C2×Dic3C22⋊Q814yes—
392C2×Dic3C22⋊Q818yes—
393C2×Dic3#12414yes—
394C2×Dic3#12418yes—
395C4×D3—416no—
396Dic6—416no—
397C24—32no—
398C24—32no—
399C24—32no—
400C24—61no—
401C24—62no—
402C24—61no—
403C24—62no—
404C24—62no—
405C24—61no—
406C24—62no—
407C24—61no—
408C24C23×D311yescharacteristic
409C2×Q8—28no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—68no—
415C2×Q8—128no—
416C2×Q8—128no—
417C2×Q8—124no—
418C2×Q8—124no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—32no—
443C22×C4—34no—
444C22×C4—32no—
445C22×C4—34no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—34no—
452C22×C4—68no—
453C22×C4—616no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—124no—
474C22×C4—128no—
475C22×C4—124no—
476C22×C4#23512yes—
477C22×C4D3×D414yes—
478C22×C4D3×D412yes—
479C22×C4D3×D412yes—
480C22×C4D3×Q812yes—
481C22×C4D3×Q812yes—
482C4⋊C4—616no—
483C4⋊C4—616no—
484C4⋊C4—128no—
485C22⋊C4—68no—
486C22⋊C4—68no—
487C22⋊C4—128no—
488C22⋊C4—128no—
489C42—68no—
490C42—124no—
491C22×C3C2×#12011yescharacteristic
492C22×C3C2×#12111yescharacteristic
493C22×C3C2×C22⋊Q811yescharacteristic
494C22×C3C2×C22⋊Q811yescharacteristic
495C22×C3C2×C22⋊Q812yes—
496C22×C3C2×#12411yescharacteristic
497C22×C3C2×#12412yes—
498C22×C3C2×C4⋊Q811yescharacteristic
499C22×C3#53211yescharacteristic
500C22×C3#53311yescharacteristic
501C22×C3#53311yescharacteristic
502C22×C3#53511yescharacteristic
503C22×C3#53611yescharacteristic
504C22×C3#54212yes—
505C22×C3#54412yes—
506C22×C3D4212yes—
507C22×C3#54612yes—
508C22×C3#54712yes—
509C22×C3#54812yes—
510C22×C3D4×Q812yes—
511C22×C3#55412yes—
512C22×C3#55512yes—
513C22×C3#55612yes—
514C2×D3—24no—
515C2×D3—28no—
516C2×D3—42no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—44no—
522C2×D3—44no—
523C2×D3—44no—
524C2×D3—42no—
525C3×C4—24no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—44no—
529C3×C4—82no—
530Dic3—24no—
531Dic3—44no—
532Dic3—42no—
533Dic3—82no—
534Dic3#39318yescomplemented
535C23—34no—
536C23—64no—
537C23—62no—
538C23—64no—
539C23—64no—
540C23—62no—
541C23—62no—
542C23—62no—
543C23—64no—
544C23—62no—
545C23—64no—
546C23—62no—
547C23—64no—
548C23—122no—
549C23—122no—
550C23—122no—
551C23—122no—
552C23—121no—
553C23—121no—
554C23—122no—
555C23—122no—
556C23—121no—
557C23—121no—
558C23C2×#23511yescharacteristic
559C23C2×D3×D412yes—
560C23C2×D3×D411yescharacteristic
561C23C2×D3×D411yescharacteristic
562C23#100412yes—
563C23C2×D3×Q811yescharacteristic
564C23C2×D3×Q811yescharacteristic
565C23#100712yes—
566C23D3×#4111yescharacteristic
567C23D3×#4112yes—
568C23D3×#4111yescharacteristic
569Q8—416no—
570Q8—1216no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—34no—
583C2×C4—38no—
584C2×C4—34no—
585C2×C4—38no—
586C2×C4—34no—
587C2×C4—34no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—64no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—1216no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4#88414yes—
630C2×C3C2×#39311yescharacteristic
631C2×C3#240811yescharacteristic
632C2×C3#240911yescharacteristic
633C2×C3#243912yes—
634C2×C3#244012yes—
635C2×C3#249112yes—
636C2×C3#249212yes—
637C2×C3#254711yescharacteristic
638C2×C3#256311yescharacteristic
639C2×C3#262811yescharacteristic
640C2×C3#285711yescharacteristic
641D3—44no—
642C22—68no—
643C22—64no—
644C22—122no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—124no—
649C22—124no—
650C22—122no—
651C22—124no—
652C22—122no—
653C22C2×#88411yescharacteristic
654C22#549012yes—
655C22#556812yes—
656C22#557811yescharacteristic
657C22#557811yescharacteristic
658C22D3×#12011yescharacteristic
659C22#559012yes—
660C22#559211yescharacteristic
661C22D3×C22⋊Q812yes—
662C22D3×C22⋊Q811yescharacteristic
663C22D3×C22⋊Q811yescharacteristic
664C22#563112yes—
665C22#563312yes—
666C22#563612yes—
667C22#564012yes—
668C22#566811yescharacteristic
669C22#567012yes—
670C22D3×#12412yes—
671C22D3×#12411yescharacteristic
672C22D3×C4⋊Q811yescharacteristic
673C22#572411yescharacteristic
674C22#572712yes—
675C22#572912yes—
676C4—24no—
677C4—38no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—64no—
682C4—82no—
683C4—124no—
684C4—122no—
685C4—242no—
686C3#1512511yescharacteristic, complemented, Sylow
687C2—124no—
688C2#7382411yescharacteristic
689C2#7650112yes—
690C2#7651712yes—
691C2#7653611yescharacteristic
692C2#7666411yescharacteristic
693C2#7679212yes—
694C2#7679911yescharacteristic
695C2D3×#39311yescharacteristic
696C2#7717211yescharacteristic
697C2#7717412yes—
698C2#7720211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1446C2
prime radicalC24×D3C23
hypercenter#1446D3
hypocenterC3#15125
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×#944C2semidirect
6. C3×#1446C2semidirect
8. #69424C2semidirect
9. #69414C2semidirect
10. Dic3⋊#95C2semidirect
11. Dic3⋊#95C2semidirect
48. #1446D3semidirect
73. C22×C4×D3Q8semidirect
91. C2×(C2×D3)⋊C4Q8semidirect
335. C22×D3C4⋊Q8semidirect
534. Dic3#393semidirect
686. C3#15125semidirect

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×#46672
C2×#46662
C3×#14462
#694282
#694242
#694142
Dic3⋊#952
#151253