Finite Group Databasebeta

← #10720456

Subgroups — #10720456

Order 768.

2058 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#883C211yesmaximal, characteristic
2C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
3C2×#4666C211yesmaximal, characteristic, complemented
4C3×#1437C211yesmaximal, characteristic, complemented
5#71550C212yesmaximal
6#71545C212yesmaximal, complemented
7#71545C211yesmaximal, characteristic, complemented
8#71536C211yesmaximal, characteristic, complemented
9#69429C212yesmaximal, complemented
10#69407C212yesmaximal, complemented
11#69405C211yesmaximal, characteristic, complemented
12#14167—31nomaximal, Sylow
13C23×#67C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2212yes—
27C2×#883—24no—
28C2×#883—24no—
29C2×#883C418yescomplemented
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C4×(C2×D3)⋊C4—24no—
40#4666—24no—
41C22×#123—31no—
42C2×C4×C22⋊C4—31no—
43C2×C4×C22⋊C4—31no—
44#1440—32no—
45#1438—32no—
46#1437—32no—
47#1437—31no—
48#1437D311yescharacteristic, complemented
49#1436—31no—
50#1436—31no—
51#1436—32no—
52C25×C3C2311yescharacteristic
53C24×D3C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C22×#67—22no—
59C22×#67—22no—
60C22×#67—42no—
61C22×#67—42no—
62C22×#67C2×C414yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C22×C4×D3—42no—
69C22×C4×D3—42no—
70C22×C4×D3C2×C414yes—
71C22×C4×D3D414yes—
72C22×C4×D3D414yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4C2×C414yes—
79C2×C3×C42—22no—
80C2×C3×C42—42no—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—42no—
86C2×#217C2×C414yes—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—24no—
94C2×(C2×D3)⋊C4—24no—
95C2×(C2×D3)⋊C4C2×C414yes—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3C2×C414yes—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4C2×C414yes—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—42no—
105C2×C4×Dic3—42no—
106C2×C4×Dic3—42no—
107#883—232no—
108C3×#95—22no—
109C3×#95—24no—
110C3⋊#95—22no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C3⋊#95—42no—
117C23×D4—31no—
118C24×C4—31no—
119C2×#123—38no—
120C2×#123—64no—
121C2×#123—64no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C22⋊C4—31no—
125C22×C22⋊C4—31no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4C2×D312yes—
130C22×C22⋊C4C2×D311yescharacteristic
131C22×C42—32no—
132C22×C42—31no—
133C22×C42C2×D311yescharacteristic
134C4×C22⋊C4—64no—
135C4×C22⋊C4—64no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95C2×D312yes—
144C2×#95C2×D311yescharacteristic
145C24×C3—21no—
146C24×C3—21no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3C22×C412yes—
154C24×C3C2411yescharacteristic
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—22no—
161C23×D3C22×C412yes—
162C23×D3C2×D412yes—
163C23×D3C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
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—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C22×C412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4114yes—
187C22×C3×C4#4114yes—
188C2×#67—28no—
189C2×#67—48no—
190C2×#67—44no—
191C2×#67—44no—
192C2×#67—48no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
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×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3C22×C412yes—
216C22×Dic3C22×C412yes—
217C22×Dic3C22×C412yes—
218C22×Dic3C2×D412yes—
219C22×Dic3C2×D412yes—
220C22×Dic3#4114yes—
221C22×Dic3#4114yes—
222C2×C4×D3—48no—
223C2×C4×D3—48no—
224C2×C4×D3—44no—
225C2×C4×D3—44no—
226C2×C4×D3C22⋊C4116yescomplemented
227C3×C22⋊C4—28no—
228C3×C22⋊C4—48no—
229C3×C42—44no—
230#217—28no—
231#217—48no—
232(C2×D3)⋊C4—216no—
233(C2×D3)⋊C4—216no—
234(C2×D3)⋊C4—28no—
235C4⋊Dic3—28no—
236Dic3⋊C4—28no—
237C4×Dic3—48no—
238C4×Dic3—44no—
239C25—31no—
240C25C22×D311yescharacteristic
241C22×D4—34no—
242C22×D4—62no—
243C22×D4—62no—
244C22×D4—122no—
245C22×D4—122no—
246C23×C4—31no—
247C23×C4—34no—
248C23×C4—34no—
249C23×C4—34no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4—31no—
254C23×C4—122no—
255C23×C4—122no—
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D312yes—
258C23×C4C22×D312yes—
259C23×C4C22×D311yescharacteristic
260#123—632no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—122no—
287C2×C22⋊C4C4×D314yes—
288C2×C42—22no—
289C2×C42—42no—
290C2×C42—64no—
291C2×C42—62no—
292C2×C42—122no—
293C2×C42—122no—
294C2×C42—122no—
295#95—22no—
296#95—24no—
297#95—64no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—62no—
302#95—122no—
303#95—122no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3C23×C411yescharacteristic
326C23×C3C22×D411yescharacteristic
327C23×C3C22×D411yescharacteristic
328C23×C3C2×#4112yes—
329C23×C3C2×#4112yes—
330C23×C3C2×#4112yes—
331C23×C3C2×#4112yes—
332C23×C3H(5,𝔽2)12yes—
333C23×C3H(5,𝔽2)12yes—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—24no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—24no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3C2×C22⋊C414yes—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4C2×C22⋊C414yes—
379#67—416no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3C2×C22⋊C414yes—
409C4×D3—416no—
410C24—22no—
411C24—21no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—32no—
419C24—32no—
420C24—32no—
421C24—62no—
422C24—62no—
423C24—62no—
424C24—62no—
425C24—62no—
426C24—62no—
427C24C2×C4×D312yes—
428C24C23×D311yescharacteristic
429C2×D4—68no—
430C2×D4—128no—
431C2×D4—124no—
432C2×D4—124no—
433C2×D4—128no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—316no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—34no—
459C22×C4—34no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—124no—
483C22×C4—128no—
484C22×C4—124no—
485C22×C4—128no—
486C22×C4C2×C4×D312yes—
487C22×C4C2×C4×D312yes—
488C22×C4#23514yes—
489C22×C4D3×D412yes—
490C22×C4D3×D412yes—
491C22×C4#23714yes—
492C4⋊C4—68no—
493C4⋊C4—68no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—68no—
497C22⋊C4—616no—
498C22⋊C4—616no—
499C22⋊C4—68no—
500C22⋊C4—128no—
501C42—44no—
502C42—124no—
503C42—128no—
504C22×C3—42no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—82no—
510C22×C3—82no—
511C22×C3—82no—
512C22×C3—82no—
513C22×C3C22×C22⋊C411yescharacteristic
514C22×C3C4×#4111yescharacteristic
515C22×C3C4×#4111yescharacteristic
516C22×C3C4×#4112yes—
517C22×C3#51812yes—
518C22×C3#52512yes—
519C22×C3#52512yes—
520C22×C3#53211yescharacteristic
521C22×C3#53211yescharacteristic
522C22×C3#53212yes—
523C22×C3#53411yescharacteristic
524C22×C3#53411yescharacteristic
525C22×C3#53412yes—
526C22×C3#55112yes—
527C22×C3#55112yes—
528C22×C3#55312yes—
529C22×C3#55312yes—
530C22×C3#55312yes—
531C22×C3#55312yes—
532C22×C3#55912yes—
533C22×C3#55912yes—
534C2×D3—24no—
535C2×D3—24no—
536C2×D3—24no—
537C2×D3—44no—
538C2×D3—44no—
539C2×D3—44no—
540C2×D3—44no—
541C2×D3—44no—
542C2×D3—44no—
543C3×C4—24no—
544C3×C4—44no—
545C3×C4—44no—
546C3×C4—42no—
547Dic3—24no—
548Dic3—42no—
549Dic3—44no—
550Dic3—44no—
551Dic3—42no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—34no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—64no—
580C23—64no—
581C23—64no—
582C23—64no—
583C23—62no—
584C23—64no—
585C23—64no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23C22×C4×D311yescharacteristic
595C23C2×#23512yes—
596C23C2×D3×D411yescharacteristic
597C23C2×D3×D411yescharacteristic
598C23C2×#23712yes—
599C23#100412yes—
600C23D3×#4112yes—
601C23D3×#4112yes—
602C23#100912yes—
603D4—1216no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—34no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—62no—
629C2×C4—64no—
630C2×C4—62no—
631C2×C4—64no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—64no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—64no—
640C2×C4—64no—
641C2×C4—64no—
642C2×C4—64no—
643C2×C4—1216no—
644C2×C4—122no—
645C2×C4—121no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4D3×C22⋊C414yes—
658C2×C3—82no—
659C2×C3#229711yescharacteristic
660C2×C3#229711yescharacteristic
661C2×C3#230811yescharacteristic
662C2×C3#237412yes—
663C2×C3#237412yes—
664C2×C3#260212yes—
665C2×C3#260212yes—
666C2×C3#260212yes—
667C2×C3#260212yes—
668D3—44no—
669C22—42no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—64no—
675C22—64no—
676C22—64no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—124no—
682C22—124no—
683C22—124no—
684C22—124no—
685C22—124no—
686C22—124no—
687C22C4×#23511yescharacteristic
688C22C2×D3×C22⋊C411yescharacteristic
689C22#549512yes—
690C22#552211yescharacteristic
691C22#553012yes—
692C22C4×#23712yes—
693C22#554512yes—
694C22#554712yes—
695C22#554812yes—
696C22#555912yes—
697C22#556512yes—
698C22#558911yescharacteristic
699C22#560112yes—
700C22#560512yes—
701C22#560612yes—
702C22#565311yescharacteristic
703C22#566112yes—
704C22#566811yescharacteristic
705C22#568112yes—
706C22#570512yes—
707C22#571611yescharacteristic
708C4—24no—
709C4—42no—
710C4—44no—
711C4—44no—
712C4—64no—
713C4—122no—
714C4—122no—
715C4—124no—
716C4—124no—
717C3#1416711yescharacteristic, complemented, Sylow
718C2—82no—
719C2—124no—
720C2#7347912yes—
721C2#7356311yescharacteristic
722C2#7487411yescharacteristic
723C2#7490412yes—
724C2#7548712yes—
725C2#7564612yes—
726C2#7589812yes—
727C2#7629011yescharacteristic
728C2#7637012yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1437C2
socle, cosocleC24×C3C24
prime radicalC23×#67C22
hypercenter#1437D3
hypocenterC3#14167
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×C4×(C2×D3)⋊C4C2semidirect
3. C2×#4666C2semidirect
4. C3×#1437C2semidirect
6. #71545C2semidirect
7. #71545C2semidirect
8. #71536C2semidirect
9. #69429C2semidirect
10. #69407C2semidirect
11. #69405C2semidirect
29. C2×#883C4semidirect
48. #1437D3semidirect
226. C2×C4×D3C22⋊C4semidirect
717. C3#14167semidirect

Maximal subgroups

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

subgroupindex
C22×#8832
C2×C4×(C2×D3)⋊C42
C2×#46662
C3×#14372
#715502
#715452
#715362
#694292
#694072
#694052
#141673