Finite Group Databasebeta

← #10720446

Subgroups — #10720446

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#898C211yesmaximal, characteristic
2C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#4648C211yesmaximal, characteristic, complemented
4C3×#1441C211yesmaximal, characteristic, complemented
5#71570C211yesmaximal, characteristic
6#71565C211yesmaximal, characteristic, complemented
7#69427C211yesmaximal, characteristic
8#69408C212yesmaximal
9#69408C212yesmaximal
10#69407C212yesmaximal, complemented
11#69406C212yesmaximal, complemented
12#14166—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×#898—24no—
27C2×#898—24no—
28C2×#898C418yescomplemented
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39Dic3×C4⋊C4—24no—
40#4648—24no—
41C22×#126—31no—
42C2×C4×C4⋊C4—31no—
43C2×#376—31no—
44#1441—31no—
45#1441D311yescharacteristic, complemented
46#1439—31no—
47#1439—31no—
48#1437—32no—
49#1437—32no—
50#1436—32no—
51#1436—32no—
52C24×D3C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4C2×C414yes—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—42no—
79C2×(C2×D3)⋊C4—42no—
80C2×(C2×D3)⋊C4C2×C414yes—
81C2×(C2×D3)⋊C4C2×C414yes—
82C2×(C2×D3)⋊C4C2×C414yes—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3C2×C414yes—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4C2×C414yes—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—42no—
96C2×C4×Dic3—42no—
97C2×C4×Dic3C2×C414yes—
98C2×C4×Dic3D414yes—
99C2×C4×Dic3D414yes—
100#898—232no—
101C3×#95—22no—
102C3×#95—24no—
103C3×#95—42no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C2×#126—38no—
112C2×#126—64no—
113C2×#126—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C4⋊C4C2×D311yescharacteristic
119C22×C22⋊C4—31no—
120C22×C22⋊C4—31no—
121C22×C22⋊C4—31no—
122C22×C22⋊C4—32no—
123C22×C22⋊C4—32no—
124C22×C42—31no—
125C22×C42—32no—
126C22×C42—32no—
127C4×C4⋊C4—64no—
128#376—64no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95C2×D312yes—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C24×C3C2411yescharacteristic
140C23×D3—21no—
141C23×D3—21no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—21no—
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3C22×C412yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4C22×C412yes—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C22×C412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4#4114yes—
181C22×C3×C4#4114yes—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3C22×C412yes—
205C22×Dic3C22×C412yes—
206C22×Dic3C22×C412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D412yes—
211C22×Dic3#4114yes—
212C22×Dic3#4114yes—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—28no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—28no—
218(C2×D3)⋊C4—48no—
219(C2×D3)⋊C4—48no—
220C4⋊Dic3—28no—
221C4⋊Dic3—28no—
222Dic3⋊C4—28no—
223C4×Dic3—216no—
224C4×Dic3—48no—
225C4×Dic3C22⋊C4116yescomplemented
226C25—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—32no—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D312yes—
235C23×C4C22×D312yes—
236C23×C4C22×D311yescharacteristic
237#126—632no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—34no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4C4×D314yes—
251C2×C22⋊C4—34no—
252C2×C22⋊C4—34no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—122no—
265C2×C22⋊C4—122no—
266C2×C42—34no—
267C2×C42—34no—
268C2×C42—34no—
269C2×C42—64no—
270C2×C42—64no—
271C2×C42—64no—
272C2×C42—64no—
273C2×C42—122no—
274C2×C42—122no—
275#95—22no—
276#95—24no—
277#95—42no—
278#95—64no—
279#95—64no—
280#95—64no—
281#95—64no—
282#95—62no—
283#95—64no—
284#95—122no—
285C23×C3C23×C411yescharacteristic
286C23×C3C22×D411yescharacteristic
287C23×C3C22×D411yescharacteristic
288C23×C3C2×#4112yes—
289C23×C3C2×#4112yes—
290C23×C3C2×#4112yes—
291C23×C3C2×#4112yes—
292C23×C3H(5,𝔽2)11yescharacteristic
293C23×C3H(5,𝔽2)11yescharacteristic
294C23×C32−1+411yescharacteristic
295C23×C32−1+411yescharacteristic
296C22×D3—22no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4C2×C22⋊C414yes—
350C2×Dic3—24no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—42no—
369C2×Dic3—41no—
370C2×Dic3—42no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3C2×C22⋊C414yes—
380C2×Dic3C2×C22⋊C414yes—
381C24—32no—
382C24—61no—
383C24—61no—
384C24—62no—
385C24—61no—
386C24—62no—
387C24—61no—
388C24—62no—
389C24—62no—
390C24—61no—
391C24—61no—
392C24C23×D311yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
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—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—34no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
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×C4×D312yes—
449C22×C4C2×C4×D312yes—
450C22×C4C2×C4×D312yes—
451C22×C4D3×D412yes—
452C22×C4D3×D412yes—
453C22×C4#23714yes—
454C22×C4#23914yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—68no—
458C4⋊C4—68no—
459C4⋊C4—68no—
460C22⋊C4—68no—
461C22⋊C4—68no—
462C22⋊C4—68no—
463C22⋊C4—128no—
464C22⋊C4—128no—
465C42—316no—
466C42—616no—
467C42—128no—
468C22×C3C22×C22⋊C411yescharacteristic
469C22×C3C4×#4112yes—
470C22×C3C4×#4112yes—
471C22×C3#52011yescharacteristic
472C22×C3#52011yescharacteristic
473C22×C3#52511yescharacteristic
474C22×C3#52511yescharacteristic
475C22×C3#52512yes—
476C22×C3#53211yescharacteristic
477C22×C3#53211yescharacteristic
478C22×C3#53212yes—
479C22×C3#53511yescharacteristic
480C22×C3#53611yescharacteristic
481C22×C3#53711yescharacteristic
482C22×C3#53711yescharacteristic
483C22×C3#55112yes—
484C22×C3#55112yes—
485C22×C3#55212yes—
486C22×C3#55212yes—
487C22×C3#55312yes—
488C22×C3#55312yes—
489C22×C3#55612yes—
490C22×C3#55612yes—
491C2×D3—42no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C3×C4—24no—
503C3×C4—42no—
504C3×C4—44no—
505C3×C4—44no—
506C3×C4—42no—
507Dic3—24no—
508Dic3—24no—
509Dic3—44no—
510Dic3—44no—
511Dic3—42no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23—64no—
516C23—64no—
517C23—122no—
518C23—122no—
519C23—121no—
520C23—122no—
521C23—121no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—121no—
526C23—122no—
527C23—121no—
528C23—122no—
529C23—122no—
530C23—121no—
531C23—121no—
532C23—122no—
533C23—121no—
534C23—121no—
535C23C22×C4×D311yescharacteristic
536C23C2×D3×D411yescharacteristic
537C23C2×D3×D411yescharacteristic
538C23C2×#23712yes—
539C23#100411yescharacteristic
540C23C2×#23912yes—
541C23#100711yescharacteristic
542C23D3×#4112yes—
543C23D3×#4112yes—
544C23#100911yescharacteristic
545C23#101011yescharacteristic
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—34no—
562C2×C4—34no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—64no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—121no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4D3×C22⋊C414yes—
608C2×C3#229711yescharacteristic
609C2×C3#229711yescharacteristic
610C2×C3#231111yescharacteristic
611C2×C3#237312yes—
612C2×C3#237312yes—
613C2×C3#260112yes—
614C2×C3#260112yes—
615C2×C3#260311yescharacteristic
616C2×C3#260311yescharacteristic
617C2×C3#260411yescharacteristic
618C2×C3#260411yescharacteristic
619D3—82no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—124no—
624C22—124no—
625C22—242no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—242no—
630C22—241no—
631C22C2×D3×C22⋊C411yescharacteristic
632C22#550411yescharacteristic
633C22#552311yescharacteristic
634C22C4×#23712yes—
635C22C4×#23912yes—
636C22#558611yescharacteristic
637C22#559612yes—
638C22#560112yes—
639C22#560711yescharacteristic
640C22#560811yescharacteristic
641C22#562712yes—
642C22#562811yescharacteristic
643C22#562911yescharacteristic
644C22#563712yes—
645C22#564212yes—
646C22#565611yescharacteristic
647C22#565812yes—
648C22#566811yescharacteristic
649C22#569312yes—
650C22#569512yes—
651C22#570212yes—
652C22#572511yescharacteristic
653C22#573212yes—
654C4—24no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—64no—
660C4—64no—
661C4—122no—
662C4—124no—
663C4—124no—
664C3#1416611yescharacteristic, complemented, Sylow
665C2—242no—
666C2#7400311yescharacteristic
667C2#7465811yescharacteristic
668C2#7497412yes—
669C2#7599112yes—
670C2#7647811yescharacteristic
671C2#7696711yescharacteristic
672C2#7719711yescharacteristic
673C2#7732712yes—
674C2#7781112yes—
675C2#7781211yescharacteristic
676C2#7820411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1441C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenter#1441D3
hypocenterC3#14166
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×C4⋊C4C2semidirect
3. C2×#4648C2semidirect
4. C3×#1441C2semidirect
6. #71565C2semidirect
10. #69407C2semidirect
11. #69406C2semidirect
28. C2×#898C4semidirect
45. #1441D3semidirect
225. C4×Dic3C22⋊C4semidirect
664. C3#14166semidirect

Maximal subgroups

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

subgroupindex
C22×#8982
C2×Dic3×C4⋊C42
C2×#46482
C3×#14412
#715702
#715652
#694272
#694082
#694072
#694062
#141663