Finite Group Databasebeta

← #10732333

Subgroups — #10732333

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4984C211yesmaximal, characteristic
3C2×#4972C211yesmaximal, characteristic
4C2×C4×Dic3⋊C4C211yesmaximal, characteristic
5#71568C212yesmaximal
6#71565C212yesmaximal
7#71563C211yesmaximal, characteristic
8#71563C211yesmaximal, characteristic
9#71535C211yesmaximal, characteristic
10#69421C212yesmaximal
11#69410C212yesmaximal
12#15194—31nomaximal, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C3×#389—24no—
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37#4984—24no—
38#4972—24no—
39C4×Dic3⋊C4—24no—
40C2×#389—31no—
41C2×#389—31no—
42C2×#389D311yescharacteristic
43C2×C4×C4⋊C4—31no—
44#1445—32no—
45#1444—32no—
46#1441—32no—
47#1441—32no—
48C4⋊#95—31no—
49C4⋊#95—31no—
50C4⋊#95—31no—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4D414yes—
70C2×C3×C4⋊C4Q814yes—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—42no—
81C2×C4⋊Dic3—42no—
82C2×C4⋊Dic3Q814yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—42no—
91C2×Dic3⋊C4Q814yes—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3D414yes—
98C3×#95—24no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4C2×D312yes—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C42—31no—
119C22×C42—31no—
120C22×C42C2×D311yescharacteristic
121#389—24no—
122#389—64no—
123#389—64no—
124C4×C4⋊C4—64no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95C2×D312yes—
136C24×C3C2411yescharacteristic
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×Q814yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4C2×Q812yes—
167C22×C3×C4C2×Q812yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
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—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×Q812yes—
210C22×Dic3C2×Q814yes—
211C22×Dic3C2×Q812yes—
212C22×Dic3#4112yes—
213C22×Dic3#4112yes—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—28no—
216C3×C4⋊C4—48no—
217C3×C42—44no—
218C4⋊Dic3—28no—
219C4⋊Dic3—48no—
220C4⋊Dic3—44no—
221Dic3⋊C4—28no—
222Dic3⋊C4—28no—
223Dic3⋊C4—48no—
224C4×Dic3—28no—
225C4×Dic3—44no—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—32no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—32no—
232C23×C4C22×D312yes—
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D312yes—
236C23×C4C22×D311yescharacteristic
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—34no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—122no—
260C2×C4⋊C4—122no—
261C2×C4⋊C4—122no—
262C2×C4⋊C4Dic614yes—
263C2×C4⋊C4#6714yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—34no—
268C2×C42—62no—
269C2×C42—62no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42—62no—
273#95—24no—
274#95—64no—
275#95—62no—
276#95—62no—
277#95—64no—
278#95—64no—
279#95—122no—
280#95—122no—
281#95—122no—
282#95—122no—
283#95—122no—
284C23×C3C22×D411yescharacteristic
285C23×C3C22×D411yescharacteristic
286C23×C3C22×Q811yescharacteristic
287C23×C3C22×Q811yescharacteristic
288C23×C3C22×Q811yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3H(5,𝔽2)11yescharacteristic
296C23×C3H(5,𝔽2)11yescharacteristic
297C23×C3H(5,𝔽2)11yescharacteristic
298C23×C32−1+411yescharacteristic
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4#12114yes—
328C2×C3×C4C22⋊Q814yes—
329C2×C3×C4C22⋊Q814yes—
330C2×C3×C4#12314yes—
331C2×C3×C4#12514yes—
332C2×C3×C4#12514yes—
333C2×C3×C4C4⋊Q814yes—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3C22⋊Q814yes—
376C24C23×D311yescharacteristic
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—34no—
407C22×C4—32no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
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×C4C2×Dic614yes—
444C22×C4C2×Dic612yes—
445C22×C4#23512yes—
446C22×C4D3×D412yes—
447C22×C4#23712yes—
448C22×C4#23712yes—
449C22×C4D3×Q812yes—
450C22×C4D3×Q812yes—
451C22×C4#23912yes—
452C22×C4C2×#6714yes—
453C22×C4C2×#6712yes—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—48no—
457C4⋊C4—68no—
458C4⋊C4—68no—
459C4⋊C4—68no—
460C4⋊C4—128no—
461C4⋊C4—128no—
462C4⋊C4—124no—
463C42—44no—
464C42—68no—
465C42—124no—
466C22×C3C2×#12111yescharacteristic
467C22×C3C2×C22⋊Q811yescharacteristic
468C22×C3C2×C22⋊Q811yescharacteristic
469C22×C3C2×C22⋊Q811yescharacteristic
470C22×C3C2×#12311yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3C2×#12511yescharacteristic
473C22×C3C2×#12511yescharacteristic
474C22×C3#52911yescharacteristic
475C22×C3#52911yescharacteristic
476C22×C3C2×C4⋊Q811yescharacteristic
477C22×C3#53311yescharacteristic
478C22×C3#53311yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3#53711yescharacteristic
481C22×C3#54011yescharacteristic
482C22×C3#54011yescharacteristic
483C22×C3#54211yescharacteristic
484C22×C3#54211yescharacteristic
485C22×C3#54611yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3D4×Q811yescharacteristic
488C22×C3#55011yescharacteristic
489C22×C3#55311yescharacteristic
490C22×C3#55311yescharacteristic
491C22×C3#55411yescharacteristic
492C22×C3#55411yescharacteristic
493C22×C3#55411yescharacteristic
494C22×C3#55411yescharacteristic
495C22×C3#55411yescharacteristic
496C22×C3#55611yescharacteristic
497C22×C3#55711yescharacteristic
498C22×C3#55711yescharacteristic
499C22×C3Q8211yescharacteristic
500C22×C3Q8211yescharacteristic
501C3×C4—44no—
502C3×C4—44no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4#39818yes—
506Dic3—24no—
507Dic3—42no—
508Dic3—44no—
509Dic3—42no—
510Dic3—42no—
511Dic3—82no—
512C23C22×Dic611yescharacteristic
513C23C2×#23511yescharacteristic
514C23C2×D3×D411yescharacteristic
515C23C2×#23711yescharacteristic
516C23C2×#23711yescharacteristic
517C23#100411yescharacteristic
518C23C2×D3×Q811yescharacteristic
519C23C2×D3×Q811yescharacteristic
520C23C2×#23911yescharacteristic
521C23#100711yescharacteristic
522C23D3×#4111yescharacteristic
523C23D3×#4111yescharacteristic
524C23#100911yescharacteristic
525C23#100911yescharacteristic
526C23C22×#6711yescharacteristic
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—34no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—121no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4#88814yes—
598C2×C4#88914yes—
599C2×C4#89014yes—
600C2×C4#89414yes—
601C2×C4#92414yes—
602C2×C4#93914yes—
603C2×C4#94614yes—
604C2×C3C2×#39811yescharacteristic
605C2×C3#245111yescharacteristic
606C2×C3#245111yescharacteristic
607C2×C3#245411yescharacteristic
608C2×C3#245411yescharacteristic
609C2×C3#250711yescharacteristic
610C2×C3#250711yescharacteristic
611C2×C3#250911yescharacteristic
612C2×C3#250911yescharacteristic
613C2×C3#253511yescharacteristic
614C2×C3#253811yescharacteristic
615C2×C3#256711yescharacteristic
616C2×C3#258411yescharacteristic
617C2×C3#262411yescharacteristic
618C2×C3#285511yescharacteristic
619C22C2×#88811yescharacteristic
620C22C2×#88911yescharacteristic
621C22C2×#89011yescharacteristic
622C22C2×#89411yescharacteristic
623C22#553911yescharacteristic
624C22#554311yescharacteristic
625C22#555711yescharacteristic
626C22#556211yescharacteristic
627C22Q8×Dic611yescharacteristic
628C22#556911yescharacteristic
629C22#557811yescharacteristic
630C22#558211yescharacteristic
631C22#560611yescharacteristic
632C22#560711yescharacteristic
633C22#561411yescharacteristic
634C22#561811yescharacteristic
635C22D3×C22⋊Q811yescharacteristic
636C22#562911yescharacteristic
637C22#563511yescharacteristic
638C22#564111yescharacteristic
639C22#569011yescharacteristic
640C22#569011yescharacteristic
641C22#569111yescharacteristic
642C22#569711yescharacteristic
643C22#572111yescharacteristic
644C22#572211yescharacteristic
645C22#572811yescharacteristic
646C22#573311yescharacteristic
647C22C2×#92411yescharacteristic
648C22C2×#93911yescharacteristic
649C22#580611yescharacteristic
650C22C2×#94611yescharacteristic
651C22Q8×#6711yescharacteristic
652C22#582911yescharacteristic
653C22#583411yescharacteristic
654C4—42no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—64no—
659C4—124no—
660C4—122no—
661C4—122no—
662C4—122no—
663C4—242no—
664C4#498218yes—
665C3#1519411yescharacteristic, complemented, Sylow
666C2#7603211yescharacteristic
667C2#7603311yescharacteristic
668C2#7604111yescharacteristic
669C2#7604211yescharacteristic
670C2#7607111yescharacteristic
671C2#7607811yescharacteristic
672C2#7624911yescharacteristic
673C2#7628711yescharacteristic
674C2C2×#498211yescharacteristic
675C2#8236311yescharacteristic
676C2#8247011yescharacteristic
677C2#8266611yescharacteristic
678C2#8277811yescharacteristic
679C2#8303011yescharacteristic
680C2#8323511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#15194
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
665. C3#15194semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49842
C2×#49722
C2×C4×Dic3⋊C42
#715682
#715652
#715632
#715352
#694212
#694102
#151943