Finite Group Databasebeta

← #10732285

Subgroups — #10732285

Order 768.

1210 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4984C211yesmaximal, characteristic
3#71567C212yesmaximal
4#71565C212yesmaximal
5#71563C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7#71563C211yesmaximal, characteristic
8#71535C211yesmaximal, characteristic
9C4×C3⋊#95C211yesmaximal, characteristic
10#69423C212yesmaximal
11#69417C212yesmaximal
12#15187—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×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C3×#389—24no—
23C2×C3×#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4984—24no—
38C2×#389—31no—
39C2×#389D311yescharacteristic
40#1445—32no—
41#1443—32no—
42#1442—32no—
43#1441—32no—
44C4⋊#95—31no—
45C4⋊#95—31no—
46C4⋊#95—31no—
47C4⋊#95—31no—
48C4×#95—31no—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4D414yes—
68C2×C3×C4⋊C4Q814yes—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—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—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—22no—
87C3×#95—24no—
88C3⋊#95—24no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—24no—
93C3⋊#95—42no—
94C3⋊#95—42no—
95C3⋊#95—42no—
96C3⋊#95—42no—
97C3⋊#95—42no—
98C3⋊#95—42no—
99C3⋊#95—42no—
100C3⋊#95Q814yes—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—32no—
104C22×C4⋊C4C2×D312yes—
105C22×C4⋊C4C2×D311yescharacteristic
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C42—31no—
108C22×C42—31no—
109C22×C42C2×D311yescharacteristic
110#389—24no—
111#389—64no—
112C2×#95—31no—
113C2×#95—31no—
114C2×#95—31no—
115C2×#95—31no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95C2×D312yes—
126C24×C3C2411yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×Q814yes—
154C22×C3×C4C2×Q812yes—
155C22×C3×C4C2×Q812yes—
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4#4112yes—
160C22×C3×C4#4112yes—
161C22×C3×C4#4112yes—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
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×Dic3C2×Q812yes—
201C22×Dic3C2×Q812yes—
202C22×Dic3#4112yes—
203C22×Dic3#4112yes—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—48no—
207C3×C42—44no—
208C4⋊Dic3—48no—
209C4⋊Dic3—44no—
210C4⋊Dic3—44no—
211C4×Dic3—44no—
212C4×Dic3—44no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4C22×D312yes—
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C23×C4C22×D312yes—
223C23×C4C22×D311yescharacteristic
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—122no—
240C2×C4⋊C4Dic614yes—
241C2×C4⋊C4#6714yes—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—62no—
246C2×C42—62no—
247C2×C42—62no—
248C2×C42—62no—
249C2×C42—62no—
250C2×C42—62no—
251#95—24no—
252#95—34no—
253#95—62no—
254#95—64no—
255#95—62no—
256#95—62no—
257#95—64no—
258#95—122no—
259#95—122no—
260#95—122no—
261#95—122no—
262#95—122no—
263#95—122no—
264#95—122no—
265C23×C3C22×D411yescharacteristic
266C23×C3C22×Q811yescharacteristic
267C23×C3C22×Q811yescharacteristic
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4111yescharacteristic
270C23×C3C2×#4111yescharacteristic
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4111yescharacteristic
274C23×C3C2×#4111yescharacteristic
275C23×C3C2×#4111yescharacteristic
276C23×C3H(5,𝔽2)11yescharacteristic
277C23×C3H(5,𝔽2)11yescharacteristic
278C23×C3H(5,𝔽2)11yescharacteristic
279C23×C32−1+411yescharacteristic
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4C22⋊Q814yes—
309C2×C3×C4#12314yes—
310C2×C3×C4#12414yes—
311C2×C3×C4#12514yes—
312C2×C3×C4#12514yes—
313C2×C3×C4#12614yes—
314C2×C3×C4#12614yes—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C24C23×D311yescharacteristic
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4C2×Dic614yes—
425C22×C4C2×Dic612yes—
426C22×C4#23512yes—
427C22×C4#23712yes—
428C22×C4#23712yes—
429C22×C4#23712yes—
430C22×C4D3×Q812yes—
431C22×C4#23912yes—
432C22×C4#23912yes—
433C22×C4C2×#6714yes—
434C22×C4C2×#6712yes—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—48no—
438C4⋊C4—128no—
439C4⋊C4—124no—
440C4⋊C4—124no—
441C42—44no—
442C42—124no—
443C42—124no—
444C22×C3C2×C22⋊Q811yescharacteristic
445C22×C3C2×#12311yescharacteristic
446C22×C3C2×#12411yescharacteristic
447C22×C3C2×#12511yescharacteristic
448C22×C3C2×#12511yescharacteristic
449C22×C3C2×#12611yescharacteristic
450C22×C3C2×#12611yescharacteristic
451C22×C3#52911yescharacteristic
452C22×C3#52911yescharacteristic
453C22×C3#52911yescharacteristic
454C22×C3#53311yescharacteristic
455C22×C3#53311yescharacteristic
456C22×C3#53311yescharacteristic
457C22×C3#54011yescharacteristic
458C22×C3#54011yescharacteristic
459C22×C3#54011yescharacteristic
460C22×C3#54211yescharacteristic
461C22×C3#54211yescharacteristic
462C22×C3#54211yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3D4×Q811yescharacteristic
466C22×C3#55011yescharacteristic
467C22×C3#55211yescharacteristic
468C22×C3#55311yescharacteristic
469C22×C3#55311yescharacteristic
470C22×C3#55311yescharacteristic
471C22×C3#55411yescharacteristic
472C22×C3#55411yescharacteristic
473C22×C3#55411yescharacteristic
474C22×C3#55411yescharacteristic
475C22×C3#55511yescharacteristic
476C22×C3#55611yescharacteristic
477C22×C3#55711yescharacteristic
478C22×C3Q8211yescharacteristic
479C3×C4—44no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—44no—
483C3×C4#40018yes—
484Dic3—42no—
485Dic3—42no—
486Dic3—42no—
487Dic3—42no—
488Dic3—82no—
489Dic3—82no—
490C23C22×Dic611yescharacteristic
491C23C2×#23511yescharacteristic
492C23C2×#23711yescharacteristic
493C23C2×#23711yescharacteristic
494C23C2×#23711yescharacteristic
495C23#100411yescharacteristic
496C23C2×D3×Q811yescharacteristic
497C23C2×#23911yescharacteristic
498C23C2×#23911yescharacteristic
499C23#100711yescharacteristic
500C23D3×#4111yescharacteristic
501C23D3×#4111yescharacteristic
502C23#100911yescharacteristic
503C23#100911yescharacteristic
504C23C22×#6711yescharacteristic
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4#89014yes—
576C2×C4#89014yes—
577C2×C4#89814yes—
578C2×C4#89814yes—
579C2×C4#92414yes—
580C2×C4#93514yes—
581C2×C4#94714yes—
582C2×C3C2×#40011yescharacteristic
583C2×C3#242011yescharacteristic
584C2×C3#242311yescharacteristic
585C2×C3#245111yescharacteristic
586C2×C3#245411yescharacteristic
587C2×C3#249011yescharacteristic
588C2×C3#249211yescharacteristic
589C2×C3#249411yescharacteristic
590C2×C3#249711yescharacteristic
591C2×C3#250711yescharacteristic
592C2×C3#250711yescharacteristic
593C2×C3#250811yescharacteristic
594C2×C3#250911yescharacteristic
595C2×C3#264411yescharacteristic
596C2×C3#286311yescharacteristic
597C22C2×#89011yescharacteristic
598C22C2×#89011yescharacteristic
599C22C2×#89811yescharacteristic
600C22C2×#89811yescharacteristic
601C22#553911yescharacteristic
602C22#554311yescharacteristic
603C22#555711yescharacteristic
604C22#556211yescharacteristic
605C22Q8×Dic611yescharacteristic
606C22#556911yescharacteristic
607C22#557811yescharacteristic
608C22#558211yescharacteristic
609C22#560011yescharacteristic
610C22#561611yescharacteristic
611C22#561811yescharacteristic
612C22#561911yescharacteristic
613C22#562211yescharacteristic
614C22#563511yescharacteristic
615C22#563911yescharacteristic
616C22#564311yescharacteristic
617C22#569111yescharacteristic
618C22#569511yescharacteristic
619C22#569511yescharacteristic
620C22#569711yescharacteristic
621C22#572811yescharacteristic
622C22#573011yescharacteristic
623C22#573211yescharacteristic
624C22#573311yescharacteristic
625C22C2×#92411yescharacteristic
626C22C2×#93511yescharacteristic
627C22#580611yescharacteristic
628C22C2×#94711yescharacteristic
629C22Q8×#6711yescharacteristic
630C22#582911yescharacteristic
631C22#583411yescharacteristic
632C4—42no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—122no—
637C4—122no—
638C4—122no—
639C4—122no—
640C4—242no—
641C4—242no—
642C4#498618yes—
643C3#1518711yescharacteristic, complemented, Sylow
644C2#7603211yescharacteristic
645C2#7603511yescharacteristic
646C2#7603711yescharacteristic
647C2#7604411yescharacteristic
648C2#7619211yescharacteristic
649C2#7623311yescharacteristic
650C2#7623911yescharacteristic
651C2#7625211yescharacteristic
652C2C2×#498611yescharacteristic
653C2#8236311yescharacteristic
654C2#8247011yescharacteristic
655C2#8264311yescharacteristic
656C2#8275411yescharacteristic
657C2#8306111yescharacteristic
658C2#8325811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#15187
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
643. C3#15187semidirect

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
#715672
#715652
#715632
#715352
C4×C3⋊#952
#694232
#694172
#151873