Finite Group Databasebeta

← #10732272

Subgroups — #10732272

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2C2×#5229C211yesmaximal, characteristic
3C2×#5229C211yesmaximal, characteristic
4C2×#4928C211yesmaximal, characteristic
5#71563C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7C4×C3⋊#95C211yesmaximal, characteristic
8#69421C212yesmaximal
9#69418C212yesmaximal
10#69417C212yesmaximal
11#69413C212yesmaximal
12#15186—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C3×#391—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24#5229—24no—
25#5229—24no—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2211yescharacteristic
39#4928—24no—
40C2×#391—31no—
41C2×#391—31no—
42C2×#391D311yescharacteristic
43C2×#389—31no—
44#1445—32no—
45#1443—32no—
46#1443—32no—
47#1442—32no—
48C4⋊#95—31no—
49C4⋊#95—31no—
50C4×#95—31no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8D414yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D414yes—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—42no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C3×#95—24no—
91C3×#95—24no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95Q814yescomplemented
105C23×Q8C2×D311yescharacteristic
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C42—31no—
111C22×C42—31no—
112C22×C42C2×D311yescharacteristic
113#391—24no—
114#391—64no—
115#391—64no—
116#389—64no—
117C2×#95—31no—
118C2×#95—31no—
119C2×#95—31no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95C2×D312yes—
131C2×#95C2×D312yes—
132C24×C3C2411yescharacteristic
133C2×C3×Q8—28no—
134C2×C3×Q8—28no—
135C2×C3×Q8—28no—
136C2×C3×Q8—44no—
137C2×C3×Q8—44no—
138C2×C3×Q8—44no—
139C2×C3×Q8—44no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
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—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
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×C4C2×D412yes—
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×Q812yes—
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
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—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×Q812yes—
214C22×Dic3C2×Q812yes—
215C22×Dic3#4112yes—
216C22×Dic3#4112yes—
217C3×C4⋊C4—44no—
218C3×C42—44no—
219C4⋊Dic3—28no—
220C4⋊Dic3—44no—
221Dic3⋊C4—48no—
222C4×Dic3—44no—
223C4×Dic3—44no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8#6714yes—
231C23×C4—32no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D312yes—
241C23×C4C22×D311yescharacteristic
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—122no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—62no—
258C2×C42—62no—
259C2×C42—62no—
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—62no—
263#95—24no—
264#95—24no—
265#95—34no—
266#95—62no—
267#95—62no—
268#95—64no—
269#95—64no—
270#95—64no—
271#95—64no—
272#95—64no—
273#95—62no—
274#95—64no—
275#95—64no—
276#95—122no—
277#95—122no—
278C23×C3C22×D411yescharacteristic
279C23×C3C22×D411yescharacteristic
280C23×C3C22×Q811yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3H(5,𝔽2)11yescharacteristic
290C23×C32−1+411yescharacteristic
291C23×C32−1+411yescharacteristic
292C23×C32−1+411yescharacteristic
293C3×Q8—416no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4#12114yes—
323C2×C3×C4#12314yes—
324C2×C3×C4#12314yes—
325C2×C3×C4#12314yes—
326C2×C3×C4#12414yes—
327C2×C3×C4#12614yes—
328C2×C3×C4#12614yes—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C24C23×D311yescharacteristic
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—34no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4#23512yes—
447C22×C4#23512yes—
448C22×C4D3×D412yes—
449C22×C4#23712yes—
450C22×C4#23712yes—
451C22×C4#23712yes—
452C22×C4D3×Q812yes—
453C22×C4#23912yes—
454C22×C4C2×#6714yes—
455C22×C4C2×#6712yes—
456C4⋊C4—44no—
457C4⋊C4—68no—
458C4⋊C4—128no—
459C4⋊C4—124no—
460C42—44no—
461C42—124no—
462C42—124no—
463C22×C3C2×#12111yescharacteristic
464C22×C3C2×#12311yescharacteristic
465C22×C3C2×#12311yescharacteristic
466C22×C3C2×#12311yescharacteristic
467C22×C3C2×#12411yescharacteristic
468C22×C3C2×#12611yescharacteristic
469C22×C3C2×#12611yescharacteristic
470C22×C3#52911yescharacteristic
471C22×C3#52911yescharacteristic
472C22×C3#52911yescharacteristic
473C22×C3#53311yescharacteristic
474C22×C3#53311yescharacteristic
475C22×C3#53311yescharacteristic
476C22×C3#54111yescharacteristic
477C22×C3#54111yescharacteristic
478C22×C3#54111yescharacteristic
479C22×C3#54211yescharacteristic
480C22×C3#54211yescharacteristic
481C22×C3#54211yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3#54711yescharacteristic
484C22×C3#54811yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3#54811yescharacteristic
487C22×C3D4×Q811yescharacteristic
488C22×C3D4×Q811yescharacteristic
489C22×C3#55011yescharacteristic
490C22×C3#55211yescharacteristic
491C22×C3#55211yescharacteristic
492C22×C3#55411yescharacteristic
493C22×C3#55611yescharacteristic
494C22×C3#55611yescharacteristic
495C22×C3#55611yescharacteristic
496C22×C3#55711yescharacteristic
497C22×C3#55911yescharacteristic
498C3×C4—44no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—82no—
502C3×C4#39718yes—
503Dic3—42no—
504Dic3—42no—
505Dic3—44no—
506Dic3—42no—
507Dic3—42no—
508Dic3—82no—
509C23C2×#23511yescharacteristic
510C23C2×#23511yescharacteristic
511C23C2×D3×D411yescharacteristic
512C23C2×#23711yescharacteristic
513C23C2×#23711yescharacteristic
514C23C2×#23711yescharacteristic
515C23C2×D3×Q811yescharacteristic
516C23C2×#23911yescharacteristic
517C23#100711yescharacteristic
518C23#100711yescharacteristic
519C23D3×#4111yescharacteristic
520C23D3×#4111yescharacteristic
521C23#100911yescharacteristic
522C23#101011yescharacteristic
523C23C22×#6711yescharacteristic
524Q8—416no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—122no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—122no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—122no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—122no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4#87914yes—
596C2×C4#87914yes—
597C2×C4#88314yes—
598C2×C4#88514yes—
599C2×C4#92914yes—
600C2×C4#93514yes—
601C2×C4#93914yes—
602C2×C3C2×#39711yescharacteristic
603C2×C3#242111yescharacteristic
604C2×C3#242311yescharacteristic
605C2×C3#242611yescharacteristic
606C2×C3#242911yescharacteristic
607C2×C3#244611yescharacteristic
608C2×C3#244611yescharacteristic
609C2×C3#244711yescharacteristic
610C2×C3#244811yescharacteristic
611C2×C3#248911yescharacteristic
612C2×C3#249211yescharacteristic
613C2×C3#251111yescharacteristic
614C2×C3#251411yescharacteristic
615C2×C3#264511yescharacteristic
616C2×C3#286311yescharacteristic
617C22C2×#87911yescharacteristic
618C22C2×#87911yescharacteristic
619C22C2×#88311yescharacteristic
620C22C2×#88511yescharacteristic
621C22#557811yescharacteristic
622C22#557911yescharacteristic
623C22#558011yescharacteristic
624C22#558111yescharacteristic
625C22#558111yescharacteristic
626C22#558211yescharacteristic
627C22#558311yescharacteristic
628C22#558411yescharacteristic
629C22#562211yescharacteristic
630C22#562411yescharacteristic
631C22#562511yescharacteristic
632C22#563011yescharacteristic
633C22#563111yescharacteristic
634C22#563211yescharacteristic
635C22#563311yescharacteristic
636C22#564311yescharacteristic
637C22#567111yescharacteristic
638C22#568011yescharacteristic
639C22#568111yescharacteristic
640C22#568211yescharacteristic
641C22#572111yescharacteristic
642C22#572811yescharacteristic
643C22#573111yescharacteristic
644C22#573211yescharacteristic
645C22C2×#92911yescharacteristic
646C22C2×#93511yescharacteristic
647C22C2×#93911yescharacteristic
648C22Q8×#6711yescharacteristic
649C22#581711yescharacteristic
650C22#583111yescharacteristic
651C22#583411yescharacteristic
652C4—42no—
653C4—42no—
654C4—44no—
655C4—82no—
656C4—122no—
657C4—122no—
658C4—122no—
659C4—122no—
660C4—124no—
661C4—242no—
662C4#495218yes—
663C3#1518611yescharacteristic, complemented, Sylow
664C2#7650211yescharacteristic
665C2#7650511yescharacteristic
666C2#7652311yescharacteristic
667C2#7652611yescharacteristic
668C2#7656011yescharacteristic
669C2#7660811yescharacteristic
670C2#7663111yescharacteristic
671C2#7663311yescharacteristic
672C2C2×#495211yescharacteristic
673C2#8249311yescharacteristic
674C2#8249411yescharacteristic
675C2#8275511yescharacteristic
676C2#8275811yescharacteristic
677C2#8296311yescharacteristic
678C2#8323411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#391C2
hypercenterC2×#391D3
hypocenterC3#15186
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
104. C3⋊#95Q8semidirect
663. C3#15186semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×#52292
C2×#49282
#715632
C4×C3⋊#952
#694212
#694182
#694172
#694132
#151863