Finite Group Databasebeta

← #10720234

Subgroups — #10720234

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 688 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×Dic3×C4⋊C4C211yesmaximal, characteristic
5C2×#4928C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7C4×C3⋊#95C211yesmaximal, characteristic
8#69417C212yesmaximal
9#69413C212yesmaximal
10#69406C212yesmaximal
11#69402C212yesmaximal
12#14142—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×Dic3C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C3×#391—24no—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25#5229—24no—
26#5229C418yescomplemented
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2211yescharacteristic
39Dic3×C4⋊C4—24no—
40#4928—24no—
41C2×#391—31no—
42C2×#391—31no—
43C2×#391D311yescharacteristic
44C2×#389—31no—
45C2×C4×C4⋊C4—31no—
46#1443—32no—
47#1442—32no—
48#1436—32no—
49C4⋊#95—31no—
50#1434—32no—
51C4×#95—31no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8C2×C414yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D414yes—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3C2×C414yes—
81C2×C4⋊Dic3D414yes—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—42no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—42no—
93C2×C4×Dic3C2×C414yes—
94C3×#95—24no—
95C3×#95—24no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—42no—
105C3⋊#95C2×C418yes—
106C3⋊#95C2×C418yes—
107C3⋊#95Q814yescomplemented
108C23×Q8C2×D311yescharacteristic
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42—31no—
114C22×C42—31no—
115C22×C42—32no—
116C22×C42C2×D311yescharacteristic
117#391—24no—
118#391—38no—
119#391—64no—
120#389—64no—
121C4×C4⋊C4—64no—
122C2×#95—31no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
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×#95C2×D312yes—
135C2×#95C2×D312yes—
136C24×C3C2411yescharacteristic
137C2×C3×Q8—28no—
138C2×C3×Q8—28no—
139C2×C3×Q8—28no—
140C2×C3×Q8—44no—
141C2×C3×Q8—44no—
142C2×C3×Q8—44no—
143C2×C3×Q8—44no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4C22×C414yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
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—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3C22×C412yes—
214C22×Dic3C22×C414yes—
215C22×Dic3C22×C412yes—
216C22×Dic3C2×D414yes—
217C22×Dic3C2×Q812yes—
218C22×Dic3C2×Q812yes—
219C22×Dic3#4112yes—
220C22×Dic3#4112yes—
221C3×C4⋊C4—28no—
222C3×C42—44no—
223C4⋊Dic3—28no—
224C4⋊Dic3—28no—
225Dic3⋊C4—48no—
226C4×Dic3—28no—
227C4×Dic3—48no—
228C4×Dic3—44no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8C4×D314yes—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—32no—
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4C22×D312yes—
243C23×C4C22×D311yescharacteristic
244C23×C4C22×D311yescharacteristic
245C23×C4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—34no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—122no—
258C2×C4⋊C4#6714yes—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—34no—
263C2×C42—62no—
264C2×C42—62no—
265C2×C42—64no—
266C2×C42—62no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—64no—
270C2×C42—122no—
271#95—24no—
272#95—24no—
273#95—38no—
274#95—38no—
275#95—34no—
276#95—62no—
277#95—62no—
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×C3C22×Q811yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3H(5,𝔽2)11yescharacteristic
297C23×C32−1+411yescharacteristic
298C23×C32−1+411yescharacteristic
299C23×C32−1+411yescharacteristic
300C3×Q8—416no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
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—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4#11714yes—
330C2×C3×C4C4×D414yes—
331C2×C3×C4C4×D414yes—
332C2×C3×C4#12114yes—
333C2×C3×C4#12314yes—
334C2×C3×C4#12414yes—
335C2×C3×C4#12614yes—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—22no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—42no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—41no—
370C2×Dic3—42no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3C4×Q814yes—
378C24C23×D311yescharacteristic
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—34no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4C2×C4×D314yes—
453C22×C4C2×C4×D312yes—
454C22×C4#23512yes—
455C22×C4D3×D412yes—
456C22×C4#23712yes—
457C22×C4#23712yes—
458C22×C4#23712yes—
459C22×C4D3×Q812yes—
460C22×C4#23912yes—
461C22×C4C2×#6714yes—
462C22×C4C2×#6712yes—
463C4⋊C4—28no—
464C4⋊C4—68no—
465C4⋊C4—68no—
466C4⋊C4—128no—
467C42—44no—
468C42—68no—
469C42—128no—
470C42—124no—
471C22×C3C2×#11711yescharacteristic
472C22×C3C2×C4×D411yescharacteristic
473C22×C3C2×C4×D411yescharacteristic
474C22×C3C2×C4×Q811yescharacteristic
475C22×C3C4×#4111yescharacteristic
476C22×C3#51911yescharacteristic
477C22×C3#52011yescharacteristic
478C22×C3C2×#12111yescharacteristic
479C22×C3C2×#12311yescharacteristic
480C22×C3C2×#12411yescharacteristic
481C22×C3C2×#12611yescharacteristic
482C22×C3#52911yescharacteristic
483C22×C3#52911yescharacteristic
484C22×C3#53311yescharacteristic
485C22×C3#53311yescharacteristic
486C22×C3#54111yescharacteristic
487C22×C3#54111yescharacteristic
488C22×C3#54211yescharacteristic
489C22×C3#54211yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54611yescharacteristic
492C22×C3#54711yescharacteristic
493C22×C3#54711yescharacteristic
494C22×C3#54811yescharacteristic
495C22×C3#54811yescharacteristic
496C22×C3D4×Q811yescharacteristic
497C22×C3D4×Q811yescharacteristic
498C22×C3#55211yescharacteristic
499C22×C3#55211yescharacteristic
500C22×C3#55411yescharacteristic
501C22×C3#55611yescharacteristic
502C22×C3#55611yescharacteristic
503C22×C3#55611yescharacteristic
504C22×C3#55711yescharacteristic
505C22×C3#55911yescharacteristic
506C3×C4—42no—
507C3×C4—42no—
508C3×C4—44no—
509C3×C4—44no—
510C3×C4#38818yes—
511Dic3—24no—
512Dic3—42no—
513Dic3—44no—
514Dic3—44no—
515Dic3—42no—
516Dic3—42no—
517C23C22×C4×D311yescharacteristic
518C23C2×#23511yescharacteristic
519C23C2×D3×D411yescharacteristic
520C23C2×#23711yescharacteristic
521C23C2×#23711yescharacteristic
522C23C2×#23711yescharacteristic
523C23C2×D3×Q811yescharacteristic
524C23C2×#23911yescharacteristic
525C23#100711yescharacteristic
526C23#100711yescharacteristic
527C23D3×#4111yescharacteristic
528C23D3×#4111yescharacteristic
529C23#100911yescharacteristic
530C23#101011yescharacteristic
531C23C22×#6711yescharacteristic
532Q8—416no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—34no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—62no—
574C2×C4—64no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—122no—
593C2×C4—121no—
594C2×C4—122no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4#87714yes—
604C2×C4#87914yes—
605C2×C4#67⋊C414yes—
606C2×C4#88514yes—
607C2×C4C4×#6714yes—
608C2×C4#93514yes—
609C2×C4#93814yes—
610C2×C3C2×#38811yescharacteristic
611C2×C3#229511yescharacteristic
612C2×C3#232011yescharacteristic
613C2×C3#236011yescharacteristic
614C2×C3#236211yescharacteristic
615C2×C3#236511yescharacteristic
616C2×C3#236711yescharacteristic
617C2×C3#242711yescharacteristic
618C2×C3#242911yescharacteristic
619C2×C3#244611yescharacteristic
620C2×C3#244711yescharacteristic
621C2×C3#248911yescharacteristic
622C2×C3#249211yescharacteristic
623C2×C3#251111yescharacteristic
624C2×C3#251411yescharacteristic
625C22C2×#87711yescharacteristic
626C22C2×#87911yescharacteristic
627C22C2×#67⋊C411yescharacteristic
628C22C2×#88511yescharacteristic
629C22C4×D3×Q811yescharacteristic
630C22#557211yescharacteristic
631C22C4×#23911yescharacteristic
632C22#557411yescharacteristic
633C22#557911yescharacteristic
634C22#558111yescharacteristic
635C22#558111yescharacteristic
636C22#558411yescharacteristic
637C22#562211yescharacteristic
638C22#562411yescharacteristic
639C22#562511yescharacteristic
640C22#563011yescharacteristic
641C22#563111yescharacteristic
642C22#563211yescharacteristic
643C22#563311yescharacteristic
644C22#564311yescharacteristic
645C22#567111yescharacteristic
646C22#568011yescharacteristic
647C22#568111yescharacteristic
648C22#568211yescharacteristic
649C22#572111yescharacteristic
650C22#572811yescharacteristic
651C22#573111yescharacteristic
652C22#573211yescharacteristic
653C22C2×C4×#6711yescharacteristic
654C22C2×#93511yescharacteristic
655C22C2×#93811yescharacteristic
656C22Q8×#6711yescharacteristic
657C22#581711yescharacteristic
658C22#582411yescharacteristic
659C22#582911yescharacteristic
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—64no—
665C4—122no—
666C4—124no—
667C4—122no—
668C4—122no—
669C4—124no—
670C4#494418yes—
671C3#1414211yescharacteristic, complemented, Sylow
672C2#7647711yescharacteristic
673C2#7648311yescharacteristic
674C2#7649211yescharacteristic
675C2#7649311yescharacteristic
676C2#7652311yescharacteristic
677C2#7652611yescharacteristic
678C2#7663111yescharacteristic
679C2#7663311yescharacteristic
680C2C2×#494411yescharacteristic
681C2#8246411yescharacteristic
682C2#8249011yescharacteristic
683C2#8275611yescharacteristic
684C2#8275711yescharacteristic
685C2#8296211yescharacteristic
686C2#8323411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#391C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#391D3
hypocenterC3#14142
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
26. #5229C4semidirect
107. C3⋊#95Q8semidirect
671. C3#14142semidirect

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×Dic3×C4⋊C42
C2×#49282
#715632
C4×C3⋊#952
#694172
#694132
#694062
#694022
#141423