Finite Group Databasebeta

← #10731944

Subgroups — #10731944

Order 768.

1410 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4982C212yesmaximal
3C2×#4984C211yesmaximal, characteristic
4C2×#4972C211yesmaximal, characteristic
5C2×C4×Dic3⋊C4C211yesmaximal, characteristic
6#71567C212yesmaximal
7#71563C211yesmaximal, characteristic
8#71563C211yesmaximal, characteristic
9#71535C211yesmaximal, characteristic
10#69414C212yesmaximal
11#69411C212yesmaximal
12#15149—31nomaximal, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C3×#389—24no—
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37#4984—24no—
38#4982—28no—
39#4972—24no—
40C4×Dic3⋊C4—24no—
41C2×#398—32no—
42C2×#389—31no—
43C2×#389—31no—
44C2×#389D311yescharacteristic
45C2×C4×C4⋊C4—31no—
46#1442—32no—
47#1442—32no—
48#1441—32no—
49C4⋊#95—31no—
50C4⋊#95—31no—
51C4⋊#95—31no—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
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—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C4⋊C4D414yes—
71C2×C3×C4⋊C4Q814yes—
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⋊Dic3—24no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3Q814yes—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—42no—
92C2×Dic3⋊C4D414yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3D414yes—
99C3×#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—42no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4C2×D312yes—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C4⋊C4C2×D311yescharacteristic
119C22×C42—31no—
120C22×C42—31no—
121C22×C42C2×D311yescharacteristic
122#398—68no—
123#389—24no—
124#389—64no—
125#389—64no—
126C4×C4⋊C4—64no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D312yes—
138C24×C3C2411yescharacteristic
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×Q814yes—
168C22×C3×C4C2×Q812yes—
169C22×C3×C4C2×Q812yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D412yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D412yes—
213C22×Dic3C2×Q814yes—
214C22×Dic3#4112yes—
215C22×Dic3#4112yes—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—48no—
219C3×C42—44no—
220C4⋊Dic3—28no—
221C4⋊Dic3—44no—
222C4⋊Dic3—48no—
223Dic3⋊C4—28no—
224Dic3⋊C4—28no—
225Dic3⋊C4—48no—
226C4×Dic3—28no—
227C4×Dic3—44no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—32no—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
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—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—122no—
264C2×C4⋊C4Dic614yes—
265C2×C4⋊C4#6714yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—34no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42—62no—
273C2×C42—62no—
274C2×C42—62no—
275#95—24no—
276#95—64no—
277#95—62no—
278#95—62no—
279#95—64no—
280#95—64no—
281#95—64no—
282#95—64no—
283#95—64no—
284#95—64no—
285#95—122no—
286C23×C3C22×D411yescharacteristic
287C23×C3C22×D411yescharacteristic
288C23×C3C22×D411yescharacteristic
289C23×C3C22×Q811yescharacteristic
290C23×C3C22×Q811yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4111yescharacteristic
297C23×C3H(5,𝔽2)11yescharacteristic
298C23×C32−1+411yescharacteristic
299C23×C32−1+411yescharacteristic
300C23×C32−1+411yescharacteristic
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4#12114yes—
330C2×C3×C4C22⋊Q814yes—
331C2×C3×C4C22⋊Q814yes—
332C2×C3×C4#12314yes—
333C2×C3×C4#12514yes—
334C2×C3×C4#12514yes—
335C2×C3×C4C4⋊Q814yes—
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—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3#12114yes—
378C24C23×D311yescharacteristic
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—34no—
409C22×C4—32no—
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—61no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4C2×Dic614yes—
446C22×C4C2×Dic612yes—
447C22×C4#23512yes—
448C22×C4D3×D412yes—
449C22×C4D3×D412yes—
450C22×C4#23712yes—
451C22×C4D3×Q812yes—
452C22×C4#23912yes—
453C22×C4#23912yes—
454C22×C4C2×#6714yes—
455C22×C4C2×#6712yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—68no—
460C4⋊C4—68no—
461C4⋊C4—68no—
462C4⋊C4—128no—
463C4⋊C4—128no—
464C4⋊C4—124no—
465C42—44no—
466C42—68no—
467C42—124no—
468C22×C3C2×#12111yescharacteristic
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×C22⋊Q811yescharacteristic
471C22×C3C2×C22⋊Q811yescharacteristic
472C22×C3C2×#12311yescharacteristic
473C22×C3#52511yescharacteristic
474C22×C3C2×#12511yescharacteristic
475C22×C3C2×#12511yescharacteristic
476C22×C3#52911yescharacteristic
477C22×C3#52911yescharacteristic
478C22×C3C2×C4⋊Q811yescharacteristic
479C22×C3#53211yescharacteristic
480C22×C3#53211yescharacteristic
481C22×C3#53611yescharacteristic
482C22×C3#53711yescharacteristic
483C22×C3#54111yescharacteristic
484C22×C3#54111yescharacteristic
485C22×C3#54211yescharacteristic
486C22×C3#54211yescharacteristic
487C22×C3D4211yescharacteristic
488C22×C3#54711yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3#54811yescharacteristic
491C22×C3#54811yescharacteristic
492C22×C3D4×Q811yescharacteristic
493C22×C3D4×Q811yescharacteristic
494C22×C3#55211yescharacteristic
495C22×C3#55211yescharacteristic
496C22×C3#55411yescharacteristic
497C22×C3#55411yescharacteristic
498C22×C3#55511yescharacteristic
499C22×C3#55711yescharacteristic
500C22×C3#55711yescharacteristic
501C22×C3#55711yescharacteristic
502C22×C3#55711yescharacteristic
503C3×C4—44no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—44no—
507C3×C4#39818yescomplemented
508Dic3—24no—
509Dic3—42no—
510Dic3—44no—
511Dic3—42no—
512Dic3—42no—
513Dic3—82no—
514C23C22×Dic611yescharacteristic
515C23C2×#23511yescharacteristic
516C23C2×D3×D411yescharacteristic
517C23C2×D3×D411yescharacteristic
518C23C2×#23711yescharacteristic
519C23#100411yescharacteristic
520C23C2×D3×Q811yescharacteristic
521C23C2×#23911yescharacteristic
522C23C2×#23911yescharacteristic
523C23#100711yescharacteristic
524C23D3×#4111yescharacteristic
525C23D3×#4111yescharacteristic
526C23#101011yescharacteristic
527C23#101011yescharacteristic
528C23C22×#6711yescharacteristic
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—34no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—121no—
581C2×C4—122no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—122no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4#88814yes—
600C2×C4#88914yes—
601C2×C4#89014yes—
602C2×C4#89414yes—
603C2×C4#92414yes—
604C2×C4#93914yes—
605C2×C4#94614yes—
606C2×C3C2×#39811yescharacteristic
607C2×C3#245211yescharacteristic
608C2×C3#245211yescharacteristic
609C2×C3#245311yescharacteristic
610C2×C3#245311yescharacteristic
611C2×C3#250611yescharacteristic
612C2×C3#250611yescharacteristic
613C2×C3#250811yescharacteristic
614C2×C3#250811yescharacteristic
615C2×C3#253611yescharacteristic
616C2×C3#253711yescharacteristic
617C2×C3#254611yescharacteristic
618C2×C3#258411yescharacteristic
619C2×C3#262511yescharacteristic
620C2×C3#285411yescharacteristic
621C22C2×#88811yescharacteristic
622C22C2×#88911yescharacteristic
623C22C2×#89011yescharacteristic
624C22C2×#89411yescharacteristic
625C22D4×Dic611yescharacteristic
626C22#553911yescharacteristic
627C22#554811yescharacteristic
628C22#556111yescharacteristic
629C22#556911yescharacteristic
630C22#557011yescharacteristic
631C22#558011yescharacteristic
632C22#558311yescharacteristic
633C22D3×#12111yescharacteristic
634C22#560811yescharacteristic
635C22#561111yescharacteristic
636C22#561611yescharacteristic
637C22#562711yescharacteristic
638C22#562811yescharacteristic
639C22#563911yescharacteristic
640C22#564211yescharacteristic
641C22#568511yescharacteristic
642C22#568511yescharacteristic
643C22#569111yescharacteristic
644C22#569811yescharacteristic
645C22#572511yescharacteristic
646C22#572711yescharacteristic
647C22#573011yescharacteristic
648C22#573311yescharacteristic
649C22C2×#92411yescharacteristic
650C22D4×#6711yescharacteristic
651C22C2×#93911yescharacteristic
652C22C2×#94611yescharacteristic
653C22#581711yescharacteristic
654C22#582411yescharacteristic
655C22#583111yescharacteristic
656C4—42no—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—64no—
661C4—124no—
662C4—122no—
663C4—122no—
664C4—122no—
665C4—242no—
666C4#498218yescomplemented
667C3#1514911yescharacteristic, complemented, Sylow
668C2#7602811yescharacteristic
669C2#7602911yescharacteristic
670C2#7603711yescharacteristic
671C2#7603811yescharacteristic
672C2#7606011yescharacteristic
673C2#7607911yescharacteristic
674C2#7610211yescharacteristic
675C2#7628611yescharacteristic
676C2C2×#498211yescharacteristic
677C2#8236211yescharacteristic
678C2#8247111yescharacteristic
679C2#8266711yescharacteristic
680C2#8277711yescharacteristic
681C2#8302911yescharacteristic
682C2#8323611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#15149
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
507. C3×C4#398semidirect
666. C4#4982semidirect
667. C3#15149semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49822
C2×#49842
C2×#49722
C2×C4×Dic3⋊C42
#715672
#715632
#715352
#694142
#694112
#151493