Finite Group Databasebeta

← #10717813

Subgroups — #10717813

Order 768.

1630 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×#4670C212yesmaximal
3C3×C4⋊#95C211yesmaximal, characteristic, complemented
4#71570C212yesmaximal
5#71564C212yesmaximal, complemented
6#71535C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#95C211yesmaximal, characteristic
8(C2×D3)⋊#95C211yesmaximal, characteristic
9#69415C212yesmaximal, complemented
10#69401C211yesmaximal, characteristic, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#13948—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
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⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C4×(C2×D3)⋊C4C418yescomplemented
37#4670—28no—
38C2×#388—32no—
39C2×C4×C22⋊C4—31no—
40#1442—32no—
41#1439—32no—
42#1436—32no—
43C4⋊#95—31no—
44C4⋊#95D311yescharacteristic, complemented
45#1434—31no—
46C22⋊#95—31no—
47C22⋊#95—31no—
48C4×#95—31no—
49C24×D3C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×Dic3C2312yes—
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—42no—
63C22×C4×D3—42no—
64C22×C4×D3C2×C414yes—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C42—42no—
69C2×C3×C42C2×C414yes—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—42no—
77C2×(C2×D3)⋊C4—42no—
78C2×(C2×D3)⋊C4C2×C414yes—
79C2×(C2×D3)⋊C4C2×C414yes—
80C2×(C2×D3)⋊C4D414yes—
81C2×(C2×D3)⋊C4Q814yes—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3—42no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—42no—
88C2×C4×Dic3—42no—
89C2×C4×Dic3C2×C414yes—
90C3×#95—24no—
91C3×#95—22no—
92C3×#95—22no—
93C3⋊#95—22no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—42no—
99C3⋊#95—42no—
100C3⋊#95—42no—
101C3⋊#95C2×C414yes—
102C3⋊#95C2×C414yes—
103C24×C4—31no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4C2×D312yes—
106C22×C22⋊C4—31no—
107C22×C22⋊C4—31no—
108C22×C22⋊C4—32no—
109C22×C22⋊C4—32no—
110C22×C42—31no—
111C22×C42—32no—
112C22×C42C2×D311yescharacteristic
113#388—68no—
114C4×C22⋊C4—38no—
115C2×#95—31no—
116C2×#95—31no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95C2×D311yescharacteristic
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D312yes—
128C24×C3C2411yescharacteristic
129C23×D3—22no—
130C23×D3—21no—
131C23×D3—21no—
132C23×D3—22no—
133C23×D3—21no—
134C23×D3—21no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3C22×C412yes—
138C23×D3C2×D412yes—
139C23×D3C2×Q812yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C22×C412yes—
160C22×C3×C4C22×C412yes—
161C22×C3×C4C22×C412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×Q812yes—
164C22×C3×C4#4114yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3C22×C412yes—
193C22×Dic3C22×C412yes—
194C22×Dic3C22×C412yes—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×Q812yes—
197C22×Dic3#4114yes—
198C2×C4×D3—28no—
199C2×C4×D3—44no—
200C2×C4×D3—44no—
201C2×C4×D3—48no—
202C2×C4×D3—48no—
203C3×C4⋊C4—48no—
204C3×C42—44no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—48no—
207(C2×D3)⋊C4—48no—
208(C2×D3)⋊C4C4⋊C4116yescomplemented
209C4⋊Dic3—48no—
210C4×Dic3—44no—
211C4×Dic3—48no—
212C25—31no—
213C23×C4—31no—
214C23×C4—32no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—34no—
218C23×C4—32no—
219C23×C4—62no—
220C23×C4—62no—
221C23×C4—122no—
222C23×C4—122no—
223C23×C4C22×D312yes—
224C23×C4C22×D312yes—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—122no—
234C2×C22⋊C4—34no—
235C2×C22⋊C4—34no—
236C2×C22⋊C4—34no—
237C2×C22⋊C4—34no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—64no—
243C2×C22⋊C4—64no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42—34no—
247C2×C42—42no—
248C2×C42—64no—
249C2×C42—64no—
250C2×C42—122no—
251C2×C42—122no—
252C2×C42C4×D314yes—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—34no—
257#95—34no—
258#95—64no—
259#95—64no—
260#95—62no—
261#95—64no—
262#95—62no—
263#95—122no—
264#95—122no—
265#95—122no—
266C23×C3C23×C411yescharacteristic
267C23×C3C22×D411yescharacteristic
268C23×C3C22×Q811yescharacteristic
269C23×C3C2×#4112yes—
270C23×C3C2×#4111yescharacteristic
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4111yescharacteristic
274C23×C3C2×#4112yes—
275C23×C3H(5,𝔽2)12yes—
276C23×C3H(5,𝔽2)12yes—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—24no—
288C22×D3—24no—
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3C2×C4⋊C414yes—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
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—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4C2×C4⋊C414yes—
327C2×C3×C4#11714yes—
328C2×C3×C4#11714yes—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—42no—
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—
357C2×Dic3C2×C4⋊C414yes—
358C4×D3—416no—
359C24—32no—
360C24—32no—
361C24—32no—
362C24—62no—
363C24—61no—
364C24—61no—
365C24—62no—
366C24—61no—
367C24—61no—
368C24—62no—
369C24—62no—
370C24C23×D311yescharacteristic
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—34no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—68no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—124no—
421C22×C4—124no—
422C22×C4—128no—
423C22×C4—128no—
424C22×C4C2×C4×D312yes—
425C22×C4C2×C4×D312yes—
426C22×C4C2×C4×D312yes—
427C22×C4#23514yes—
428C22×C4D3×D412yes—
429C22×C4#23712yes—
430C22×C4#23712yes—
431C22×C4D3×Q812yes—
432C22×C4#23912yes—
433C22×C4#23912yes—
434C4⋊C4—48no—
435C4⋊C4—128no—
436C22⋊C4—316no—
437C22⋊C4—68no—
438C22⋊C4—128no—
439C22⋊C4—128no—
440C42—44no—
441C42—124no—
442C42—128no—
443C22×C3C22×C4⋊C411yescharacteristic
444C22×C3C2×#11711yescharacteristic
445C22×C3C2×#11711yescharacteristic
446C22×C3C4×#4111yescharacteristic
447C22×C3C4×#4111yescharacteristic
448C22×C3#51811yescharacteristic
449C22×C3#51811yescharacteristic
450C22×C3#52912yes—
451C22×C3#52912yes—
452C22×C3#53211yescharacteristic
453C22×C3#53211yescharacteristic
454C22×C3#53311yescharacteristic
455C22×C3#53311yescharacteristic
456C22×C3#53812yes—
457C22×C3#53812yes—
458C22×C3#54612yes—
459C22×C3#54612yes—
460C22×C3#55112yes—
461C22×C3#55312yes—
462C22×C3#55312yes—
463C22×C3#55412yes—
464C22×C3#55412yes—
465C22×C3#55912yes—
466C2×D3—24no—
467C2×D3—24no—
468C2×D3—24no—
469C2×D3—42no—
470C2×D3—44no—
471C2×D3—44no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—44no—
476C2×D3—44no—
477C3×C4—24no—
478C3×C4—24no—
479C3×C4—24no—
480C3×C4—44no—
481C3×C4—82no—
482Dic3—24no—
483Dic3—42no—
484Dic3—42no—
485Dic3—44no—
486Dic3—82no—
487C23—34no—
488C23—64no—
489C23—62no—
490C23—62no—
491C23—62no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23—64no—
500C23—64no—
501C23—64no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23C22×C4×D311yescharacteristic
511C23C2×#23512yes—
512C23C2×D3×D411yescharacteristic
513C23C2×#23711yescharacteristic
514C23C2×#23711yescharacteristic
515C23#100412yes—
516C23C2×D3×Q811yescharacteristic
517C23C2×#23911yescharacteristic
518C23C2×#23911yescharacteristic
519C23D3×#4112yes—
520C23#100912yes—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—34no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—1216no—
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—122no—
576C2×C4—122no—
577C2×C4D3×C4⋊C414yes—
578C2×C4#89214yes—
579C2×C4#89214yes—
580C2×C3#228511yescharacteristic
581C2×C3#229811yescharacteristic
582C2×C3#231211yescharacteristic
583C2×C3#234911yescharacteristic
584C2×C3#234911yescharacteristic
585C2×C3#235311yescharacteristic
586C2×C3#235711yescharacteristic
587C2×C3#255712yes—
588C2×C3#255912yes—
589C2×C3#258712yes—
590C2×C3#258912yes—
591D3—44no—
592C22—64no—
593C22—64no—
594C22—64no—
595C22—124no—
596C22—122no—
597C22—124no—
598C22—124no—
599C22—124no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22C4×#23511yescharacteristic
604C22#548611yescharacteristic
605C22#549512yes—
606C22C2×D3×C4⋊C411yescharacteristic
607C22C2×#89211yescharacteristic
608C22C2×#89211yescharacteristic
609C22#551412yes—
610C22#552111yescharacteristic
611C22#552211yescharacteristic
612C22#553812yes—
613C22#555012yes—
614C22#557912yes—
615C22#558012yes—
616C22#561012yes—
617C22#561712yes—
618C22#563612yes—
619C22#564412yes—
620C22#568112yes—
621C22#569512yes—
622C22#571611yescharacteristic
623C22#572511yescharacteristic
624C22#572811yescharacteristic
625C22#572811yescharacteristic
626C4—24no—
627C4—24no—
628C4—24no—
629C4—44no—
630C4—64no—
631C4—82no—
632C4—122no—
633C4—124no—
634C4—122no—
635C4—242no—
636C3#1394811yescharacteristic, complemented, Sylow
637C2—124no—
638C2#7333611yescharacteristic
639C2#7362011yescharacteristic
640C2#7377011yescharacteristic
641C2#7400411yescharacteristic
642C2#7401111yescharacteristic
643C2#7416011yescharacteristic
644C2#7431811yescharacteristic
645C2#7607012yes—
646C2#7608612yes—
647C2#7635712yes—
648C2#7655412yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4⋊#95C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenterC4⋊#95D3
hypocenterC3#13948
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C3×C4⋊#95C2semidirect
5. #71564C2semidirect
6. #71535C2semidirect
9. #69415C2semidirect
10. #69401C2semidirect
11. Dic3×#95C2semidirect
36. C4×(C2×D3)⋊C4C4semidirect
44. C4⋊#95D3semidirect
208. (C2×D3)⋊C4C4⋊C4semidirect
636. C3#13948semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C2×D3)⋊C42
C2×#46702
C3×C4⋊#952
#715702
#715642
#715352
(C2×D3)⋊#952
#694152
#694012
Dic3×#952
#139483