Finite Group Databasebeta

← #10717367

Subgroups — #10717367

Order 768.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×#4667C211yesmaximal, characteristic
3C2×#4648C211yesmaximal, characteristic, complemented
4C3×#1434C211yesmaximal, characteristic, complemented
5#71534C211yesmaximal, characteristic, complemented
6#69429C212yesmaximal
7(C2×D3)⋊#95C211yesmaximal, characteristic
8#69414C212yesmaximal, complemented
9#69408C212yesmaximal
10#69406C212yesmaximal, complemented
11#69402C211yesmaximal, characteristic, complemented
12#13898—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C4×(C2×D3)⋊C4C418yescomplemented
37#4667—24no—
38#4648—24no—
39C2×#380—31no—
40C2×C4×C22⋊C4—31no—
41C2×#376—31no—
42#1442—32no—
43#1440—32no—
44#1437—32no—
45#1436—32no—
46#1434—31no—
47#1434—31no—
48#1434D311yescharacteristic, complemented
49C22⋊#95—31no—
50C24×D3C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—24no—
64C22×C4×D3—42no—
65C22×C4×D3C2×C414yes—
66C2×C3×C42—42no—
67C2×C3×C42C2×C414yes—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—42no—
76C2×(C2×D3)⋊C4—42no—
77C2×(C2×D3)⋊C4C2×C414yes—
78C2×(C2×D3)⋊C4C2×C414yes—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—42no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3C2×C414yes—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—24no—
91C3×#95—42no—
92C3⋊#95—22no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—42no—
97C3⋊#95—42no—
98C3⋊#95—42no—
99C3⋊#95C2×C414yes—
100C3⋊#95C2×C414yes—
101C3⋊#95D414yes—
102C24×C4—31no—
103C22×C4⋊C4—32no—
104C22×C22⋊C4—31no—
105C22×C22⋊C4—31no—
106C22×C22⋊C4—32no—
107C22×C22⋊C4—32no—
108C22×C42—31no—
109C22×C42—32no—
110C22×C42C2×D311yescharacteristic
111#380—64no—
112C4×C22⋊C4—38no—
113#376—64no—
114C2×#95—31no—
115C2×#95—31no—
116C2×#95—31no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95C2×D311yescharacteristic
125C2×#95C2×D311yescharacteristic
126C2×#95C2×D312yes—
127C2×#95C2×D312yes—
128C24×C3C2411yescharacteristic
129C23×D3—21no—
130C23×D3—22no—
131C23×D3—22no—
132C23×D3—21no—
133C23×D3—22no—
134C23×D3—21no—
135C23×D3—22no—
136C23×D3—21no—
137C23×D3C22×C412yes—
138C23×D3#4112yes—
139C23×D3#4112yes—
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—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
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×C4C22×C412yes—
164C22×C3×C4C22×C412yes—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4#4114yes—
168C22×C3×C4#4112yes—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3C22×C412yes—
195C22×Dic3C22×C412yes—
196C22×Dic3C22×C412yes—
197C22×Dic3C2×D414yes—
198C22×Dic3#4114yes—
199C22×Dic3#4114yes—
200C2×C4×D3—28no—
201C2×C4×D3—28no—
202C2×C4×D3—44no—
203C2×C4×D3—48no—
204C2×C4×D3—48no—
205C3×C42—44no—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—48no—
209(C2×D3)⋊C4—48no—
210Dic3⋊C4—48no—
211C4×Dic3—216no—
212C4×Dic3—28no—
213C25—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—32no—
217C23×C4—31no—
218C23×C4—34no—
219C23×C4—32no—
220C23×C4—64no—
221C23×C4—62no—
222C23×C4—62no—
223C23×C4—122no—
224C23×C4C22×D312yes—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D312yes—
228C23×C4C22×D311yescharacteristic
229C2×C4⋊C4—64no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—122no—
232C2×C22⋊C4—34no—
233C2×C22⋊C4—34no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—122no—
242C2×C22⋊C4—122no—
243C2×C42—34no—
244C2×C42—42no—
245C2×C42—64no—
246C2×C42—64no—
247C2×C42—64no—
248C2×C42—62no—
249C2×C42—62no—
250C2×C42C4×D314yes—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—34no—
255#95—34no—
256#95—34no—
257#95—42no—
258#95—64no—
259#95—64no—
260#95—64no—
261#95—62no—
262#95—122no—
263#95—122no—
264#95—122no—
265C23×C3C23×C411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3C2×#4112yes—
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4112yes—
270C23×C3C2×#4112yes—
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3H(5,𝔽2)12yes—
274C23×C3H(5,𝔽2)11yescharacteristic
275C23×C32−1+411yescharacteristic
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—24no—
283C22×D3—24no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—24no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—24no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3#11714yes—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
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×C4—42no—
327C2×C3×C4C4×D414yes—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
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—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
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×Dic3#11714yes—
358C2×Dic3#11714yes—
359C4×D3—416no—
360C24—32no—
361C24—32no—
362C24—32no—
363C24—62no—
364C24—61no—
365C24—62no—
366C24—61no—
367C24—62no—
368C24—62no—
369C24—61no—
370C24—61no—
371C24C23×D311yescharacteristic
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—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—32no—
400C22×C4—34no—
401C22×C4—68no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—68no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—124no—
429C22×C4—128no—
430C22×C4—128no—
431C22×C4C2×C4×D312yes—
432C22×C4C2×C4×D312yes—
433C22×C4C2×C4×D312yes—
434C22×C4#23514yes—
435C22×C4D3×D412yes—
436C22×C4#23912yes—
437C4⋊C4—128no—
438C22⋊C4—68no—
439C22⋊C4—68no—
440C22⋊C4—128no—
441C22⋊C4—128no—
442C42—44no—
443C42—616no—
444C42—68no—
445C22×C3C2×#11711yescharacteristic
446C22×C3C2×#11711yescharacteristic
447C22×C3C2×#11711yescharacteristic
448C22×C3C2×C4×D411yescharacteristic
449C22×C3C4×#4111yescharacteristic
450C22×C3#51811yescharacteristic
451C22×C3#52011yescharacteristic
452C22×C3#52512yes—
453C22×C3#52912yes—
454C22×C3#52912yes—
455C22×C3#52912yes—
456C22×C3#54012yes—
457C22×C3#54212yes—
458C22×C3#54611yescharacteristic
459C22×C3#54612yes—
460C22×C3#54811yescharacteristic
461C22×C3#55112yes—
462C22×C3#55112yes—
463C22×C3#55211yescharacteristic
464C22×C3#55212yes—
465C22×C3#55311yescharacteristic
466C22×C3#55312yes—
467C22×C3#55512yes—
468C2×D3—24no—
469C2×D3—24no—
470C2×D3—24no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—44no—
475C2×D3—44no—
476C2×D3—44no—
477C2×D3—44no—
478C2×D3—42no—
479C3×C4—24no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—44no—
483C3×C4—82no—
484Dic3—24no—
485Dic3—24no—
486Dic3—42no—
487Dic3—44no—
488Dic3—82no—
489C23—34no—
490C23—62no—
491C23—62no—
492C23—62no—
493C23—62no—
494C23—64no—
495C23—64no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23—62no—
502C23—64no—
503C23—64no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23C22×C4×D311yescharacteristic
513C23C2×#23512yes—
514C23C2×D3×D411yescharacteristic
515C23#100412yes—
516C23C2×#23911yescharacteristic
517C23D3×#4112yes—
518C23D3×#4112yes—
519C23D3×#4111yescharacteristic
520C23D3×#4111yescharacteristic
521C23#100911yescharacteristic
522C23#101011yescharacteristic
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—34no—
537C2×C4—34no—
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—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—1216no—
568C2×C4—122no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4#89314yes—
584C2×C3#228311yescharacteristic
585C2×C3#229011yescharacteristic
586C2×C3#231911yescharacteristic
587C2×C3#234211yescharacteristic
588C2×C3#234611yescharacteristic
589C2×C3#235211yescharacteristic
590C2×C3#235311yescharacteristic
591C2×C3#247612yes—
592C2×C3#247712yes—
593C2×C3#253012yes—
594C2×C3#253112yes—
595D3—44no—
596C22—64no—
597C22—64no—
598C22—64no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—124no—
603C22—124no—
604C22—124no—
605C22—124no—
606C22—122no—
607C22C4×#23511yescharacteristic
608C22#548611yescharacteristic
609C22C2×#89311yescharacteristic
610C22D3×#11711yescharacteristic
611C22D3×#11711yescharacteristic
612C22D3×#11711yescharacteristic
613C22#552311yescharacteristic
614C22#553112yes—
615C22#553212yes—
616C22#554712yes—
617C22#555612yes—
618C22#555812yes—
619C22#558012yes—
620C22#561012yes—
621C22#563712yes—
622C22#566212yes—
623C22#566312yes—
624C22#567611yescharacteristic
625C22#567711yescharacteristic
626C22#569411yescharacteristic
627C22#569511yescharacteristic
628C22#570612yes—
629C22#570612yes—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—64no—
635C4—64no—
636C4—82no—
637C4—124no—
638C4—122no—
639C4—242no—
640C3#1389811yescharacteristic, complemented, Sylow
641C2—124no—
642C2#7339211yescharacteristic
643C2#7357611yescharacteristic
644C2#7375411yescharacteristic
645C2#7400911yescharacteristic
646C2#7405511yescharacteristic
647C2#7422911yescharacteristic
648C2#7425811yescharacteristic
649C2#7535512yes—
650C2#7535612yes—
651C2#7586412yes—
652C2#7586512yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1434C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenter#1434D3
hypocenterC3#13898
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. C2×#4648C2semidirect
4. C3×#1434C2semidirect
5. #71534C2semidirect
8. #69414C2semidirect
10. #69406C2semidirect
11. #69402C2semidirect
36. C4×(C2×D3)⋊C4C4semidirect
48. #1434D3semidirect
640. C3#13898semidirect

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×#46672
C2×#46482
C3×#14342
#715342
#694292
(C2×D3)⋊#952
#694142
#694082
#694062
#694022
#138983