Finite Group Databasebeta

← #10716890

Subgroups — #10716890

Order 768.

1814 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C2×C4×#217C211yesmaximal, characteristic
3C2×C4×#217C211yesmaximal, characteristic
4C2×C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#4932C211yesmaximal, characteristic, complemented
6#71549C212yesmaximal
7#71547C212yesmaximal
8C4×C3⋊#95C211yesmaximal, characteristic, complemented
9C4×C3⋊#95C211yesmaximal, characteristic, complemented
10#69418C212yesmaximal, complemented
11#69414C212yesmaximal, complemented
12#13842—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C3×C4×C22⋊C4—24no—
26C2×C3×#95C2212yes—
27C4×#217—24no—
28C4×#217C418yescomplemented
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2211yescharacteristic
39#4932—24no—
40C4×Dic3⋊C4—24no—
41C2×C4×C4⋊C4—31no—
42C2×C4×C22⋊C4—31no—
43C2×C4×C22⋊C4—31no—
44C2×C4×C22⋊C4D311yescharacteristic
45C2×#376—31no—
46#1443—32no—
47#1442—32no—
48#1440—32no—
49#1439—32no—
50C4×#95—31no—
51C4×#95—31no—
52C25×C3C2311yescharacteristic
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4C2×C414yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2312yes—
68C23×Dic3C2312yes—
69C23×Dic3C2311yescharacteristic
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42D414yes—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217C2×C414yes—
85C2×#217C2×C414yes—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3C2×C414yes—
99C2×C4×Dic3C2×C414yes—
100C2×C4×Dic3D414yes—
101C3×#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95C2×C414yes—
111C3⋊#95C2×C414yes—
112C24×C4C2×D311yescharacteristic
113C22×C4⋊C4—32no—
114C22×C22⋊C4—31no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4C2×D312yes—
118C22×C42—31no—
119C22×C42—31no—
120C22×C42—31no—
121C22×C42—31no—
122C22×C42C2×D311yescharacteristic
123C22×C42C2×D311yescharacteristic
124C4×C4⋊C4—64no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—38no—
127C4×C22⋊C4—64no—
128#376—64no—
129C2×#95—31no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95C2×D312yes—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—21no—
146C24×C3—21no—
147C24×C3—21no—
148C24×C3—21no—
149C24×C3C22×C412yes—
150C24×C3#4114yes—
151C24×C3C2411yescharacteristic
152C22×C3×C4—22no—
153C22×C3×C4—21no—
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×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—28no—
171C22×C3×C4—28no—
172C22×C3×C4—28no—
173C22×C3×C4—44no—
174C22×C3×C4—44no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4C22×C412yes—
178C22×C3×C4C22×C412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4114yes—
183C22×C3×C4#4112yes—
184C22×C3×C4#4112yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—21no—
202C22×Dic3—22no—
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×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3C22×C412yes—
215C22×Dic3C22×C412yes—
216C22×Dic3C22×C412yes—
217C22×Dic3C22×C412yes—
218C22×Dic3C2×D414yes—
219C22×Dic3#4114yes—
220C22×Dic3#4114yes—
221C3×C22⋊C4—216no—
222C3×C42—28no—
223C3×C42—44no—
224#217—216no—
225#217—28no—
226#217—28no—
227Dic3⋊C4—216no—
228C4×Dic3—28no—
229C4×Dic3—28no—
230C4×Dic3—28no—
231C4×Dic3—44no—
232C25C22×D311yescharacteristic
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—31no—
240C23×C4—32no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4C4×D314yes—
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D312yes—
248C23×C4C22×D312yes—
249C23×C4C22×D311yescharacteristic
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—34no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—34no—
269C2×C42—34no—
270C2×C42—34no—
271C2×C42—62no—
272C2×C42—62no—
273C2×C42—62no—
274C2×C42—62no—
275C2×C42—62no—
276C2×C42—62no—
277C2×C42—62no—
278C2×C42—62no—
279C2×C42—62no—
280C2×C42#6714yes—
281#95—24no—
282#95—34no—
283#95—34no—
284#95—64no—
285#95—64no—
286#95—64no—
287#95—64no—
288#95—64no—
289#95—64no—
290#95—64no—
291#95—64no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3#11714yes—
315C23×C3C23×C411yescharacteristic
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D411yescharacteristic
318C23×C3C2×#4112yes—
319C23×C3C2×#4111yescharacteristic
320C23×C3C2×#4112yes—
321C23×C3C2×#4111yescharacteristic
322C23×C3C2×#4111yescharacteristic
323C23×C3C2×#4111yescharacteristic
324C23×C3C2×#4112yes—
325C23×C3C2×#4112yes—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—416no—
348C2×C3×C4#11714yes—
349C2×C3×C4C4×D414yes—
350C2×C3×C4C4×D414yes—
351C2×C3×C4#12114yes—
352C2×C3×C4#12314yes—
353C2×C3×C4#12414yes—
354C2×C3×C4#12614yes—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3#11714yes—
389C24—22no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—22no—
397C24—21no—
398C24C2×C4×D312yes—
399C24#23514yes—
400C24C23×D311yescharacteristic
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—32no—
423C22×C4—34no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4C2×C4×D312yes—
461C22×C4C2×C4×D312yes—
462C22×C4#23514yes—
463C22×C4#23512yes—
464C22×C4D3×D412yes—
465C22×C4#23712yes—
466C22×C4#23712yes—
467C22×C4#23712yes—
468C22×C4C2×#6714yes—
469C22×C4C2×#6712yes—
470C4⋊C4—616no—
471C22⋊C4—216no—
472C22⋊C4—68no—
473C22⋊C4—616no—
474C22⋊C4—68no—
475C42—28no—
476C42—44no—
477C42—68no—
478C42—68no—
479C42—68no—
480C42—124no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3C2×#11711yescharacteristic
493C22×C3C2×#11711yescharacteristic
494C22×C3C2×#11711yescharacteristic
495C22×C3C2×C4×D411yescharacteristic
496C22×C3C2×C4×D411yescharacteristic
497C22×C3C4×#4111yescharacteristic
498C22×C3C4×#4111yescharacteristic
499C22×C3C2×#12111yescharacteristic
500C22×C3C2×#12311yescharacteristic
501C22×C3#52512yes—
502C22×C3#52512yes—
503C22×C3#52512yes—
504C22×C3C2×#12411yescharacteristic
505C22×C3C2×#12611yescharacteristic
506C22×C3#52912yes—
507C22×C3#52912yes—
508C22×C3#52912yes—
509C22×C3#52912yes—
510C22×C3#52912yes—
511C22×C3#52912yes—
512C22×C3#52912yes—
513C22×C3#52912yes—
514C22×C3#53212yes—
515C3×C4—44no—
516C3×C4—44no—
517C3×C4—42no—
518C3×C4#38818yes—
519Dic3—24no—
520Dic3—44no—
521Dic3—44no—
522Dic3—42no—
523Dic3—42no—
524Dic3—42no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23#87214yes—
548C23C22×C4×D311yescharacteristic
549C23C2×#23512yes—
550C23C2×#23512yes—
551C23C2×#23511yescharacteristic
552C23C2×D3×D411yescharacteristic
553C23C2×#23711yescharacteristic
554C23C2×#23711yescharacteristic
555C23C2×#23711yescharacteristic
556C23D3×#4112yes—
557C23D3×#4112yes—
558C23C22×#6711yescharacteristic
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—34no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—416no—
582C2×C4—62no—
583C2×C4—64no—
584C2×C4—62no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4#87714yes—
616C2×C4#87914yes—
617C2×C4#67⋊C414yes—
618C2×C4#88314yes—
619C2×C4C4×#6714yes—
620C2×C4#93614yes—
621C2×C4#93914yes—
622C2×C3—44no—
623C2×C3#228311yescharacteristic
624C2×C3C2×#38811yescharacteristic
625C2×C3#229011yescharacteristic
626C2×C3#229011yescharacteristic
627C2×C3#229011yescharacteristic
628C2×C3#229311yescharacteristic
629C2×C3#229811yescharacteristic
630C2×C3#239212yes—
631C2×C3#239512yes—
632C2×C3#239812yes—
633C2×C3#240112yes—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22C2×#87211yescharacteristic
646C22C4×#23511yescharacteristic
647C22#547912yes—
648C22C2×#87711yescharacteristic
649C22C2×#87911yescharacteristic
650C22C2×#67⋊C411yescharacteristic
651C22C2×#88311yescharacteristic
652C22D3×#11711yescharacteristic
653C22#552211yescharacteristic
654C22#552812yes—
655C22#553812yes—
656C22#553812yes—
657C22#553812yes—
658C22#553812yes—
659C22#553812yes—
660C22#554712yes—
661C22#554812yes—
662C22#555712yes—
663C22C2×C4×#6711yescharacteristic
664C22#579112yes—
665C22C2×#93611yescharacteristic
666C22C2×#93911yescharacteristic
667C22#582812yes—
668C4—44no—
669C4—42no—
670C4—44no—
671C4—64no—
672C4—124no—
673C4—122no—
674C4—124no—
675C4—122no—
676C4—122no—
677C4#494518yes—
678C3#1384211yescharacteristic, complemented, Sylow
679C2—44no—
680C2#7343711yescharacteristic
681C2#7343811yescharacteristic
682C2#7343911yescharacteristic
683C2#7344011yescharacteristic
684C2#7348212yes—
685C2#7358212yes—
686C2#8206511yescharacteristic
687C2C2×#494511yescharacteristic
688C2#8226811yescharacteristic
689C2#8234512yes—
690C2#8234812yes—
subgroupG/H
centerC23×C4C22×D3
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC2×C3×C4×C22⋊C4C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#13842
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4×Dic3⋊C4C2semidirect
5. C2×#4932C2semidirect
8. C4×C3⋊#95C2semidirect
9. C4×C3⋊#95C2semidirect
10. #69418C2semidirect
11. #69414C2semidirect
28. C4×#217C4semidirect
678. C3#13842semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C2×C4×#2172
C2×C4×Dic3⋊C42
C2×#49322
#715492
#715472
C4×C3⋊#952
#694182
#694142
#138423