Finite Group Databasebeta

← #10945098

Subgroups — #10945098

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5163C211yesmaximal, characteristic
2C4×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×#4791C211yesmaximal, characteristic, complemented
4#76424C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic, complemented
6#72048C211yesmaximal, characteristic, complemented
7#71633C211yesmaximal, characteristic, complemented
8#70240C211yesmaximal, characteristic, complemented
9#70161C211yesmaximal, characteristic
10#70156C211yesmaximal, characteristic, complemented
11#69637C211yesmaximal, characteristic, complemented
12#68800C211yesmaximal, characteristic, complemented
13#68792C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic, complemented
15#68605C211yesmaximal, characteristic, complemented
16#33025—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#940C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#885—21no—
25C2×#885—21no—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5163—24no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4988C2211yescharacteristic
40Dic3×C4⋊C4—21no—
41Dic3×C4⋊C4—21no—
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4957—21no—
47#4957—21no—
48#4939C2211yescharacteristic
49C4×C4⋊Dic3—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51C42×Dic3—21no—
52C42×Dic3C2211yescharacteristic
53#4791—24no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#475—31no—
63C2×#460—31no—
64#2858—31no—
65C42×Q8—31no—
66#1988—31no—
67#1982—31no—
68#1926—31no—
69#1926—31no—
70#1926—31no—
71#1926—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1880D311yescharacteristic, complemented
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×Dic6—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102Q8×Dic3D414yes—
103C2×#215—22no—
104C2×#215—22no—
105C2×#215—22no—
106#940—22no—
107#940—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3D411yescharacteristic
125C2×C4×Dic3D411yescharacteristic
126C2×C4×Dic3D411yescharacteristic
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131#888—22no—
132#888—22no—
133#885—44no—
134#885—44no—
135C3×#103—22no—
136C3×#102—22no—
137#835—22no—
138#832—22no—
139#832—22no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#821—22no—
144#821—22no—
145#817—22no—
146Dic3⋊C8—22no—
147Dic3⋊C8—22no—
148#809—22no—
149#808—22no—
150C22×SD16—31no—
151C22×SD16C2×D311yescharacteristic
152C2×C4⋊Q8—31no—
153C2×#127—31no—
154C2×#124—61no—
155C2×#124—61no—
156C2×C4×Q8—31no—
157C2×C4×Q8—31no—
158C2×C4×Q8C2×D311yescharacteristic
159#475—64no—
160#460—64no—
161C2×#106—31no—
162C2×#106—31no—
163C2×C4⋊C8—31no—
164C2×C4⋊C8—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#102C2×D311yescharacteristic
173#390—31no—
174#390—31no—
175#390—31no—
176#390—61no—
177#390—61no—
178#390C2×D311yescharacteristic
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185C43—31no—
186C43—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193#336—31no—
194#336C2×D311yescharacteristic
195C24×C3—21no—
196C23×D3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×D412yes—
201C2×C3×Q8#4112yes—
202C2×C3×Q8#4112yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—42no—
220C2×#67—42no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C3×SD16—44no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C42—24no—
252C3×C42—21no—
253C3×C42—21no—
254#217—42no—
255#217—42no—
256#215—44no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3C2×D412yes—
268C4⋊Dic3#4112yes—
269Dic3⋊C4—42no—
270C4×Dic3—24no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—24no—
274C4×Dic3—22no—
275C4×Dic3—41no—
276C4×Dic3—41no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280C4×Dic3SD1614yescomplemented
281C4×Dic3C2×D412yes—
282C4×Dic3#4112yes—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C22×Q8—61no—
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C22×D4—61no—
293C22×D4—61no—
294C22×D4C22×D311yescharacteristic
295C2×SD16—22no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16—62no—
299C2×SD16—62no—
300C2×SD16—62no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C4⋊Q8—62no—
306C4⋊Q8—62no—
307#127—62no—
308#127—62no—
309#124—124no—
310#124—124no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—34no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C42—21no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—61no—
342C2×C42—61no—
343C2×C42—61no—
344C2×C42—61no—
345C2×C42—61no—
346C2×C42C22×D311yescharacteristic
347#106—62no—
348#106—62no—
349C4⋊C8—62no—
350C4⋊C8—62no—
351#103—22no—
352#103—62no—
353#102—22no—
354#102—62no—
355#102—62no—
356#102—62no—
357#102—62no—
358#102—62no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3C22×D411yescharacteristic
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—22no—
380C3×Q8#12414yescomplemented
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4C2×SD1612yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406#67—84no—
407#67—84no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3#12112yes—
425C2×Dic3#12112yes—
426C2×Dic3C2×SD1612yes—
427C2×Dic3C2×SD1612yes—
428C2×Dic3AGL(1,ℤ/8)12yes—
429D12—41no—
430D12—41no—
431D12—41no—
432D12—41no—
433Dic6—84no—
434C3×C8—41no—
435C3×C8—41no—
436C3⋊C8—41no—
437C3⋊C8—41no—
438C24—21no—
439C24—61no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—122no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8#23512yes—
447C2×Q8D3×D412yes—
448C2×Q8#23712yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—122no—
464C2×D4—122no—
465C2×D4—122no—
466C2×D4—122no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4D3×D411yescharacteristic
478C22×C4D3×D411yescharacteristic
479C22×C4C23×D311yescharacteristic
480SD16—44no—
481SD16—124no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—32no—
499C4⋊C4—32no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—122no—
507C4⋊C4#23712yes—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—34no—
520C42—32no—
521C42—32no—
522C42—62no—
523C42—64no—
524C42—64no—
525C42—61no—
526C42—61no—
527C42—121no—
528C42—121no—
529C42—122no—
530C42—122no—
531C42—122no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3C2×#12111yescharacteristic
540C22×C3#52511yescharacteristic
541C22×C3#52511yescharacteristic
542C22×C3C22×SD1611yescharacteristic
543C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57511yescharacteristic
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C3×C4—22no—
554C3×C4—24no—
555C3×C4—41no—
556C3×C4—42no—
557C3×C4—81no—
558C3×C4C2×#12411yescharacteristic
559C3×C4#52911yescharacteristic
560C3×C4#52911yescharacteristic
561C3×C4#53211yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—41no—
565Dic3—42no—
566Dic3—42no—
567Dic3—81no—
568Dic3#46014yes—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23C2×D3×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8—244no—
588Q8#88514yescomplemented
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—121no—
594D4—121no—
595D4—121no—
596D4—121no—
597D4—244no—
598D4—244no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—32no—
609C2×C4—32no—
610C2×C4—32no—
611C2×C4—32no—
612C2×C4—32no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4D3×SD1612yes—
635C2×C4C2×#23511yescharacteristic
636C2×C4C2×D3×D411yescharacteristic
637C2×C4C2×#23711yescharacteristic
638C2×C4C2×#23711yescharacteristic
639C2×C4D3×#4111yescharacteristic
640C2×C4D3×#4111yescharacteristic
641C8—41no—
642C8—41no—
643C8—121no—
644C8—121no—
645C2×C3—161no—
646C2×C3#285011yescharacteristic
647C2×C3C2×#46011yescharacteristic
648C2×C3#304011yescharacteristic
649C2×C3#310311yescharacteristic
650C2×C3#310311yescharacteristic
651C2×C3#311511yescharacteristic
652C2×C3#311511yescharacteristic
653D3—161no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—241no—
662C22—241no—
663C22—241no—
664C22—241no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22#554711yescharacteristic
669C22#558611yescharacteristic
670C22D3×#12111yescharacteristic
671C22C2×D3×SD1611yescharacteristic
672C22#576811yescharacteristic
673C22D3×AGL(1,ℤ/8)11yescharacteristic
674C22#577711yescharacteristic
675C4—24no—
676C4—22no—
677C4—34no—
678C4—41no—
679C4—42no—
680C4—62no—
681C4—62no—
682C4—81no—
683C4—121no—
684C4—122no—
685C4—122no—
686C4—241no—
687C4C2×#88511yescharacteristic
688C4#553811yescharacteristic
689C4#559611yescharacteristic
690C4#564911yescharacteristic
691C3#3302511yescharacteristic, complemented, Sylow
692C2—161no—
693C2—481no—
694C2#7486911yescharacteristic
695C2#7965311yescharacteristic
696C2#7981111yescharacteristic
697C2#8002811yescharacteristic
698C2#8007111yescharacteristic
699C2D3×#46011yescharacteristic
700C2#8059911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1880D3
hypocenterC3#33025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. C4×Q8×Dic3C2semidirect
3. C2×#4791C2semidirect
5. C3×#1880C2semidirect
6. #72048C2semidirect
7. #71633C2semidirect
8. #70240C2semidirect
10. #70156C2semidirect
11. #69637C2semidirect
12. #68800C2semidirect
13. #68792C2semidirect
14. #68626C2semidirect
15. #68605C2semidirect
75. #1880D3semidirect
280. C4×Dic3SD16semidirect
380. C3×Q8#124semidirect
588. Q8#885semidirect
691. C3#33025semidirect

Maximal subgroups

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

subgroupindex
C2×#51632
C4×Q8×Dic32
C2×#47912
#764242
C3×#18802
#720482
#716332
#702402
#701612
#701562
#696372
#688002
#687922
#686262
#686052
#330253