Finite Group Databasebeta

← #10957960

Subgroups — #10957960

Order 768.

2714 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5524C211yesmaximal, characteristic
2C3×#2314C211yesmaximal, characteristic, complemented
3#82231C211yesmaximal, characteristic, complemented
4#75836C214yesmaximal, complemented
5#75489C214yesmaximal, complemented
6#75300C214yesmaximal, complemented
7#75298C214yesmaximal, complemented
8#74360C212yesmaximal, complemented
9#74229C212yesmaximal, complemented
10#74003C212yesmaximal, complemented
11D3×#380C212yesmaximal, complemented
12#73771C212yesmaximal, complemented
13C4⋊C4×D12C212yesmaximal, complemented
14#34360—31nomaximal, Sylow
15C22×#235C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18#5524C4116yescomplemented
19C2×#897C2212yes—
20C2×#895C2212yes—
21C2×#893C2214yes—
22C2×#892C2212yescomplemented
23C2×D3×C4⋊C4C2212yes—
24C2×D3×C4⋊C4C2212yes—
25C2×D3×C4⋊C4C2212yescomplemented
26C2×#886C2212yes—
27C2×#882C2212yes—
28C2×D3×C22⋊C4C2214yes—
29C2×C4×D12C2214yes—
30C2×C4×D12C2212yes—
31C2×C4×D12C2212yes—
32C3×#388C2214yes—
33C3×#387C2214yescomplemented
34C3×#384C2211yescharacteristic, complemented
35C3×#382C2211yescharacteristic, complemented
36C3×#380C2212yes—
37C3×C4×C4⋊C4C2212yescomplemented
38#5212C2212yes—
39#4989C2214yes—
40#4989C2214yescomplemented
41#4987C2212yes—
42#4987C2212yes—
43#4987C2212yescomplemented
44#4987C2212yescomplemented
45#4980C2214yescomplemented
46#4972C2214yescomplemented
47#4957C2214yes—
48#4956C2214yes—
49#4951C2214yes—
50#4942C2214yes—
51#4940C2214yescomplemented
52C4×(C2×D3)⋊C4C2214yes—
53#4936C2211yescharacteristic, complemented
54#4935C2211yescharacteristic, complemented
55C4×C4⋊Dic3C2212yescomplemented
56#4928C2214yescomplemented
57#4669C2214yescomplemented
58#4668C2214yes—
59#4668C2214yescomplemented
60#4667C2214yes—
61#4666C2214yes—
62#4665C2214yes—
63#4665C2214yes—
64#4665C2214yescomplemented
65D3×#95C2214yes—
66#4654C2214yescomplemented
67#4653C2214yescomplemented
68#4646C2214yescomplemented
69C2×#525—31no—
70#2376—34no—
71#2358—34no—
72#2352—32no—
73#2352—32no—
74D4×C4⋊C4—32no—
75D4×C4⋊C4—32no—
76#2329—34no—
77#2325—34no—
78#2314—31no—
79#2314D311yescharacteristic, complemented
80#2311—32no—
81C2×#380—32no—
82C24×D3C2311yescharacteristic
83C23×C3×C4C2311yescharacteristic
84C22×#67C2312yes—
85C23×Dic3C2311yescharacteristic
86C2×#235—42no—
87C2×#235—42no—
88C2×#235C2×C418yescomplemented
89C22×D12C2312yes—
90C22×D12C2312yes—
91C22×D12C2311yescharacteristic
92C22×C4×D3—42no—
93C22×C4×D3—42no—
94C22×C4×D3C2×C418yescomplemented
95C22×C4×D3C2312yes—
96C22×C4×D3C2312yes—
97C22×C4×D3C2314yes—
98C22×C4×D3C2314yes—
99C22×C4×D3C2314yes—
100C22×C4×D3C2312yes—
101C22×Dic6C2311yescharacteristic
102C3×#117C2×C418yescomplemented
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C42C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×#217C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2312yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×C4×Dic3C2314yes—
131C2×C4×Dic3C2314yes—
132#897C2×C4116yescomplemented
133#895C2×C4116yescomplemented
134#893—48no—
135#892—44no—
136D3×C4⋊C4—44no—
137D3×C4⋊C4—44no—
138D3×C4⋊C4—44no—
139#886C2×C4116yescomplemented
140#882C2×C4116yescomplemented
141D3×C22⋊C4—48no—
142C4×D12—48no—
143C4×D12C2×C4116yescomplemented
144C4×D12C2×C4116yescomplemented
145C3×#95C2314yes—
146C3×#95C2314yes—
147C3×#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C22×#41—31no—
156C24×C4—31no—
157#525—316no—
158C2×#123—32no—
159C2×C22⋊Q8—32no—
160C2×#121—32no—
161C2×#120—32no—
162C2×C4×D4—34no—
163C2×C4×D4—32no—
164C2×C4×D4—32no—
165C2×C4×D4—34no—
166C2×#117—32no—
167C2×#117C2×D311yescharacteristic
168C22×C4⋊C4—32no—
169C22×C4⋊C4—32no—
170C22×C4⋊C4—32no—
171C22×C22⋊C4—34no—
172#391—34no—
173#390—34no—
174#389—34no—
175#389—34no—
176#388—34no—
177#388—34no—
178#388—34no—
179#388C2×D314yes—
180#387—34no—
181#387—34no—
182#387—34no—
183#387C2×D314yescomplemented
184#386—34no—
185#386—34no—
186#386—34no—
187#386—34no—
188#385—34no—
189#385—34no—
190#385—34no—
191#385—34no—
192#384—31no—
193#384C2×D311yescharacteristic, complemented
194#382—31no—
195#382C2×D311yescharacteristic, complemented
196#380—32no—
197#380—32no—
198#380—32no—
199#380—32no—
200#380—32no—
201#380—34no—
202#380—34no—
203#380C2×D312yes—
204C4×C4⋊C4—32no—
205C4×C4⋊C4C2×D312yescomplemented
206C4×C22⋊C4—34no—
207C4×C22⋊C4—34no—
208C2×#95—34no—
209C24×C3C2411yescharacteristic
210C23×D3—24no—
211C23×D3—24no—
212C23×D3—41no—
213C23×D3—41no—
214C23×D3C22×C414yes—
215C23×D3C2412yes—
216C23×D3C2412yes—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4C22×C414yes—
220C22×C3×C4C2414yes—
221C22×C3×C4C2414yes—
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2412yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2411yescharacteristic
226C2×#67—42no—
227C2×#67—42no—
228C2×#67C22×C418yes—
229C22×Dic3—41no—
230C22×Dic3—41no—
231C22×Dic3C22×C414yes—
232C22×Dic3C2414yes—
233C22×Dic3C2414yes—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236#235—416no—
237C2×D12—24no—
238C2×D12—24no—
239C2×D12—24no—
240C2×D12—24no—
241C2×D12—41no—
242C2×D12—41no—
243C2×D12C22×C418yes—
244C2×D12C22×C418yes—
245C2×D12C22×C414yes—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×C4×D3—24no—
249C2×C4×D3—24no—
250C2×C4×D3—28no—
251C2×C4×D3—28no—
252C2×C4×D3—416no—
253C2×C4×D3—42no—
254C2×C4×D3—42no—
255C2×C4×D3—44no—
256C2×C4×D3—44no—
257C2×C4×D3—44no—
258C2×C4×D3—44no—
259C2×C4×D3—44no—
260C2×C4×D3—44no—
261C2×C4×D3—44no—
262C2×C4×D3C22×C418yes—
263C2×C4×D3C22×C418yes—
264C2×C4×D3C22×C418yes—
265C2×Dic6—41no—
266C2×Dic6—41no—
267C2×Dic6C22×C414yes—
268C3×C4⋊C4—24no—
269C3×C4⋊C4—24no—
270C3×C4⋊C4—44no—
271C3×C4⋊C4C22×C418yes—
272C3×C22⋊C4—44no—
273C3×C22⋊C4C22×C418yes—
274C3×C42—44no—
275C3×C42C22×C414yes—
276C3×C42C22×C414yes—
277#217—44no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—44no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—44no—
283(C2×D3)⋊C4C22×C418yes—
284(C2×D3)⋊C4C22×C418yes—
285(C2×D3)⋊C4C22×C418yes—
286(C2×D3)⋊C4C22×C418yes—
287C4⋊Dic3—24no—
288C4⋊Dic3—24no—
289C4⋊Dic3—44no—
290C4⋊Dic3C22×C418yes—
291C4⋊Dic3C22×C418yes—
292Dic3⋊C4—44no—
293Dic3⋊C4—44no—
294Dic3⋊C4—44no—
295C4×Dic3—44no—
296C4×Dic3—44no—
297C25—31no—
298C2×#41—38no—
299C2×#41—122no—
300C2×#41—122no—
301C22×Q8—31no—
302C22×D4—32no—
303C22×D4—32no—
304C22×D4—32no—
305C22×D4—31no—
306C23×C4—31no—
307C23×C4—32no—
308C23×C4—32no—
309C23×C4—38no—
310C23×C4—32no—
311C23×C4—34no—
312C23×C4—34no—
313C23×C4—34no—
314C23×C4—122no—
315C23×C4—122no—
316C23×C4C22×D311yescharacteristic
317#123—316no—
318C22⋊Q8—316no—
319#121—316no—
320#120—316no—
321C4×D4—316no—
322C4×D4—316no—
323C4×D4—128no—
324C4×D4—128no—
325#117—124no—
326#117C4×D318yescomplemented
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—124no—
336C2×C4⋊C4—124no—
337C2×C4⋊C4—124no—
338C2×C4⋊C4C22×D312yes—
339C2×C4⋊C4C22×D312yes—
340C2×C4⋊C4C22×D314yes—
341C2×C4⋊C4C22×D312yes—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4—128no—
353C2×C22⋊C4C22×D314yes—
354C2×C22⋊C4C22×D312yes—
355C2×C42—34no—
356C2×C42—34no—
357C2×C42C22×D314yes—
358C2×C42C22×D311yescharacteristic
359C2×C42C22×D311yescharacteristic
360#95—34no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95C22×D314yes—
368#95C22×D314yes—
369#95C22×D314yes—
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—22no—
374C23×C3C23×C412yes—
375C23×C3C2511yescharacteristic
376C22×D3—22no—
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—22no—
382C22×D3—22no—
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—22no—
386C22×D3—216no—
387C22×D3—44no—
388C22×D3—48no—
389C22×D3—44no—
390C22×D3—44no—
391C22×D3C23×C414yes—
392C22×D3C23×C414yes—
393C22×D3H(5,𝔽2)14yes—
394C22×D32−1+414yes—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—28no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—24no—
407C2×C3×C4—24no—
408C2×C3×C4C23×C414yes—
409C2×C3×C4C23×C414yes—
410C2×C3×C4C23×C412yes—
411C2×C3×C4C23×C412yes—
412C2×C3×C4H(5,𝔽2)12yes—
413C2×C3×C4H(5,𝔽2)12yes—
414#67—48no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—24no—
420C2×Dic3—24no—
421C2×Dic3—24no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—44no—
426C2×Dic3C23×C414yes—
427C2×Dic3C23×C414yes—
428D12—216no—
429D12—216no—
430D12—44no—
431C4×D3—216no—
432C4×D3—216no—
433C4×D3—48no—
434C4×D3—88no—
435C4×D3—88no—
436C4×D3—88no—
437Dic6—44no—
438C24—32no—
439C24—32no—
440C24—34no—
441C24—64no—
442C24—64no—
443C24—121no—
444C24—121no—
445C24C23×D311yescharacteristic
446#41—1216no—
447C2×Q8—34no—
448C2×Q8—121no—
449C2×Q8—121no—
450C2×D4—38no—
451C2×D4—38no—
452C2×D4—38no—
453C2×D4—34no—
454C2×D4—64no—
455C2×D4—64no—
456C2×D4—64no—
457C2×D4—64no—
458C2×D4—121no—
459C2×D4—121no—
460C2×D4—122no—
461C2×D4—122no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—32no—
465C22×C4—32no—
466C22×C4—34no—
467C22×C4—38no—
468C22×C4—38no—
469C22×C4—38no—
470C22×C4—34no—
471C22×C4—34no—
472C22×C4—68no—
473C22×C4—68no—
474C22×C4—64no—
475C22×C4—64no—
476C22×C4—64no—
477C22×C4—64no—
478C22×C4—1216no—
479C22×C4—122no—
480C22×C4—122no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4—124no—
484C22×C4—124no—
485C22×C4—124no—
486C22×C4—124no—
487C22×C4—124no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4C2×C4×D314yes—
491C22×C4C23×D314yes—
492C22×C4C23×D314yes—
493C22×C4C23×D311yescharacteristic
494C22×C4C23×D312yes—
495C22×C4C23×D312yes—
496C22×C4C23×D311yescharacteristic
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—38no—
500C4⋊C4—38no—
501C4⋊C4—44no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—124no—
505C4⋊C4—124no—
506C4⋊C4—124no—
507C4⋊C4—124no—
508C4⋊C4C2×C4×D318yes—
509C22⋊C4—38no—
510C22⋊C4—38no—
511C22⋊C4—38no—
512C22⋊C4—38no—
513C22⋊C4—44no—
514C22⋊C4—68no—
515C22⋊C4—68no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C22⋊C4—124no—
520C22⋊C4C2×C4×D318yes—
521C42—44no—
522C42—124no—
523C42—124no—
524C42C2×C4×D314yes—
525C42C2×C4×D314yes—
526C22×C3—22no—
527C22×C3—24no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3C24×C411yescharacteristic
532C22×C3C2×H(5,𝔽2)11yescharacteristic
533C22×C3C2×H(5,𝔽2)11yescharacteristic
534C22×C3C2×H(5,𝔽2)12yes—
535C22×C3C2×2−1+412yes—
536C2×D3—24no—
537C2×D3—24no—
538C2×D3—24no—
539C2×D3—24no—
540C2×D3—24no—
541C2×D3—24no—
542C2×D3—24no—
543C2×D3—24no—
544C2×D3—24no—
545C2×D3—416no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3#52018yes—
551C3×C4—22no—
552C3×C4—24no—
553C3×C4—24no—
554C3×C4—44no—
555C3×C4—44no—
556C3×C4#51814yes—
557Dic3—24no—
558Dic3—24no—
559Dic3—44no—
560Dic3—44no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—22no—
565C23—34no—
566C23—34no—
567C23—34no—
568C23—34no—
569C23—616no—
570C23—62no—
571C23—62no—
572C23—62no—
573C23—62no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—62no—
580C23—124no—
581C23—128no—
582C23—124no—
583C23—124no—
584C23C22×C4×D312yes—
585C23C24×D311yescharacteristic
586Q8—124no—
587D4—616no—
588D4—616no—
589D4—124no—
590D4—128no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—34no—
605C2×C4—34no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—616no—
611C2×C4—616no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—128no—
619C2×C4—124no—
620C2×C4—248no—
621C2×C4—248no—
622C2×C4—248no—
623C2×C4C22×C4×D314yes—
624C2×C4C22×C4×D314yes—
625C2×C4C22×C4×D312yes—
626C2×C4C22×C4×D312yes—
627C2×C4#100912yes—
628C2×C4#100912yes—
629C2×C3—42no—
630C2×C3C2×#51811yescharacteristic
631C2×C3C2×#52012yes—
632C2×C3#353612yes—
633C2×C3#353711yescharacteristic
634C2×C3#353811yescharacteristic
635D3—28no—
636D3—44no—
637C22—24no—
638C22—22no—
639C22—38no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—64no—
644C22—64no—
645C22—64no—
646C22—64no—
647C22—64no—
648C22—64no—
649C22—64no—
650C22—64no—
651C22—64no—
652C22—1216no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22C23×C4×D311yescharacteristic
658C22C2×#100911yescharacteristic
659C22C2×#100911yescharacteristic
660C22D3×H(5,𝔽2)12yes—
661C22D3×2−1+412yes—
662C4—24no—
663C4—24no—
664C4—22no—
665C4—44no—
666C4—44no—
667C4—64no—
668C4—64no—
669C4—124no—
670C4—124no—
671C4#552114yes—
672C3#3436011yescharacteristic, complemented, Sylow
673C2—42no—
674C2—68no—
675C2—124no—
676C2C2×#552111yescharacteristic
677C2D3×#52012yes—
678C2#8649111yescharacteristic
679C2#8651511yescharacteristic
680C2#8656112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2314C2
socle, cosocleC23×C3C25
prime radicalC2×#5524C2
hypercenter#2314D3
hypocenterC3#34360
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. C3×#2314C2semidirect
3. #82231C2semidirect
4. #75836C2semidirect
5. #75489C2semidirect
6. #75300C2semidirect
7. #75298C2semidirect
8. #74360C2semidirect
9. #74229C2semidirect
10. #74003C2semidirect
11. D3×#380C2semidirect
12. #73771C2semidirect
13. C4⋊C4×D12C2semidirect
18. #5524C4semidirect
22. C2×#892C22semidirect
25. C2×D3×C4⋊C4C22semidirect
33. C3×#387C22semidirect
34. C3×#384C22semidirect
35. C3×#382C22semidirect
37. C3×C4×C4⋊C4C22semidirect
40. #4989C22semidirect
43. #4987C22semidirect
44. #4987C22semidirect
45. #4980C22semidirect
46. #4972C22semidirect
51. #4940C22semidirect
53. #4936C22semidirect
54. #4935C22semidirect
55. C4×C4⋊Dic3C22semidirect
56. #4928C22semidirect
57. #4669C22semidirect
59. #4668C22semidirect
64. #4665C22semidirect
66. #4654C22semidirect
67. #4653C22semidirect
68. #4646C22semidirect
79. #2314D3semidirect
88. C2×#235C2×C4semidirect
94. C22×C4×D3C2×C4semidirect
102. C3×#117C2×C4semidirect
132. #897C2×C4semidirect
133. #895C2×C4semidirect
139. #886C2×C4semidirect
140. #882C2×C4semidirect
143. C4×D12C2×C4semidirect
144. C4×D12C2×C4semidirect
183. #387C2×D3semidirect
193. #384C2×D3semidirect
195. #382C2×D3semidirect
205. C4×C4⋊C4C2×D3semidirect
326. #117C4×D3semidirect
672. C3#34360semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
C3×#23142
#822312
#758362
#754892
#753002
#752982
#743602
#742292
#740032
D3×#3802
#737712
C4⋊C4×D122
#343603