Finite Group Databasebeta

← #10802425

Subgroups — #10802425

Order 768.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#946C211yesmaximal, characteristic
2C2×#4815C211yesmaximal, characteristic
3C4×#892C211yesmaximal, characteristic
4C3×#1763C211yesmaximal, characteristic, complemented
5#71631C211yesmaximal, characteristic, complemented
6#70269C211yesmaximal, characteristic
7#70241C211yesmaximal, characteristic, complemented
8#70224C211yesmaximal, characteristic, complemented
9#69814C211yesmaximal, characteristic, complemented
10#69807C211yesmaximal, characteristic
11#69481C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic
13#68873C211yesmaximal, characteristic
14#68818C211yesmaximal, characteristic, complemented
15#68790C211yesmaximal, characteristic, complemented
16#21583—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×C42×D3C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33#4989—21no—
34#4987C2211yescharacteristic
35#4978—21no—
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×(C2×D3)⋊C4—21no—
42C4×(C2×D3)⋊C4—21no—
43C4×C4⋊Dic3C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4—21no—
46C42×Dic3—21no—
47C2×#803C2211yescharacteristic
48#4815C418yescomplemented
49#4666—21no—
50#4648—21no—
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#485—31no—
62C4×C22⋊Q8—31no—
63C4×#117—31no—
64#1928—31no—
65#1927—31no—
66#1921—31no—
67#1920—31no—
68#1873—31no—
69#1854—31no—
70#1825—31no—
71#1776—31no—
72#1776—31no—
73#1763—31no—
74#1763D311yescharacteristic, complemented
75#1749—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C4×D3D411yescharacteristic
78C22×C4×D3D411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#946—22no—
92#946C2×C414yes—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#892—22no—
115#892—22no—
116#892—22no—
117#892C2×C414yes—
118C42×D3—22no—
119C42×D3—22no—
120C3×#103—22no—
121C3×#103C2×C414yescomplemented
122C3×#97—22no—
123C3×#95—21no—
124#835—22no—
125#835C2×C414yescomplemented
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134(C2×D3)⋊C8—22no—
135(C2×D3)⋊C8C2×C414yes—
136#818—22no—
137#817C2×C414yescomplemented
138#808—22no—
139#807C2×C414yescomplemented
140#803—22no—
141C2×C22⋊Q8—31no—
142C2×C4×Q8C2×D311yescharacteristic
143C2×#117—31no—
144C2×#117—31no—
145C22×C42—31no—
146#485—38no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157C2×#97—31no—
158C2×#97C2×D311yescharacteristic
159#391—31no—
160#389—31no—
161#387—61no—
162#385—61no—
163#380—31no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—61no—
169C4×C4⋊C4C2×D311yescharacteristic
170C4×C22⋊C4—61no—
171C4×C22⋊C4—61no—
172C4×C22⋊C4—61no—
173#376—61no—
174C43—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#336—31no—
184#336—31no—
185C23×D3C2×D411yescharacteristic
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8C22×C412yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—41no—
214C2×C4×D3—41no—
215C2×C4×D3—41no—
216C2×C4×D3C22×C412yes—
217C2×C4×D3#4112yes—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8C22×C412yes—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4C22×C412yes—
230C3×C4⋊C4#4112yes—
231C3×C42—42no—
232C3×C42—42no—
233C3×C42#4112yes—
234C3×C42#4112yes—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3C22×C412yes—
245C4⋊Dic3C22×C412yes—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—24no—
248Dic3⋊C4—42no—
249C4×Dic3—24no—
250C4×Dic3—22no—
251C4×Dic3—22no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—24no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8C22×C412yes—
265C22×Q8C22×D311yescharacteristic
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C22⋊Q8—34no—
274C22⋊Q8—62no—
275C4×Q8—22no—
276C4×Q8—22no—
277#117—34no—
278#117—62no—
279#117—62no—
280#117—62no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42—62no—
303C2×C42—62no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—34no—
310#107—62no—
311#106—34no—
312#106—62no—
313#103—22no—
314#103—34no—
315#103—62no—
316#103C4×D314yescomplemented
317C22⋊C8—34no—
318C22⋊C8—62no—
319#97—22no—
320#97—62no—
321#95—21no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3C22×D411yescharacteristic
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3C4×D412yes—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4#12312yes—
359C2×C3×C4#12312yes—
360C2×C3×C4C23×C411yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3C4×D412yes—
386C2×Dic3#13712yes—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—22no—
391C4×D3—44no—
392C4×D3—44no—
393C3×C8—22no—
394C3×C8—41no—
395C3⋊C8—22no—
396C3⋊C8—41no—
397C24—31no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8C2×C4×D312yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—31no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—121no—
425C22×C4—121no—
426C22×C4—121no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—32no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8C2×C4×D312yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—32no—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—42no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4C2×C4×D312yes—
458C4⋊C4#23512yes—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—122no—
463C42—42no—
464C42—42no—
465C42—64no—
466C42—64no—
467C42—62no—
468C42—62no—
469C42—61no—
470C42—61no—
471C42—122no—
472C42—122no—
473C42—122no—
474C42—122no—
475C42#23512yes—
476C42#23712yes—
477C22×C3C2×C4×D411yescharacteristic
478C22×C3C2×#12311yescharacteristic
479C22×C3#52511yescharacteristic
480C22×C3C2×#13711yescharacteristic
481C22×C3#57511yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57511yescharacteristic
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4C4×#4111yescharacteristic
498C3×C4C4×#4111yescharacteristic
499C3×C4#52911yescharacteristic
500C3×C4#52911yescharacteristic
501Dic3—22no—
502Dic3—22no—
503Dic3—41no—
504Dic3—42no—
505Dic3—42no—
506Dic3—42no—
507Dic3—81no—
508C23—32no—
509C23—61no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23C2×D3×D411yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4#88312yes—
564C2×C4#88312yes—
565C2×C4C22×C4×D311yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23711yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—22no—
572C8—41no—
573C8—62no—
574C8—121no—
575C2×C3C4×#12311yescharacteristic
576C2×C3#294311yescharacteristic
577C2×C3C4×#13711yescharacteristic
578C2×C3#309311yescharacteristic
579C2×C3#309511yescharacteristic
580C2×C3#311111yescharacteristic
581C2×C3#311611yescharacteristic
582D3—42no—
583C22—62no—
584C22—62no—
585C22—62no—
586C22—121no—
587C22—121no—
588C22—121no—
589C22—121no—
590C22C2×#88311yescharacteristic
591C22C4×D3×D411yescharacteristic
592C22#554711yescharacteristic
593C22#577311yescharacteristic
594C22D3×#13711yescharacteristic
595C22#577711yescharacteristic
596C22#577711yescharacteristic
597C4—22no—
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—62no—
603C4—62no—
604C4—81no—
605C4—121no—
606C4—122no—
607C4—122no—
608C4—122no—
609C4—241no—
610C4C4×#23511yescharacteristic
611C4#552811yescharacteristic
612C4C4×#23711yescharacteristic
613C4#555711yescharacteristic
614C3#2158311yescharacteristic, complemented, Sylow
615C2—122no—
616C2C4×#88311yescharacteristic
617C2#7889311yescharacteristic
618C2#7896511yescharacteristic
619C2#7977211yescharacteristic
620C2#7978311yescharacteristic
621C2#7982911yescharacteristic
622C2#7984711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1763D3
hypocenterC3#21583
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. C3×#1763C2semidirect
5. #71631C2semidirect
7. #70241C2semidirect
8. #70224C2semidirect
9. #69814C2semidirect
11. #69481C2semidirect
14. #68818C2semidirect
15. #68790C2semidirect
48. #4815C4semidirect
74. #1763D3semidirect
121. C3×#103C2×C4semidirect
125. #835C2×C4semidirect
137. #817C2×C4semidirect
139. #807C2×C4semidirect
316. #103C4×D3semidirect
614. C3#21583semidirect

Maximal subgroups

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

subgroupindex
C4×#9462
C2×#48152
C4×#8922
C3×#17632
#716312
#702692
#702412
#702242
#698142
#698072
#694812
#688772
#688732
#688182
#687902
#215833