Finite Group Databasebeta

← #10798331

Subgroups — #10798331

Order 768.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic
2#82317C211yesmaximal, characteristic
3#76483C211yesmaximal, characteristic, complemented
4C3×#1877C211yesmaximal, characteristic
5#72047C211yesmaximal, characteristic
6#71606C211yesmaximal, characteristic
7#70242C211yesmaximal, characteristic, complemented
8#70220C211yesmaximal, characteristic, complemented
9#70167C211yesmaximal, characteristic
10#70152C211yesmaximal, characteristic
11#69667C211yesmaximal, characteristic, complemented
12#68818C211yesmaximal, characteristic, complemented
13#68785C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68598C211yesmaximal, characteristic, complemented
16#21130—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#391C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C4×#217—21no—
31C2×C3⋊#95—21no—
32Dic3×SD16C418yescomplemented
33C2×#817C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42Dic3×C22⋊C4—21no—
43Dic3×C22⋊C4—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45#4932C2211yescharacteristic
46#4928C2211yescharacteristic
47#4654—21no—
48#4653—21no—
49#4651—21no—
50#4650—21no—
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×SD16—31no—
62#2320—31no—
63#2275—31no—
64#1928—31no—
65#1928—31no—
66#1920—31no—
67#1920—31no—
68#1877—31no—
69#1877D311yescharacteristic
70#1862—31no—
71#1862—31no—
72C4×#107—31no—
73#1763—31no—
74#1762—31no—
75#1746—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×Dic3—21no—
80C23×Dic3—21no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16C2×C414yes—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94Q8×Dic3—22no—
95Q8×Dic3C2×C414yescomplemented
96D4×Dic3—22no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99D4×Dic3C2×C414yes—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#103—22no—
117C3×#102—22no—
118C3×#95—21no—
119C3×#95—21no—
120#835—22no—
121#835C2×C414yescomplemented
122#832—22no—
123#832C2×C414yes—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#817C2×C414yescomplemented
137#815—22no—
138C8×Dic3C2×C414yescomplemented
139#807—22no—
140#807—22no—
141C22×SD16C2×D311yescharacteristic
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4C2×D311yescharacteristic
146C4×SD16—38no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#102—31no—
154C2×#102—31no—
155C2×#102C2×D311yescharacteristic
156C2×#97—31no—
157C2×C4×C8—31no—
158#391—31no—
159#391C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#387—61no—
163#387—61no—
164#383—61no—
165#382—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—61no—
171C4×C22⋊C4—61no—
172C4×C22⋊C4—61no—
173#376—31no—
174C2×#95—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C24×C3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8C22×C412yes—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C22×C412yes—
196C2×C3×D4#4112yes—
197C2×C3×D4#4112yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—21no—
213C22×Dic3—81no—
214C22×Dic3—81no—
215C22×Dic3—81no—
216C22×Dic3—81no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C3×SD16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8C22×C412yes—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4#4112yes—
232C3×C22⋊C4—24no—
233C3×C42—21no—
234C3×C42—21no—
235#217—24no—
236#217—24no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—24no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3C22×C412yes—
244C4⋊Dic3C22×C412yes—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—24no—
251C4×Dic3—81no—
252C4×Dic3—81no—
253C4×Dic3—81no—
254C4×Dic3—81no—
255C4×Dic3C22×C412yes—
256C4×Dic3#4112yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C22×C412yes—
263C22×Q8C22×D311yescharacteristic
264C22×D4C22×D311yescharacteristic
265C23×C4—21no—
266C23×C4—61no—
267C23×C4—61no—
268C2×SD16—22no—
269C2×SD16—22no—
270C2×SD16C4×D314yes—
271C22×C8—31no—
272C22×C8—31no—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4×Q8—34no—
276C4×Q8—62no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—34no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—61no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304#107—62no—
305#106—34no—
306#103—22no—
307#103—34no—
308#103—62no—
309#102—22no—
310#102—34no—
311#102—62no—
312#97—62no—
313C4×C8—34no—
314#95—21no—
315#95—21no—
316#95—31no—
317#95—31no—
318#95—31no—
319#95—31no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C22×D411yescharacteristic
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×Q8—41no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4#11714yes—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
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—41no—
359C2×C3×C4—84no—
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—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—84no—
389C2×Dic3—84no—
390C2×Dic3C4×D412yes—
391C2×Dic3C4×D412yes—
392C3×C8—22no—
393C3×C8—41no—
394C3⋊C8—22no—
395C3⋊C8—41no—
396C24—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×C4×D312yes—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C2×C4×D312yes—
407C2×D4#23712yes—
408C2×D4#23712yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—64no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—64no—
422C22×C4—61no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4—241no—
428C22×C4—241no—
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4C23×D311yescharacteristic
432SD16—28no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—32no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8C2×C4×D312yes—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—32no—
449C4⋊C4—32no—
450C4⋊C4—32no—
451C4⋊C4—64no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4#23712yes—
460C22⋊C4—24no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—64no—
468C42—61no—
469C42—61no—
470C42—241no—
471C42—241no—
472C42—241no—
473C42—241no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3C2×C4×D411yescharacteristic
482C22×C3#52511yescharacteristic
483C22×C3#52511yescharacteristic
484C22×C3#57611yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57711yescharacteristic
487C22×C3#57811yescharacteristic
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4C2×#11711yescharacteristic
494C3×C4C4×#4111yescharacteristic
495C3×C4#52911yescharacteristic
496C3×C4#52911yescharacteristic
497Dic3—22no—
498Dic3—22no—
499Dic3—41no—
500Dic3—42no—
501Dic3—41no—
502Dic3—42no—
503Dic3—42no—
504Dic3—81no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C2×D3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517D4—22no—
518D4—22no—
519D4—22no—
520D4#87714yes—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—32no—
530C2×C4—32no—
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—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—84no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—244no—
562C2×C4—244no—
563C2×C4C22×C4×D311yescharacteristic
564C2×C4C2×#23711yescharacteristic
565C2×C4C2×#23711yescharacteristic
566C2×C4C2×#23711yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C8—22no—
570C8—41no—
571C8—62no—
572C8—121no—
573C2×C3—44no—
574C2×C3#229011yescharacteristic
575C2×C3#294911yescharacteristic
576C2×C3#295111yescharacteristic
577C2×C3#311811yescharacteristic
578C2×C3#311811yescharacteristic
579C2×C3#312011yescharacteristic
580C2×C3#312011yescharacteristic
581C22—24no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22C4×D3×D411yescharacteristic
589C22#558611yescharacteristic
590C22#560611yescharacteristic
591C22#576211yescharacteristic
592C22#577011yescharacteristic
593C22#577411yescharacteristic
594C22#577911yescharacteristic
595C4—41no—
596C4—42no—
597C4—41no—
598C4—42no—
599C4—62no—
600C4—62no—
601C4—81no—
602C4—121no—
603C4—122no—
604C4—121no—
605C4—122no—
606C4—122no—
607C4—241no—
608C4C2×#87711yescharacteristic
609C4C4×#23711yescharacteristic
610C4#560011yescharacteristic
611C4#564911yescharacteristic
612C3#2113011yescharacteristic, complemented, Sylow
613C2—44no—
614C2#7465711yescharacteristic
615C2#7964911yescharacteristic
616C2#7977811yescharacteristic
617C2#8013611yescharacteristic
618C2#8021611yescharacteristic
619C2#8041211yescharacteristic
620C2#8050911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1877D3
hypocenterC3#21130
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #76483C2semidirect
7. #70242C2semidirect
8. #70220C2semidirect
11. #69667C2semidirect
12. #68818C2semidirect
13. #68785C2semidirect
14. #68623C2semidirect
15. #68598C2semidirect
32. Dic3×SD16C4semidirect
95. Q8×Dic3C2×C4semidirect
121. #835C2×C4semidirect
136. #817C2×C4semidirect
138. C8×Dic3C2×C4semidirect
612. C3#21130semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
#823172
#764832
C3×#18772
#720472
#716062
#702422
#702202
#701672
#701522
#696672
#688182
#687852
#686232
#685982
#211303