Finite Group Databasebeta

← #10814911

Subgroups — #10814911

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic
2#76489C211yesmaximal, characteristic, complemented
3#76302C211yesmaximal, characteristic
4C3×#1968C211yesmaximal, characteristic
5#72049C211yesmaximal, characteristic
6#70554C211yesmaximal, characteristic, complemented
7Dic3×#106C211yesmaximal, characteristic, complemented
8#70241C211yesmaximal, characteristic, complemented
9#70221C211yesmaximal, characteristic, complemented
10#70174C211yesmaximal, characteristic
11#70153C211yesmaximal, characteristic
12C8×Dic3⋊C4C211yesmaximal, characteristic, complemented
13C4×#817C211yesmaximal, characteristic, complemented
14#69666C211yesmaximal, characteristic, complemented
15#69637C211yesmaximal, characteristic
16#22805—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×#877—21no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#390C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33Dic3×SD16C418yescomplemented
34C2×#817C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#4975C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4945—21no—
46#4944—21no—
47#4942—21no—
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50C4×Dic3⋊C4C2211yescharacteristic
51#4654—21no—
52#4653—21no—
53#4650—21no—
54#4649—21no—
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×SD16—31no—
62#2355—31no—
63#2354—31no—
64#1968—31no—
65#1968D311yescharacteristic
66#1927—31no—
67#1927—31no—
68#1926—31no—
69#1926—31no—
70#1844—31no—
71#1803—31no—
72#1802—31no—
73C4×#106—31no—
74C4×#106—31no—
75C8×C4⋊C4—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×Dic3—21no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16C2×C414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92Q8×Dic3—22no—
93Q8×Dic3C2×C414yes—
94D4×Dic3—22no—
95D4×Dic3C2×C414yes—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4D411yescharacteristic
106C2×Dic3⋊C4D411yescharacteristic
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
116#877—44no—
117C3×#106—22no—
118C3×#103—41no—
119C3×#102—41no—
120C3×C4×C8—22no—
121C3×#95—21no—
122C3×#95—21no—
123#835—41no—
124#835C2×C414yes—
125#832—41no—
126#832C2×C414yes—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#817—22no—
136#817C2×C414yes—
137Dic3⋊C8—22no—
138C8×Dic3—22no—
139C8×Dic3C2×C414yes—
140#808—41no—
141#808—41no—
142C22×SD16C2×D311yescharacteristic
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—61no—
146C4×SD16—38no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#106C2×D311yescharacteristic
152C2×C4⋊C8—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103C2×D311yescharacteristic
156C2×#102—31no—
157C2×#102—31no—
158C2×#102C2×D311yescharacteristic
159C2×C4×C8—31no—
160C2×C4×C8—31no—
161C2×C4×C8C2×D311yescharacteristic
162#391—31no—
163#391C2×D311yescharacteristic
164#390—31no—
165#390C2×D311yescharacteristic
166#388—61no—
167#388—61no—
168#387—61no—
169#387—61no—
170#384—31no—
171#384—31no—
172#382—61no—
173#380—61no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—31no—
179C4×C4⋊C4—61no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8C22×C412yes—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4C22×C412yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—24no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C3×SD16—28no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8C22×C412yes—
221C2×C3×C8#4112yes—
222C2×C3×C8#4112yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—81no—
229C3×C42—21no—
230C3×C42—21no—
231#217—24no—
232#217—42no—
233#217—81no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—24no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3C22×C412yes—
242C4⋊Dic3C22×C412yes—
243Dic3⋊C4—22no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248Dic3⋊C4SD1614yescomplemented
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—24no—
252C4×Dic3—42no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C4×Dic3—81no—
256C4×Dic3C22×C412yes—
257C4×Dic3#4112yes—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8C22×C412yes—
264C22×Q8C22×D311yescharacteristic
265C22×D4C22×D311yescharacteristic
266C23×C4—61no—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×SD16C4×D314yes—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C4×Q8—34no—
275C4×Q8—62no—
276C4×D4—34no—
277C4×D4—62no—
278#117—124no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
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—21no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#106—22no—
306#106—34no—
307#106—62no—
308#106—121no—
309#106—121no—
310C4⋊C8—62no—
311#103—34no—
312#103—41no—
313#103—121no—
314#102—34no—
315#102—41no—
316#102—121no—
317C4×C8—22no—
318C4×C8—34no—
319C4×C8—62no—
320#95—21no—
321#95—21no—
322#95—31no—
323#95—31no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—41no—
345C3×D4—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
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—41no—
360C2×C3×C4C2×SD1612yes—
361C2×C3×C4C23×C411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×C3×C42−1+411yescharacteristic
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—48no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3C4×D412yes—
388C2×Dic3C4×D412yes—
389C2×Dic3C2×SD1612yes—
390C2×Dic3C2×SD1612yes—
391C2×Dic3#13512yes—
392C3×C8—41no—
393C3×C8#11714yescomplemented
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×D4—21no—
407C2×D4—21no—
408C2×D4C2×C4×D312yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—64no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—28no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8C2×C4×D312yes—
437C2×C8#23712yes—
438C2×C8#23712yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—34no—
444C4⋊C4—32no—
445C4⋊C4—32no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—122no—
455C4⋊C4—122no—
456C4⋊C4—241no—
457C4⋊C4—241no—
458C22⋊C4—42no—
459C22⋊C4—64no—
460C22⋊C4—81no—
461C22⋊C4—122no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—64no—
468C42—61no—
469C42—61no—
470C42—122no—
471C42—122no—
472C42—241no—
473C42—241no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3C2×C4×D411yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3#55011yescharacteristic
484C22×C3C22×SD1611yescharacteristic
485C22×C3C2×#13511yescharacteristic
486C22×C3#57711yescharacteristic
487C22×C3#57711yescharacteristic
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4C2×#11711yescharacteristic
494C3×C4#52011yescharacteristic
495C3×C4#55511yescharacteristic
496C3×C4#55611yescharacteristic
497Dic3—22no—
498Dic3—22no—
499Dic3—41no—
500Dic3—42no—
501Dic3—42no—
502Dic3—81no—
503Dic3—81no—
504Dic3C4×SD1614yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C2×D3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517D4—22no—
518D4—22no—
519D4—41no—
520D4—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
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—128no—
550C2×C4—121no—
551C2×C4—121no—
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×C4D3×SD1612yes—
561C2×C4C22×C4×D311yescharacteristic
562C2×C4C2×#23711yescharacteristic
563C2×C4C2×#23711yescharacteristic
564C2×C4#100411yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4#101011yescharacteristic
567C8—41no—
568C8—62no—
569C8—121no—
570C8#87714yescomplemented
571C2×C3—82no—
572C2×C3#235411yescharacteristic
573C2×C3C2×C4×SD1611yescharacteristic
574C2×C3#295211yescharacteristic
575C2×C3#333511yescharacteristic
576C2×C3#334011yescharacteristic
577C2×C3#339211yescharacteristic
578C2×C3#339411yescharacteristic
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22C4×D3×D411yescharacteristic
587C22#561111yescharacteristic
588C22#571811yescharacteristic
589C22C2×D3×SD1611yescharacteristic
590C22#576211yescharacteristic
591C22D3×#13511yescharacteristic
592C22#577011yescharacteristic
593C4—22no—
594C4—34no—
595C4—41no—
596C4—42no—
597C4—62no—
598C4—62no—
599C4—81no—
600C4—81no—
601C4—121no—
602C4—122no—
603C4—122no—
604C4—241no—
605C4—241no—
606C4C2×#87711yescharacteristic
607C4#554611yescharacteristic
608C4#561611yescharacteristic
609C4#567111yescharacteristic
610C3#2280511yescharacteristic, complemented, Sylow
611C2—82no—
612C2#7630211yescharacteristic
613C2C4×D3×SD1611yescharacteristic
614C2#7965011yescharacteristic
615C2#8064811yescharacteristic
616C2#8065011yescharacteristic
617C2#8175511yescharacteristic
618C2#8181411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1968C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1968D3
hypocenterC3#22805
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #76489C2semidirect
6. #70554C2semidirect
7. Dic3×#106C2semidirect
8. #70241C2semidirect
9. #70221C2semidirect
12. C8×Dic3⋊C4C2semidirect
13. C4×#817C2semidirect
14. #69666C2semidirect
33. Dic3×SD16C4semidirect
248. Dic3⋊C4SD16semidirect
393. C3×C8#117semidirect
570. C8#877semidirect
610. C3#22805semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
#764892
#763022
C3×#19682
#720492
#705542
Dic3×#1062
#702412
#702212
#701742
#701532
C8×Dic3⋊C42
C4×#8172
#696662
#696372
#228053