Finite Group Databasebeta

← #10801122

Subgroups — #10801122

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×2−1+4C211yesmaximal, characteristic, complemented
2C2×#5245C211yesmaximal, characteristic
3#78866C212yesmaximal, complemented
4#76484C211yesmaximal, characteristic, complemented
5C3×#1793C211yesmaximal, characteristic, complemented
6#70366C212yesmaximal, complemented
7#70354C212yesmaximal, complemented
8#70297C212yesmaximal, complemented
9#70104C211yesmaximal, characteristic, complemented
10Dic3×#101C211yesmaximal, characteristic, complemented
11#69672C211yesmaximal, characteristic, complemented
12#21475—31nomaximal, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14#5827—22no—
15#5827—22no—
16#5827C2212yes—
17Dic3×#41—22no—
18Dic3×#41—22no—
19Dic3×#41C2212yes—
20C2×#949C2212yes—
21C2×#947C2211yescharacteristic
22#5812—22no—
23#5812C2211yescharacteristic
24C2×Q8×Dic3—22no—
25C2×Q8×Dic3C2211yescharacteristic
26C2×#893C2211yescharacteristic
27C2×C3×#104C2212yes—
28C3×#417C2212yes—
29C2×C3×#101C2211yescharacteristic
30C3×#391C2211yescharacteristic
31#5245C418yescomplemented
32C2×#837C2212yes—
33C2×#836C2211yescharacteristic
34#5131C2212yes—
35Dic3×M16C2212yes—
36#4975C2211yescharacteristic
37#4964C2212yes—
38#4939C2211yescharacteristic
39#4932C2211yescharacteristic
40#4817—22no—
41#4788—22no—
42#4785—22no—
43#4756—22no—
44#4669—22no—
45#4668—22no—
46#4555C2212yes—
47#4554C2212yes—
48#4523C2212yes—
49C4×2−1+4—31no—
50C2×#454—31no—
51C4×AGL(1,ℤ/8)—32no—
52#2326—31no—
53#1888—31no—
54#1883—32no—
55#1859—32no—
56#1857—31no—
57#1793D311yescharacteristic, complemented
58#1759—32no—
59C4×#101—31no—
60C3×2−1+4—24no—
61C3×2−1+4—22no—
62C3×2−1+4C2×C414yescomplemented
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41C2312yes—
66C22×C3×Q8—22no—
67C22×C3×Q8C2311yescharacteristic
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C2×C3×M16C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73#949—24no—
74#949C2×C418yescomplemented
75#947—22no—
76#947C2×C414yescomplemented
77Q8×Dic3—22no—
78Q8×Dic3—24no—
79Q8×Dic3—22no—
80Q8×Dic3—22no—
81Q8×Dic3—22no—
82Q8×Dic3—22no—
83Q8×Dic3—22no—
84Q8×Dic3—22no—
85Q8×Dic3D412yes—
86Q8×Dic3D412yes—
87C2×#215—22no—
88D4×Dic3—22no—
89D4×Dic3—24no—
90D4×Dic3—22no—
91D4×Dic3—22no—
92D4×Dic3—22no—
93D4×Dic3—24no—
94D4×Dic3—22no—
95D4×Dic3D412yes—
96D4×Dic3D412yes—
97C2×#213—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—22no—
101#926—24no—
102#926—24no—
103#926—22no—
104#926—22no—
105#926—22no—
106#926C2312yes—
107C2×C4⋊Dic3—24no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3D412yes—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—22no—
112C2×C4×Dic3—24no—
113C2×C4×Dic3—22no—
114C2×C4×Dic3D412yes—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×#208C2312yes—
118#893—22no—
119#893—22no—
120C3×#104—24no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#101—22no—
124C3×#95—22no—
125#837C2×C418yescomplemented
126#836C2×C414yescomplemented
127C3⋊#95—21no—
128C3⋊#95—21no—
129#821—22no—
130#821—22no—
131#815—22no—
132C8×Dic3—22no—
133#808—22no—
134#807—22no—
135C2×2−1+4C2×D311yescharacteristic
136C2×AGL(1,ℤ/8)—32no—
137C2×#124—31no—
138#520—32no—
139#520—62no—
140#520—62no—
141#519—31no—
142#519—62no—
143C4×#41—32no—
144C4×#41—62no—
145C4×#41—62no—
146C2×C4×Q8—31no—
147C2×C4×Q8—62no—
148C2×C4×D4—31no—
149#454—38no—
150#442—62no—
151#440—62no—
152C4×SD16—62no—
153C4×D8—62no—
154#428—32no—
155C2×#104—32no—
156C2×#104C2×D312yes—
157#418—32no—
158#417C2×D312yes—
159C2×#101—31no—
160C2×#101C2×D311yescharacteristic
161C4×M16—32no—
162#391C2×D311yescharacteristic
163#390—31no—
164#388—62no—
165#385—62no—
166#384—31no—
167#376—31no—
168#343—32no—
169#339—32no—
170#339—32no—
171C23×D3—21no—
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C3×#41—22no—
177C3×#41—22no—
178C3×#41C22×C414yes—
179C3×#41#4114yes—
180C2×C3×Q8—22no—
181C2×C3×Q8—28no—
182C2×C3×Q8—28no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8C22×C412yes—
186C2×C3×Q8C2×D412yes—
187C2×C3×Q8#4112yes—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4C2×D412yes—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—24no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—21no—
207C2×D12C22×C412yes—
208C2×C4×D3—42no—
209C2×C4×D3—82no—
210C3×M16—22no—
211C3×M16—22no—
212C2×C3×C8—22no—
213C3×C4⋊C4—22no—
214C3×C42—22no—
215#217—24no—
216#217—22no—
217#217—22no—
218#217—22no—
219#215—28no—
220#213—28no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—41no—
223(C2×D3)⋊C4—41no—
224(C2×D3)⋊C4—82no—
225C4⋊Dic3—24no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3—24no—
231C4⋊Dic3—22no—
232C4⋊Dic3C2×D412yes—
233C4⋊Dic3C2×D412yes—
234Dic3⋊C4—82no—
235C4×Dic3—24no—
236C4×Dic3—24no—
237C4×Dic3—22no—
238C4×Dic3—21no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241C4×Dic3—22no—
242C4×Dic3—22no—
243C4×Dic3C22×C412yes—
244C4×Dic3C2×D412yes—
245C4×Dic3C2×D412yes—
246#208—22no—
247#208C22×C414yes—
248C2×C3⋊C8—22no—
2492−1+4—24no—
2502−1+4—22no—
2512−1+4C4×D314yescomplemented
252C2×#41—22no—
253C2×#41—22no—
254C2×#41C22×D312yes—
255C22×Q8—22no—
256C22×Q8C22×D311yescharacteristic
257C22×D4—31no—
258C23×C4—61no—
259AGL(1,ℤ/8)—38no—
260AGL(1,ℤ/8)—64no—
261C2×SD16—62no—
262C2×D8—62no—
263C2×M16—32no—
264C2×M16C22×D312yes—
265#124—34no—
266#124—62no—
267C4×Q8—32no—
268C4×Q8—32no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8—62no—
274C4×Q8—64no—
275C4×Q8—62no—
276C4×Q8—62no—
277C4×D4—32no—
278C4×D4—32no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—64no—
284C4×D4—62no—
285C4×D4—62no—
286C4×D4—64no—
287C4×D4—62no—
288#117—32no—
289#117—62no—
290#117—62no—
291#117—64no—
292#117—64no—
293#117—62no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—64no—
298C2×C4⋊C4—62no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C22⋊C4—62no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—62no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—32no—
306C2×C42—62no—
307C2×C42—64no—
308C2×C42C22×D311yescharacteristic
309#107—62no—
310#106—62no—
311#104—24no—
312#104—38no—
313#103—22no—
314#102—22no—
315#102—62no—
316#102—62no—
317#101—22no—
318#101—34no—
319#97—62no—
320C4×C8—62no—
321#95—22no—
322#95—61no—
323#95—61no—
324C23×C3—22no—
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C3×Q8—28no—
332C3×Q8—24no—
333C3×Q8—22no—
334C3×Q8—24no—
335C3×Q8C4×D414yes—
336C3×D4—28no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—22no—
340C3×D4C4×D414yes—
341C2×C3×C4—216no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4C4×D414yes—
353C2×C3×C4C23×C411yescharacteristic
354C2×C3×C4C22×D412yes—
355C2×C3×C4C2×#4112yes—
356C2×C3×C4C2×#4111yescharacteristic
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—44no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—82no—
368C2×Dic3—84no—
369C2×Dic3#12011yescharacteristic
370C2×Dic3#12011yescharacteristic
371C2×Dic3#12012yes—
372D12—24no—
373D12—42no—
374C4×D3—84no—
375C3×C8—42no—
376C3⋊C8—24no—
377C24—61no—
378#41—28no—
379#41—28no—
380#41—28no—
381#41—28no—
382#41—22no—
383#41—22no—
384#41C2×C4×D314yes—
385#41#23714yes—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8C2×C4×D312yes—
392C2×Q8D3×D412yes—
393C2×Q8#23712yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—32no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4—61no—
400C2×D4D3×D412yes—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—32no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—64no—
412C22×C4—122no—
413C22×C4—242no—
414C22×C4D3×D412yes—
415C22×C4C23×D311yescharacteristic
416SD16—68no—
417D8—68no—
418M16—22no—
419M16—22no—
420M16—34no—
421M16—62no—
422C2×C8—22no—
423C2×C8—62no—
424C4⋊C4—22no—
425C4⋊C4—32no—
426C4⋊C4—32no—
427C4⋊C4—64no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—64no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—242no—
435C22⋊C4—62no—
436C22⋊C4—62no—
437C22⋊C4—64no—
438C22⋊C4—62no—
439C22⋊C4—68no—
440C22⋊C4—121no—
441C22⋊C4—121no—
442C22⋊C4—242no—
443C42—22no—
444C42—32no—
445C42—32no—
446C42—32no—
447C42—62no—
448C42—62no—
449C42—62no—
450C42—62no—
451C42—64no—
452C42—64no—
453C42—62no—
454C42—61no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—24no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3C2×C4×D411yescharacteristic
461C22×C3C2×#12011yescharacteristic
462C22×C3#52511yescharacteristic
463C2×D3—44no—
464C2×D3—42no—
465C2×D3—82no—
466C2×D3—81no—
467C2×D3—81no—
468C3×C4—24no—
469C3×C4—24no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4C2×C4×D412yes—
473C3×C4#52512yes—
474Dic3—21no—
475Dic3—21no—
476Dic3—22no—
477Dic3—42no—
478Dic3—42no—
479Dic3—42no—
480Dic3—42no—
481C23—22no—
482C23—62no—
483C23—122no—
484C23—122no—
485C23—121no—
486C23—121no—
487C23C2×D3×D411yescharacteristic
488Q8—28no—
489Q8—24no—
490Q8—22no—
491Q8—24no—
492Q8#67⋊C414yes—
493D4—28no—
494D4—24no—
495D4—24no—
496D4—22no—
497D4—64no—
498D4—122no—
499D4#67⋊C414yes—
500C2×C4—216no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—32no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—124no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—244no—
525C2×C4—242no—
526C2×C4—244no—
527C2×C4#67⋊C414yes—
528C2×C4C22×C4×D311yescharacteristic
529C2×C4C2×D3×D412yes—
530C2×C4C2×#23712yes—
531C2×C4C2×#23711yescharacteristic
532C8—42no—
533C8—64no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—44no—
537C2×C3C4×#12011yescharacteristic
538C2×C3#302011yescharacteristic
539C2×C3#302111yescharacteristic
540D3—82no—
541C22—21no—
542C22—21no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—124no—
547C22—122no—
548C22—242no—
549C22—241no—
550C22—241no—
551C22C2×#67⋊C411yescharacteristic
552C22#558611yescharacteristic
553C22D3×#12011yescharacteristic
554C4—24no—
555C4—24no—
556C4—62no—
557C4—61no—
558C4—61no—
559C4—81no—
560C4—81no—
561C4—122no—
562C4—122no—
563C4—122no—
564C4—122no—
565C4C2×#67⋊C412yes—
566C4#558612yes—
567C3#2147511yescharacteristic, complemented, Sylow
568C2—21no—
569C2—21no—
570C2—44no—
571C2—242no—
572C2#7381811yescharacteristic
573C2#8028611yescharacteristic
574C2#8029911yescharacteristic
subgroupG/H
centerC22D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1793C2
socleC22×C3C2×#120
prime radicalC2×#5245C2
cosocleC22×C3×C4C24
hypercenter#1793D3
hypocenterC3#21475
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. Dic3×2−1+4C2semidirect
3. #78866C2semidirect
4. #76484C2semidirect
5. C3×#1793C2semidirect
6. #70366C2semidirect
7. #70354C2semidirect
8. #70297C2semidirect
9. #70104C2semidirect
10. Dic3×#101C2semidirect
11. #69672C2semidirect
31. #5245C4semidirect
57. #1793D3semidirect
62. C3×2−1+4C2×C4semidirect
74. #949C2×C4semidirect
76. #947C2×C4semidirect
125. #837C2×C4semidirect
126. #836C2×C4semidirect
251. 2−1+4C4×D3semidirect
567. C3#21475semidirect

Maximal subgroups

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

subgroupindex
Dic3×2−1+42
C2×#52452
#788662
#764842
C3×#17932
#703662
#703542
#702972
#701042
Dic3×#1012
#696722
#214753