Finite Group Databasebeta

← #10828463

Subgroups — #10828463

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×#4863C212yesmaximal, complemented
3D3×#389C211yesmaximal, characteristic, complemented
4C3×#2058C211yesmaximal, characteristic, complemented
5#71783C211yesmaximal, characteristic, complemented
6#70584C212yesmaximal, complemented
7#69627C211yesmaximal, characteristic, complemented
8#68895C212yesmaximal, complemented
9#68880C212yesmaximal, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#24061—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16#5725—28no—
17C2×#895—21no—
18C2×#895C2212yes—
19C2×#893—21no—
20C2×#893C2212yes—
21C2×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×#874C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C3×#106C2212yes—
27C3×#389C2211yescharacteristic
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31#4984C2211yescharacteristic
32#4976—21no—
33#4972C2211yescharacteristic
34C2×#809C2212yes—
35C2×#809C2212yes—
36C2×#808C2212yes—
37#4928C2211yescharacteristic
38#4863—28no—
39D3×#95—21no—
40#4647—21no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#481—32no—
48C2×#389—31no—
49#2058—31no—
50#2058D311yescharacteristic, complemented
51#2041—31no—
52#2041—31no—
53#1874—32no—
54#1864—32no—
55#1864—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—44no—
60C2×#239—44no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×C4×D3—21no—
66C22×C4×D3—22no—
67C22×C4×D3—21no—
68C22×C4×D3D411yescharacteristic
69C22×C4×D3D411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—24no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79#947—28no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#895—28no—
94#895—24no—
95#895D418yes—
96#893—28no—
97#893—24no—
98#893—24no—
99D3×C4⋊C4—22no—
100D3×C4⋊C4—22no—
101D3×C4⋊C4—24no—
102D3×C4⋊C4—44no—
103#874—24no—
104C42×D3—24no—
105C3×#106—24no—
106C3×#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—24no—
111#821—24no—
112(C2×D3)⋊C8—24no—
113#809—24no—
114#809—24no—
115#808—24no—
116C22×#41—61no—
117#532—68no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×#127—31no—
120C2×#124—61no—
121C2×#121—32no—
122C2×#121—61no—
123C2×C4×D4—32no—
124C2×C4×D4—61no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—61no—
128C22×C42—31no—
129#481—68no—
130C2×#106—32no—
131C2×#106C2×D312yes—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×C22⋊C8—32no—
137#389—31no—
138#389—31no—
139#389—31no—
140#389—61no—
141#389—61no—
142#389C2×D311yescharacteristic
143C2×#95—61no—
144#340—32no—
145#340C2×D312yes—
146#336—32no—
147#336—31no—
148#336C2×D311yescharacteristic
149C23×D3—22no—
150C23×D3C2×D411yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—42no—
153C2×C3×Q8—42no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165#239—432no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—24no—
171C2×D12—24no—
172C2×D12—28no—
173C2×D12—22no—
174C2×D12—42no—
175C2×D12—42no—
176C2×D12—44no—
177C2×D12—44no—
178C2×D12—42no—
179C2×D12—42no—
180C2×D12C2×D414yes—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—22no—
186C2×C4×D3—24no—
187C2×C4×D3—28no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—41no—
191C2×C4×D3—44no—
192C2×C4×D3—42no—
193C2×C4×D3—42no—
194C2×C4×D3—44no—
195C2×C4×D3—42no—
196C2×C4×D3—42no—
197C2×C4×D3—42no—
198C2×C4×D3SD1614yes—
199C2×C4×D3C2×D414yes—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D414yes—
209C3×C4⋊C4#4112yescomplemented
210C3×C42—22no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—28no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—42no—
216C4⋊Dic3#4112yescomplemented
217Dic3⋊C4—24no—
218Dic3⋊C4—42no—
219Dic3⋊C4—42no—
220Dic3⋊C4—42no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×#41—68no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—21no—
231C22×D4—32no—
232C22×D4—62no—
233C22×D4—61no—
234C22×D4—61no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—62no—
239C23×C4—61no—
240C23×C4—61no—
241C22×C8—32no—
242C22×C8C22×D312yes—
243C4⋊Q8—24no—
244#127—64no—
245#124—68no—
246#121—38no—
247#121—64no—
248#121—68no—
249C4×D4—64no—
250C4×D4—68no—
251C4×D4—64no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—124no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—31no—
273C2×C42—64no—
274C2×C42C22×D311yescharacteristic
275#106—24no—
276#106—64no—
277#102—64no—
278#102—64no—
279#102—64no—
280#102—64no—
281C22⋊C8—64no—
282#95—21no—
283#95—61no—
284#95—61no—
285#95—61no—
286C23×C3C22×D411yescharacteristic
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—24no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3#12012yes—
299C22×D3C2×SD1612yes—
300C22×D3AGL(1,ℤ/8)12yes—
301C3×Q8—48no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4C2×SD1612yes—
316C2×C3×C4AGL(1,ℤ/8)12yes—
317C2×C3×C4C22×D412yes—
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C42−1+412yes—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—22no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3#12012yes—
335C2×Dic3C2×SD1612yes—
336D12—24no—
337D12—24no—
338D12—48no—
339D12—48no—
340D12—416no—
341D12—42no—
342D12—42no—
343C4×D3—28no—
344C4×D3—24no—
345C4×D3—24no—
346C4×D3—416no—
347C4×D3—44no—
348C4×D3—48no—
349C4×D3—84no—
350C3×C8—42no—
351C3⋊C8—42no—
352C24—31no—
353C24—62no—
354#41—1232no—
355C2×Q8—24no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×D4—34no—
359C2×D4—68no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—124no—
370C2×D4—124no—
371C2×D4—122no—
372C2×D4—122no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—31no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—64no—
385C22×C4—68no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—121no—
393C22×C4—124no—
394C22×C4—124no—
395C22×C4—122no—
396C22×C4—122no—
397C22×C4—122no—
398C22×C4—122no—
399C22×C4—122no—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—32no—
414C4⋊C4—42no—
415C4⋊C4—64no—
416C4⋊C4—61no—
417C4⋊C4—62no—
418C4⋊C4—122no—
419C4⋊C4—122no—
420C4⋊C4—122no—
421C4⋊C4—122no—
422C4⋊C4D3×D414yes—
423C4⋊C4#23912yescomplemented
424C22⋊C4—68no—
425C22⋊C4—68no—
426C42—22no—
427C42—62no—
428C42—62no—
429C42—62no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#52511yescharacteristic
432C22×C3#53611yescharacteristic
433C22×C3C22×SD1611yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
436C22×C3#57511yescharacteristic
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—82no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3#45014yes—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—44no—
455C3×C4—81no—
456C3×C4#54712yes—
457C3×C4#54712yes—
458Dic3—22no—
459Dic3—22no—
460Dic3—42no—
461Dic3—41no—
462Dic3—81no—
463C23—32no—
464C23—32no—
465C23—32no—
466C23—61no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—64no—
471C23—122no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23C2×D3×D411yescharacteristic
478Q8—48no—
479D4—64no—
480D4—64no—
481D4—128no—
482D4—128no—
483D4—1216no—
484D4—122no—
485D4—122no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—68no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—124no—
513C2×C4—1216no—
514C2×C4—128no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—244no—
521C2×C4D3×SD1612yes—
522C2×C4#91412yes—
523C2×C4C2×D3×D412yes—
524C2×C4C2×#23911yescharacteristic
525C2×C4#100712yes—
526C2×C4D3×#4111yescharacteristic
527C8—42no—
528C8—122no—
529C2×C3#240611yescharacteristic
530C2×C3C2×#45011yescharacteristic
531C2×C3#300611yescharacteristic
532C2×C3#310311yescharacteristic
533C2×C3#310911yescharacteristic
534C2×C3#322711yescharacteristic
535C2×C3#323411yescharacteristic
536D3—24no—
537D3—84no—
538C22—34no—
539C22—62no—
540C22—62no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—62no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—242no—
549C22—242no—
550C22—242no—
551C22—242no—
552C22D3×#12011yescharacteristic
553C22#562711yescharacteristic
554C22#572411yescharacteristic
555C22C2×D3×SD1611yescharacteristic
556C22C2×#91411yescharacteristic
557C22D3×AGL(1,ℤ/8)11yescharacteristic
558C22#577711yescharacteristic
559C4—22no—
560C4—22no—
561C4—44no—
562C4—42no—
563C4—62no—
564C4—62no—
565C4—81no—
566C4—122no—
567C4—121no—
568C4—241no—
569C4#563112yes—
570C4#572612yes—
571C3#2406111yescharacteristic, complemented, Sylow
572C2—64no—
573C2—244no—
574C2#7724911yescharacteristic
575C2#8004011yescharacteristic
576C2D3×#45011yescharacteristic
577C2#8099611yescharacteristic
578C2#8104611yescharacteristic
579C2#8189411yescharacteristic
580C2#8197811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2058C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenter#2058D3
hypocenterC3#24061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
2. C2×#4863C2semidirect
3. D3×#389C2semidirect
4. C3×#2058C2semidirect
5. #71783C2semidirect
6. #70584C2semidirect
7. #69627C2semidirect
8. #68895C2semidirect
9. #68880C2semidirect
10. #68664C2semidirect
11. #68629C2semidirect
50. #2058D3semidirect
209. C3×C4⋊C4#41semidirect
216. C4⋊Dic3#41semidirect
423. C4⋊C4#239semidirect
571. C3#24061semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#48632
D3×#3892
C3×#20582
#717832
#705842
#696272
#688952
#688802
#686642
#686292
#240613