Finite Group Databasebeta

← #10823451

Subgroups — #10823451

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×#4817C212yesmaximal, complemented
3#76479C211yesmaximal, characteristic, complemented
4C3×#2024C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#70272C212yesmaximal, complemented
7#69688C211yesmaximal, characteristic, complemented
8#68822C212yesmaximal, complemented
9#68782C212yesmaximal, complemented
10#68629C211yesmaximal, characteristic, complemented
11#68600C211yesmaximal, characteristic, complemented
12#23576—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16C2×#946C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C22×#215C2212yes—
19#5725—28no—
20C2×#895—21no—
21C2×#895C2212yes—
22C2×#893—21no—
23C2×#893C2212yes—
24C2×#874C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C3×#103C2212yes—
27C3×#391C2211yescharacteristic
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#815C2212yes—
32#4987C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34C2×#809C2212yes—
35C2×#807C2212yes—
36#4928C2211yescharacteristic
37#4817—28no—
38#4668—21no—
39#4666—21no—
40#4655—21no—
41#4649—21no—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#440—32no—
48C4×C22⋊Q8—31no—
49#2057—31no—
50#2041—31no—
51#2033—32no—
52#2024D311yescharacteristic, complemented
53#2014—31no—
54#2000—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C2×#239—28no—
60C2×#239—44no—
61C2×#239—44no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×C4×D3—22no—
67C22×C4×D3D411yescharacteristic
68C22×C4×D3D411yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C42C2311yescharacteristic
76#947—28no—
77#946—22no—
78#946—22no—
79Q8×Dic3—24no—
80C2×#215—24no—
81C2×#215—24no—
82C2×#215—24no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98#895—28no—
99#895—24no—
100#895—24no—
101#893—28no—
102#893—24no—
103#893D418yes—
104#874—24no—
105C42×D3—24no—
106C3×#103—24no—
107C3×#95—21no—
108C3×#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112#821—24no—
113#821—24no—
114(C2×D3)⋊C8—24no—
115#815—24no—
116#809—24no—
117#807—24no—
118C22×#41—61no—
119C22×SD16—32no—
120#532—68no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#127—31no—
123C2×#124—61no—
124C2×C22⋊Q8—31no—
125C2×#121—32no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130C22×C42—31no—
131#440—68no—
132C2×#107—32no—
133C2×#103C2×D312yes—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#391C2×D311yescharacteristic
140#389—31no—
141#387—61no—
142#385—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C4⋊C4—61no—
146C4×C22⋊C4—61no—
147#340—32no—
148#340C2×D312yes—
149#336—31no—
150#336C2×D311yescharacteristic
151C23×D3—22no—
152C23×D3C2×D411yescharacteristic
153C2×C3×Q8—24no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—42no—
158C2×C3×Q8—42no—
159C2×C3×Q8#4112yescomplemented
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3C2×D411yescharacteristic
169C22×Dic3C2×D411yescharacteristic
170C22×Dic3C2×D411yescharacteristic
171#239—432no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—24no—
177C2×D12—24no—
178C2×D12—28no—
179C2×D12—22no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12C2×D414yes—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×C4×D3—28no—
197C2×C4×D3—41no—
198C2×C4×D3—44no—
199C2×C4×D3—44no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4C2×D414yes—
207C3×C42—22no—
208#215—216no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—42no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—42no—
216C4⋊Dic3#4112yescomplemented
217Dic3⋊C4—24no—
218Dic3⋊C4—42no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—42no—
222C4×Dic3—42no—
223C4×Dic3C2×D414yes—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8#4114yescomplemented
227C2×#41—68no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—21no—
231C22×Q8C22×D311yescharacteristic
232C22×D4—32no—
233C22×D4—62no—
234C22×D4—61no—
235C22×D4—61no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—62no—
240C2×SD16—64no—
241C2×SD16—64no—
242C2×SD16—64no—
243C22×C8—32no—
244C22×C8C22×D312yes—
245C4⋊Q8—24no—
246#127—64no—
247#124—68no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250#121—64no—
251#121—68no—
252#121—64no—
253C4×Q8—64no—
254C4×D4—38no—
255C4×D4—64no—
256C4×D4—68no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—61no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—64no—
273C2×C42—61no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#107—64no—
277#103—24no—
278#102—64no—
279#102—64no—
280#102—64no—
281C22⋊C8—64no—
282#97—64no—
283#95—21no—
284#95—21no—
285#95—61no—
286#95—61no—
287#95—61no—
288C23×C3C22×D411yescharacteristic
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—24no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3AGL(1,ℤ/8)12yes—
303C3×Q8—24no—
304C3×Q8—48no—
305C3×Q8—42no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4AGL(1,ℤ/8)12yes—
320C2×C3×C4#13712yes—
321C2×C3×C4C22×D412yes—
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4112yes—
324C2×C3×C4C2×#4111yescharacteristic
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332C2×Dic3—21no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3#12012yes—
339C2×Dic3#12012yes—
340D12—24no—
341D12—24no—
342D12—48no—
343D12—48no—
344D12—416no—
345D12—42no—
346D12—42no—
347C4×D3—28no—
348C4×D3—44no—
349C4×D3—416no—
350C4×D3—42no—
351C4×D3—42no—
352C3×C8—42no—
353C3⋊C8—24no—
354C24—31no—
355C24—62no—
356#41—1232no—
357C2×Q8—24no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8#23912yescomplemented
364C2×D4—34no—
365C2×D4—64no—
366C2×D4—68no—
367C2×D4—64no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—122no—
374C2×D4—122no—
375C2×D4—124no—
376C2×D4—124no—
377C2×D4—122no—
378C2×D4—122no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—68no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—61no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—61no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—121no—
398C22×C4—124no—
399C22×C4—124no—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403SD16—616no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—34no—
408C2×C8—62no—
409C2×C8—62no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—32no—
414C4⋊C4—64no—
415C4⋊C4—61no—
416C4⋊C4—62no—
417C4⋊C4—122no—
418C4⋊C4—122no—
419C4⋊C4D3×D414yes—
420C22⋊C4—68no—
421C22⋊C4—68no—
422C22⋊C4—64no—
423C22⋊C4—122no—
424C42—22no—
425C42—34no—
426C42—62no—
427C42—62no—
428C42—122no—
429C42—122no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#52511yescharacteristic
432C22×C3#53211yescharacteristic
433C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3C2×#13711yescharacteristic
436C22×C3#57511yescharacteristic
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—41no—
441C2×D3—41no—
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—44no—
455C3×C4—81no—
456C3×C4#52512yes—
457C3×C4#53212yes—
458Dic3—22no—
459Dic3—22no—
460Dic3—42no—
461Dic3—41no—
462Dic3—81no—
463C23—32no—
464C23—61no—
465C23—61no—
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—24no—
479Q8—48no—
480Q8—42no—
481D4—64no—
482D4—64no—
483D4—128no—
484D4—128no—
485D4—1216no—
486D4—122no—
487D4—122no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—32no—
495C2×C4—32no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—68no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—124no—
513C2×C4—1216no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4#91412yes—
522C2×C4#91812yes—
523C2×C4C2×D3×D412yes—
524C2×C4C2×#23911yescharacteristic
525C2×C4D3×#4111yescharacteristic
526C2×C4D3×#4112yes—
527C8—42no—
528C8—64no—
529C2×C3#238811yescharacteristic
530C2×C3#300311yescharacteristic
531C2×C3#300711yescharacteristic
532C2×C3#311211yescharacteristic
533C2×C3#311511yescharacteristic
534C2×C3#318011yescharacteristic
535C2×C3#318211yescharacteristic
536D3—42no—
537D3—84no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—242no—
549C22—242no—
550C22—242no—
551C22—242no—
552C22D3×#12011yescharacteristic
553C22#562711yescharacteristic
554C22#566811yescharacteristic
555C22C2×#91411yescharacteristic
556C22C2×#91811yescharacteristic
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#562712yes—
570C4#566812yes—
571C3#2357611yescharacteristic, complemented, Sylow
572C2—122no—
573C2—244no—
574C2#7677711yescharacteristic
575C2#8014611yescharacteristic
576C2#8018411yescharacteristic
577C2#8099911yescharacteristic
578C2#8109411yescharacteristic
579C2#8159511yescharacteristic
580C2#8166811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2024C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenter#2024D3
hypocenterC3#23576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
2. C2×#4817C2semidirect
3. #76479C2semidirect
4. C3×#2024C2semidirect
5. #71782C2semidirect
6. #70272C2semidirect
7. #69688C2semidirect
8. #68822C2semidirect
9. #68782C2semidirect
10. #68629C2semidirect
11. #68600C2semidirect
52. #2024D3semidirect
159. C2×C3×Q8#41semidirect
216. C4⋊Dic3#41semidirect
226. C2×C3⋊C8#41semidirect
363. C2×Q8#239semidirect
571. C3#23576semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#48172
#764792
C3×#20242
#717822
#702722
#696882
#688222
#687822
#686292
#686002
#235763