Finite Group Databasebeta

← #10801439

Subgroups — #10801439

Order 768.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4843C211yesmaximal, characteristic
2C2×#4711C212yesmaximal, complemented
3C4×#874C211yesmaximal, characteristic, complemented
4C42×D12C211yesmaximal, characteristic, complemented
5C3×#1767C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#70457C211yesmaximal, characteristic, complemented
8#70454C211yesmaximal, characteristic, complemented
9#69812C212yesmaximal, complemented
10#68673C212yesmaximal, complemented
11#68626C212yesmaximal, complemented
12#21503—31nomaximal, Sylow
13C22×D24C2212yes—
14C2×#874C2211yescharacteristic
15C2×C4×D12—21no—
16C2×C4×D12—21no—
17C2×C4×D12C2211yescharacteristic
18C2×C4×D12C2212yes—
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C42×D3—21no—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#97C2212yes—
25C3×C43C2211yescharacteristic
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×#817C2212yes—
31#4939—21no—
32#4939—21no—
33#4939C2212yes—
34C4×(C2×D3)⋊C4—21no—
35C4×(C2×D3)⋊C4—21no—
36C4×(C2×D3)⋊C4—21no—
37#4935C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4843C418yescomplemented
42#4711—28no—
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#459—31no—
47C2×#442—32no—
48C4×#127—31no—
49C42×D4—31no—
50#1982—31no—
51#1948—31no—
52#1926—32no—
53#1879—32no—
54#1802—31no—
55#1767D311yescharacteristic, complemented
56#1762—32no—
57C22×D12—21no—
58C22×D12—21no—
59C22×D12C2312yes—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3—21no—
63C22×C4×D3—21no—
64C22×C4×D3—22no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4×Dic3—21no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3—21no—
89C2×D24—24no—
90C2×D24—24no—
91C2×D24C2×C418yescomplemented
92#874—22no—
93#874C2×C414yescomplemented
94C4×D12—22no—
95C4×D12—22no—
96C4×D12—22no—
97C4×D12—24no—
98C4×D12—24no—
99C4×D12—22no—
100C4×D12—44no—
101C4×D12—44no—
102C4×D12C2×C414yescomplemented
103C4×D12D414yescomplemented
104C42×D3—44no—
105C3×C4⋊C8—22no—
106C3×C4⋊C8C2×C414yescomplemented
107C3×#97—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111#821—24no—
112#821—24no—
113#821—24no—
114#821C2×C418yescomplemented
115#817—24no—
116C22×D8—32no—
117C2×#127—31no—
118C2×C4×D4—31no—
119C2×C4×D4—31no—
120C2×C4×D4—32no—
121C2×C4×D4—31no—
122C2×C4×D4—61no—
123C2×C4×D4—61no—
124C22×C42—61no—
125#459—38no—
126#442—68no—
127C2×#106—32no—
128C2×C4⋊C8C2×D311yescharacteristic
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#97C2×D312yes—
135#390—32no—
136#390—61no—
137#390—61no—
138#384—31no—
139C4×C4⋊C4—31no—
140C4×C4⋊C4—31no—
141C4×C4⋊C4—31no—
142C4×C22⋊C4—61no—
143C4×C22⋊C4—61no—
144C4×C22⋊C4—61no—
145C43C2×D311yescharacteristic
146#340—32no—
147#340—32no—
148#336C2×D312yes—
149C23×D3—21no—
150C23×D3—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—24no—
170C2×D12—24no—
171C2×D12—22no—
172C2×D12—42no—
173C2×D12—42no—
174C2×D12—42no—
175C2×D12—42no—
176C2×D12C22×C414yes—
177C2×D12C22×C412yes—
178C2×D12C2×D412yes—
179C2×D12#4112yes—
180C2×C4×D3—24no—
181C2×C4×D3—24no—
182C2×C4×D3—24no—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×C4×D3—42no—
187C2×C4×D3—42no—
188C2×C4×D3—42no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3—44no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8C22×C414yes—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—41no—
204C3×C42C22×C412yes—
205C3×C42C2×D412yes—
206C3×C42#4112yes—
207C3×C42#4112yes—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—44no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—44no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220C4⋊Dic3—42no—
221C4⋊Dic3C22×C412yes—
222C4⋊Dic3C2×D412yes—
223C4⋊Dic3#4112yes—
224C4×Dic3—42no—
225C4×Dic3—42no—
226C4×Dic3—42no—
227D24—216no—
228C22×D4—31no—
229C22×D4—32no—
230C22×D4—61no—
231C22×D4—61no—
232C23×C4—61no—
233C23×C4—61no—
234C23×C4—61no—
235C23×C4—62no—
236C2×D8—38no—
237C2×D8—64no—
238C2×D8—64no—
239C22×C8C22×D312yes—
240C22×C8C22×D312yes—
241#127—34no—
242#127—62no—
243C4×D4—34no—
244C4×D4—34no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4—62no—
248C4×D4—64no—
249C4×D4—62no—
250C4×D4—64no—
251C4×D4—124no—
252C4×D4—124no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C42—61no—
266C2×C42—61no—
267C2×C42—61no—
268C2×C42—124no—
269C2×C42D1211yescharacteristic
270C2×C42D1211yescharacteristic
271C2×C42D1211yescharacteristic
272C2×C42D1211yescharacteristic
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C2×C42C22×D311yescharacteristic
276#106—64no—
277C4⋊C8—22no—
278C4⋊C8C4×D314yescomplemented
279#102—38no—
280#102—64no—
281#102—64no—
282#102—64no—
283#97—24no—
284#95—61no—
285#95—61no—
286#95—61no—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—24no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4C4×D412yes—
314C2×C3×C4C4×D412yes—
315C2×C3×C4#12112yes—
316C2×C3×C4#12112yes—
317C2×C3×C4AGL(1,ℤ/8)12yes—
318C2×C3×C4AGL(1,ℤ/8)12yes—
319C2×C3×C4AGL(1,ℤ/8)12yes—
320C2×C3×C4C23×C411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338D12—24no—
339D12—22no—
340D12—24no—
341D12—22no—
342D12—22no—
343D12—48no—
344D12—42no—
345D12—42no—
346D12—48no—
347D12C4×D414yescomplemented
348C4×D3—48no—
349C4×D3—48no—
350C4×D3—48no—
351C4×D3—88no—
352C3×C8—24no—
353C3×C8—42no—
354C24—61no—
355C24—62no—
356C2×D4—32no—
357C2×D4—34no—
358C2×D4—32no—
359C2×D4—32no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—61no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—61no—
366C2×D4—62no—
367C2×D4—61no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—122no—
371C2×D4—122no—
372C2×D4—122no—
373C2×D4—122no—
374C22×C4—64no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4—62no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4—122no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—124no—
390C22×C4C2×D1211yescharacteristic
391C22×C4C2×D1211yescharacteristic
392C22×C4C2×D1211yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4C2×D1211yescharacteristic
395C22×C4C2×D1211yescharacteristic
396C22×C4C23×D311yescharacteristic
397D8—616no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8C2×C4×D314yes—
404C4⋊C4—32no—
405C4⋊C4—32no—
406C4⋊C4—32no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—122no—
413C4⋊C4—122no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—124no—
418C22⋊C4—124no—
419C22⋊C4—124no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—41no—
426C42—122no—
427C42—122no—
428C42—122no—
429C42C2×C4×D312yes—
430C42#23512yes—
431C42D3×D412yes—
432C42#23912yes—
433C22×C3C2×C4×D411yescharacteristic
434C22×C3C2×#12111yescharacteristic
435C22×C3#52511yescharacteristic
436C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3#57511yescharacteristic
440C2×D3—24no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—82no—
451C2×D3—82no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4#44214yescomplemented
460C3×C4C2×C4×D411yescharacteristic
461C3×C4C4×#4111yescharacteristic
462C3×C4#52911yescharacteristic
463C3×C4#53211yescharacteristic
464Dic3—42no—
465Dic3—42no—
466Dic3—42no—
467Dic3—82no—
468C23—62no—
469C23—62no—
470C23—62no—
471C23—64no—
472C23—121no—
473C23—121no—
474C23—121no—
475C23—121no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23C22×D1211yescharacteristic
483D4—34no—
484D4—64no—
485D4—62no—
486D4—62no—
487D4—64no—
488D4—62no—
489D4—128no—
490D4—128no—
491D4—122no—
492D4—122no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—128no—
508C2×C4—128no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—128no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—248no—
520C2×C4C4×D1212yes—
521C2×C4C4×D1212yes—
522C2×C4#89512yes—
523C2×C4#89512yes—
524C2×C4#90812yes—
525C2×C4#90812yes—
526C2×C4#90812yes—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×#23511yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23911yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C8—24no—
534C8—42no—
535C2×C3C4×#12111yescharacteristic
536C2×C3C2×#44211yescharacteristic
537C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
538C2×C3#303611yescharacteristic
539C2×C3#303811yescharacteristic
540C2×C3#310911yescharacteristic
541C2×C3#311511yescharacteristic
542D3—44no—
543D3—84no—
544C22—64no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—124no—
552C22—124no—
553C22—124no—
554C22—242no—
555C22—242no—
556C22—242no—
557C22—242no—
558C22C2×C4×D1211yescharacteristic
559C22C2×#89511yescharacteristic
560C22#552411yescharacteristic
561C22C2×#90811yescharacteristic
562C22C2×#90811yescharacteristic
563C22C2×#90811yescharacteristic
564C22#574811yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—122no—
571C4—122no—
572C4—122no—
573C4—242no—
574C4#471114yescomplemented
575C4C4×D3×D411yescharacteristic
576C4#554811yescharacteristic
577C4C4×#23911yescharacteristic
578C4#558011yescharacteristic
579C3#2150311yescharacteristic, complemented, Sylow
580C2—124no—
581C2—244no—
582C2C4×#89511yescharacteristic
583C2C2×#471111yescharacteristic
584C2C4×#90811yescharacteristic
585C2#7905511yescharacteristic
586C2#7905611yescharacteristic
587C2#7934311yescharacteristic
588C2#7934711yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1767C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
cosocleC22×C3×C4C24
hypercenter#1767D3
hypocenterC3#21503
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C2×#4711C2semidirect
3. C4×#874C2semidirect
4. C42×D12C2semidirect
5. C3×#1767C2semidirect
6. #70479C2semidirect
7. #70457C2semidirect
8. #70454C2semidirect
9. #69812C2semidirect
10. #68673C2semidirect
11. #68626C2semidirect
41. #4843C4semidirect
55. #1767D3semidirect
91. C2×D24C2×C4semidirect
93. #874C2×C4semidirect
102. C4×D12C2×C4semidirect
103. C4×D12D4semidirect
106. C3×C4⋊C8C2×C4semidirect
114. #821C2×C4semidirect
278. C4⋊C8C4×D3semidirect
347. D12C4×D4semidirect
459. C3×C4#442semidirect
574. C4#4711semidirect
579. C3#21503semidirect

Maximal subgroups

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

subgroupindex
C2×#48432
C2×#47112
C4×#8742
C42×D122
C3×#17672
#704792
#704572
#704542
#698122
#686732
#686262
#215033