Finite Group Databasebeta

← #10826971

Subgroups — #10826971

Order 768.

1948 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5791C212yesmaximal, complemented
2C2×#4733C212yesmaximal, complemented
3C2×#4734C212yesmaximal, complemented
4C3×#1779C211yesmaximal, characteristic, complemented
5#69964C211yesmaximal, characteristic, complemented
6#69962C212yesmaximal, complemented
7#69961C211yesmaximal, characteristic, complemented
8#69933C212yesmaximal, complemented
9#69931C212yesmaximal, complemented
10#23923—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×#235C2212yes—
13C2×#953—22no—
14#5791—28no—
15#5791—28no—
16C2×#930C2212yes—
17C2×#930C2212yes—
18C2×#929—22no—
19C2×C4×#67—22no—
20C2×C4×#67—22no—
21C2×#926C2212yes—
22C2×#924C2212yes—
23C2×#924C2212yes—
24C22×Dic12C2211yescharacteristic, complemented
25C22×D24C2211yescharacteristic, complemented
26C22×#204C2212yescomplemented
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×#821C2212yescomplemented
32C2×#821C2212yescomplemented
33C2×#816C2212yescomplemented
34C2×#816C2212yescomplemented
35#4734—28no—
36#4733—28no—
37#4551C2211yescharacteristic, complemented
38#4551C2212yescomplemented
39#4551C2211yescharacteristic, complemented
40C2×#525—32no—
41C2×#452—32no—
42C2×#449—32no—
43#2001—31no—
44#2000—32no—
45#1999—31no—
46#1875—32no—
47#1873—32no—
48#1779D311yescharacteristic, complemented
49C25×C3D411yescharacteristic
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—41no—
53C23×C3×C4—41no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D412yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—22no—
59C22×#67—22no—
60C22×#67—22no—
61C2×#235—24no—
62C2×#235—24no—
63C2×#235—24no—
64C2×#235—24no—
65C2×#235—42no—
66C2×#235—42no—
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×Dic6C2312yes—
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73#953—216no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77#930—24no—
78#930—24no—
79#930—24no—
80#930D418yescomplemented
81#929—216no—
82C4×#67—216no—
83C4×#67—216no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86#926—24no—
87#926—24no—
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2312yes—
90#924—24no—
91#924—24no—
92#924—24no—
93#924D418yescomplemented
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—22no—
97C2×Dic12—24no—
98C2×Dic12—41no—
99C2×D24—24no—
100C2×D24—41no—
101C2×#204—24no—
102C2×#204—24no—
103C2×#204—42no—
104C3×C22⋊C8—24no—
105C3×C22⋊C8—24no—
106C3×C22⋊C8—41no—
107C3×C22⋊C8—41no—
108#821—24no—
109#821—24no—
110#816—24no—
111#816—24no—
112C22×#41—32no—
113C24×C4C2×D311yescharacteristic
114C22×Q16—31no—
115C22×SD16—32no—
116C22×D8—31no—
117#525—68no—
118#525—68no—
119C2×#123—62no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—32no—
122C2×#121—32no—
123C2×#121—32no—
124C2×#120—62no—
125C2×C4×D4—62no—
126C2×C4×D4—62no—
127C2×#117—32no—
128#452—68no—
129#449—68no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×C22⋊C8C2×D312yes—
135C2×C22⋊C8C2×D312yes—
136C2×C22⋊C8C2×D311yescharacteristic
137C2×C22⋊C8C2×D311yescharacteristic
138#340—31no—
139#340—32no—
140#340—31no—
141C24×C3—24no—
142C24×C3—22no—
143C24×C3—42no—
144C24×C3—41no—
145C24×C3—41no—
146C24×C3C2×D412yes—
147C24×C3C2×D411yescharacteristic
148C23×D3—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—41no—
161C22×C3×C4—42no—
162C22×C3×C4—44no—
163C22×C3×C4—48no—
164C22×C3×C4—44no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2411yescharacteristic
170C2×#67—28no—
171C2×#67—28no—
172C2×#67—28no—
173C2×#67—44no—
174C2×#67—44no—
175C2×#67—44no—
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182#235—48no—
183#235—48no—
184#235—48no—
185#235—48no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12C2×D414yes—
192C2×C4×D3—28no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6C2×D414yes—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208#217—28no—
209#217—28no—
210#217—28no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—28no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3C2×D414yes—
216C4⋊Dic3C2×D414yes—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219C4×Dic3—28no—
220Dic12—44no—
221D24—44no—
222#204—48no—
223C25D1211yescharacteristic
224C2×#41—64no—
225C2×#41—64no—
226C2×#41—64no—
227C2×#41—64no—
228C2×#41—122no—
229C2×#41—122no—
230C22×Q8—32no—
231C22×D4—32no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—62no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—41no—
238C23×C4—41no—
239C23×C4—62no—
240C23×C4D1211yescharacteristic
241C23×C4D1212yes—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D312yes—
244C2×Q16—64no—
245C2×Q16—121no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C2×D8—64no—
250C2×D8—121no—
251C22×C8C22×D311yescharacteristic
252C22×C8C22×D312yes—
253C22×C8C22×D311yescharacteristic
254#123—616no—
255C22⋊Q8—38no—
256C22⋊Q8—64no—
257C22⋊Q8—64no—
258C22⋊Q8—64no—
259#121—38no—
260#121—64no—
261#121—64no—
262#121—64no—
263#120—616no—
264C4×D4—616no—
265C4×D4—616no—
266#117—64no—
267#117—64no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—62no—
278#103—64no—
279#103—64no—
280#102—64no—
281#102—64no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—44no—
297C23×C3—48no—
298C23×C3—44no—
299C23×C3—48no—
300C23×C3—44no—
301C23×C3#12014yes—
302C23×C3#13512yes—
303C23×C3C22×D411yescharacteristic
304C22×D3—24no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—44no—
320C2×C3×C4—48no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4#12014yes—
325C2×C3×C4C22×D412yes—
326C2×C3×C4C22×D412yes—
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329#67—416no—
330#67—416no—
331#67—416no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341D12—24no—
342D12—24no—
343D12—42no—
344D12—42no—
345C4×D3—416no—
346Dic6—24no—
347Dic6—24no—
348Dic6—42no—
349Dic6—42no—
350C3×C8—42no—
351C3×C8—41no—
352C3×C8—41no—
353C24—24no—
354C24—22no—
355C24—41no—
356C24—42no—
357C24—41no—
358C24—62no—
359C24C2×D1212yes—
360C24C2×D1211yescharacteristic
361#41—128no—
362#41—128no—
363#41—128no—
364#41—128no—
365C2×Q8—34no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×D4—34no—
372C2×D4—68no—
373C2×D4—68no—
374C2×D4—68no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—124no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4—124no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—68no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—124no—
407C22×C4—124no—
408C22×C4C2×D1211yescharacteristic
409C22×C4C2×D1212yes—
410C22×C4D3×D414yes—
411C22×C4D3×D414yes—
412C22×C4C23×D311yescharacteristic
413Q16—124no—
414SD16—128no—
415D8—124no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—21no—
423C4⋊C4—34no—
424C4⋊C4—34no—
425C4⋊C4—68no—
426C4⋊C4—68no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C22⋊C4—68no—
430C22⋊C4—68no—
431C22⋊C4—68no—
432C22⋊C4—68no—
433C22⋊C4—68no—
434C42—68no—
435C22×C3—24no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—416no—
441C22×C3—41no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3C2×#12012yes—
449C22×C3#53411yescharacteristic
450C22×C3C2×#13511yescharacteristic
451C22×C3#57511yescharacteristic
452C22×C3#57512yes—
453C2×D3—44no—
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—82no—
457C2×D3—82no—
458C2×D3—82no—
459C2×D3—82no—
460C3×C4—24no—
461C3×C4—41no—
462C3×C4D4211yescharacteristic
463C3×C4D4212yes—
464C3×C4D4211yescharacteristic
465Dic3—44no—
466Dic3—44no—
467Dic3—44no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—44no—
479C23—48no—
480C23—44no—
481C23—48no—
482C23—44no—
483C23—64no—
484C23—122no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23#88214yes—
491C23#90412yes—
492C23C22×D1211yescharacteristic
493Q8—64no—
494Q8—64no—
495Q8—122no—
496Q8—122no—
497D4—64no—
498D4—64no—
499D4—1216no—
500D4—1216no—
501D4—1216no—
502D4—122no—
503D4—122no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—1216no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4#88214yes—
528C2×C4C2×D3×D412yes—
529C2×C4C2×D3×D412yes—
530C2×C4#100411yescharacteristic
531C2×C4#100411yescharacteristic
532C8—41no—
533C8—42no—
534C8—41no—
535C2×C3—44no—
536C2×C3—42no—
537C2×C3#240311yescharacteristic
538C2×C3#300212yes—
539C2×C3#300512yes—
540C2×C3#319111yescharacteristic
541C2×C3#319711yescharacteristic
542D3—84no—
543C22—24no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—416no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—124no—
557C22—124no—
558C22—124no—
559C22—242no—
560C22—242no—
561C22—242no—
562C22—242no—
563C22C2×#88212yes—
564C22#549311yescharacteristic
565C22C2×#90411yescharacteristic
566C22#574812yes—
567C22#574811yescharacteristic
568C4—24no—
569C4—41no—
570C4—124no—
571C4—124no—
572C4—124no—
573C4#559011yescharacteristic
574C4#559012yes—
575C4#559011yescharacteristic
576C3#2392311yescharacteristic, complemented, Sylow
577C2—42no—
578C2—44no—
579C2—244no—
580C2#7385211yescharacteristic
581C2#7861012yes—
582C2#7861411yescharacteristic
583C2#7863812yes—
584C2#7864411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
hypercenter#1779D3
hypocenterC3#23923
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. C2×#5791C2semidirect
2. C2×#4733C2semidirect
3. C2×#4734C2semidirect
4. C3×#1779C2semidirect
5. #69964C2semidirect
6. #69962C2semidirect
7. #69961C2semidirect
8. #69933C2semidirect
9. #69931C2semidirect
24. C22×Dic12C22semidirect
25. C22×D24C22semidirect
26. C22×#204C22semidirect
31. C2×#821C22semidirect
32. C2×#821C22semidirect
33. C2×#816C22semidirect
34. C2×#816C22semidirect
37. #4551C22semidirect
38. #4551C22semidirect
39. #4551C22semidirect
48. #1779D3semidirect
80. #930D4semidirect
93. #924D4semidirect
576. C3#23923semidirect

Maximal subgroups

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

subgroupindex
C2×#57912
C2×#47332
C2×#47342
C3×#17792
#699642
#699622
#699612
#699332
#699312
#239233