Finite Group Databasebeta

← #10944233

Subgroups — #10944233

Order 768.

1708 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×C8C211yesmaximal, characteristic, complemented
2C2×#5114C211yesmaximal, characteristic, complemented
3C2×#4864C212yesmaximal, complemented
4#74049C211yesmaximal, characteristic, complemented
5#74048C211yesmaximal, characteristic
6C3×#1807C211yesmaximal, characteristic, complemented
7#70577C212yesmaximal, complemented
8#69515C211yesmaximal, characteristic, complemented
9#69514C211yesmaximal, characteristic, complemented
10#68883C212yesmaximal, complemented
11#68882C212yesmaximal, complemented
12#32929—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18C22×#204C2211yescharacteristic, complemented
19C22×D3×C8—24no—
20C22×D3×C8—28no—
21C22×D3×C8C2212yes—
22C22×D3×C8C2212yes—
23C2×#897C2212yes—
24C2×#895C2212yes—
25C2×#882—21no—
26C2×C3×#106C2212yescomplemented
27C3×#380C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5114—24no—
30C2×#821C2211yescharacteristic, complemented
31C2×#817C2211yescharacteristic, complemented
32C2×#816C2211yescharacteristic, complemented
33#4987C2212yes—
34#4987C2212yes—
35C2×#810C2212yescomplemented
36C2×#809C2212yescomplemented
37#4958—21no—
38#4955—21no—
39#4950—21no—
40#4864—28no—
41#4668—22no—
42#4665—22no—
43C3×#340C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C24×C8—31no—
47C2×#465—32no—
48C2×#465—31no—
49#2848—31no—
50#2847—31no—
51#1893—31no—
52#1893—32no—
53#1892—31no—
54#1892—32no—
55#1807—32no—
56#1807D311yescharacteristic, complemented
57C24×D3D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C23×Dic3D411yescharacteristic
61C22×D12—22no—
62C22×D12C2311yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—24no—
66C22×C4×D3—24no—
67C22×C4×D3D412yes—
68C22×C4×D3C2312yes—
69C22×Dic6C2311yescharacteristic
70C22×C3×C8—24no—
71C22×C3×C8D414yescomplemented
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×#217—21no—
79#930—22no—
80#930—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89#924—22no—
90#924—22no—
91C2×Dic3⋊C4—21no—
92C22×C3⋊C8—22no—
93C22×C3⋊C8—24no—
94C22×C3⋊C8C2312yes—
95C2×#204—24no—
96C2×#204—22no—
97C2×D3×C8—232no—
98C2×D3×C8—24no—
99C2×D3×C8—24no—
100C2×D3×C8—24no—
101C2×D3×C8—24no—
102C2×D3×C8—24no—
103C2×D3×C8D418yescomplemented
104#897—24no—
105#897—24no—
106#895—24no—
107#895—24no—
108#882—44no—
109C3×#106—24no—
110C3×#95—21no—
111C3×#95—21no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#821—22no—
115#817—22no—
116#816—22no—
117#810—24no—
118#809—24no—
119C24×C4—31no—
120C22×SD16—31no—
121C23×C8—32no—
122C23×C8—31no—
123C23×C8—32no—
124C23×C8—68no—
125C23×C8—64no—
126C23×C8C2×D311yescharacteristic
127C2×C22⋊Q8—31no—
128C2×C22⋊Q8—32no—
129C2×#121—31no—
130C2×#121—32no—
131C2×#120—61no—
132#465—68no—
133#465—64no—
134C2×#106—31no—
135C2×#106C2×D312yescomplemented
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140#386—62no—
141#386—61no—
142#385—62no—
143#385—61no—
144#380—32no—
145#380—32no—
146#380C2×D311yescharacteristic
147#380C2×D311yescharacteristic
148#379—61no—
149#340—32no—
150#340—32no—
151#340C2×D312yescomplemented
152C24×C3C2×D411yescharacteristic
153C23×D3—22no—
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—24no—
157C23×D3—21no—
158C23×D3SD1614yescomplemented
159C23×D3C2×D412yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C2×#67—42no—
169C2×#67—42no—
170C2×#67—42no—
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3C2×D412yes—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12—22no—
180C2×D12—21no—
181C2×D12—44no—
182C2×D12—44no—
183C2×D12—44no—
184C2×C4×D3—28no—
185C2×C4×D3—216no—
186C2×C4×D3—28no—
187C2×C4×D3—24no—
188C2×C4×D3—24no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—24no—
194C2×C4×D3C2×D414yes—
195C2×C4×D3#4114yes—
196C2×Dic6—22no—
197C2×Dic6—21no—
198C2×Dic6—22no—
199C2×Dic6—21no—
200C2×C3×C8—22no—
201C2×C3×C8—28no—
202C2×C3×C8C2×D412yes—
203C2×C3×C8C2×D412yes—
204C2×C3×C8C2×D414yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211#217—42no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—42no—
216(C2×D3)⋊C4—44no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—44no—
222Dic3⋊C4—42no—
223C2×C3⋊C8—28no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8C2×D414yes—
227#204—44no—
228D3×C8—216no—
229D3×C8—216no—
230C25—31no—
231C22×Q8—31no—
232C22×D4—31no—
233C22×D4—62no—
234C22×D4—61no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—64no—
239C23×C4—64no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4C22×D311yescharacteristic
243C2×SD16—62no—
244C2×SD16—64no—
245C22×C8—24no—
246C22×C8—32no—
247C22×C8—38no—
248C22×C8—64no—
249C22×C8—64no—
250C22×C8—632no—
251C22×C8—64no—
252C22×C8—62no—
253C22×C8—64no—
254C22×C8—64no—
255C22×C8—64no—
256C22×C8D1214yescomplemented
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22⋊Q8—64no—
260C22⋊Q8—64no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263#121—64no—
264#121—64no—
265#121—62no—
266#121—62no—
267#120—124no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282#106—24no—
283#106—62no—
284#103—64no—
285#103—62no—
286#102—64no—
287#102—62no—
288#95—21no—
289#95—21no—
290#95—62no—
291#95—62no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#12112yes—
295C23×C3C2×SD1612yes—
296C23×C3#13512yes—
297C23×C3C22×D411yescharacteristic
298C22×D3—28no—
299C22×D3—216no—
300C22×D3—28no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—28no—
306C22×D3—216no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3#12114yes—
313C22×D3C2×SD1614yes—
314C22×D3#13514yes—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4#12112yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D412yes—
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4112yes—
330#67—84no—
331C2×Dic3—28no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—41no—
341C2×Dic3#12114yes—
342D12—41no—
343D12—41no—
344D12—42no—
345D12—88no—
346C4×D3—216no—
347C4×D3—24no—
348C4×D3—24no—
349C4×D3—24no—
350C4×D3—24no—
351Dic6—41no—
352Dic6—41no—
353Dic6—42no—
354C3×C8—22no—
355C3×C8#12014yescomplemented
356C3⋊C8—24no—
357C24—32no—
358C24—34no—
359C24—64no—
360C24—64no—
361C24—62no—
362C24—62no—
363C24—61no—
364C24D3×D411yescharacteristic
365C2×Q8—61no—
366C2×Q8—62no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×D4—61no—
370C2×D4—62no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—122no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—34no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—68no—
389C22×C4—68no—
390C22×C4—616no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—64no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4C2×D1212yes—
402C22×C4D3×D411yescharacteristic
403C22×C4#23912yes—
404C22×C4C23×D311yescharacteristic
405SD16—124no—
406C2×C8—22no—
407C2×C8—28no—
408C2×C8—34no—
409C2×C8—616no—
410C2×C8—68no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—616no—
414C2×C8C2×D1212yes—
415C2×C8C2×D1212yes—
416C2×C8D3×D414yes—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—124no—
426C4⋊C4—122no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—124no—
430C22⋊C4—124no—
431C22⋊C4—124no—
432C22⋊C4—124no—
433C22⋊C4—122no—
434C22⋊C4—122no—
435C22×C3—22no—
436C22×C3—24no—
437C22×C3—22no—
438C22×C3—24no—
439C22×C3#46514yescomplemented
440C22×C3C2×#12111yescharacteristic
441C22×C3C2×#12112yes—
442C22×C3C22×SD1611yescharacteristic
443C22×C3C2×#13511yescharacteristic
444C22×C3C2×#13512yes—
445C2×D3—232no—
446C2×D3—24no—
447C2×D3—24no—
448C2×D3—24no—
449C2×D3—24no—
450C2×D3—24no—
451C2×D3—24no—
452C2×D3—81no—
453C2×D3—82no—
454C2×D3—81no—
455C2×D3—82no—
456C2×D3—81no—
457C2×D3#46518yescomplemented
458C3×C4—22no—
459C3×C4—82no—
460C3×C4—82no—
461C3×C4C2×#12011yescharacteristic
462C3×C4#52511yescharacteristic
463C3×C4#52512yes—
464Dic3—24no—
465Dic3—81no—
466Dic3—81no—
467Dic3—81no—
468C23—22no—
469C23—22no—
470C23—34no—
471C23—34no—
472C23—34no—
473C23—616no—
474C23—68no—
475C23—616no—
476C23—68no—
477C23—68no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—62no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—122no—
486C23—122no—
487C23#89512yes—
488C23D3×SD1612yes—
489C23#91612yes—
490C23C2×D3×D411yescharacteristic
491Q8—121no—
492Q8—121no—
493Q8—122no—
494D4—121no—
495D4—121no—
496D4—122no—
497D4—248no—
498D4—244no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—34no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—616no—
511C2×C4—68no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4#89512yes—
526C2×C4C22×D1211yescharacteristic
527C2×C4C2×D3×D412yes—
528C2×C4C2×#23911yescharacteristic
529C2×C4D3×#4112yes—
530C8—22no—
531C8—64no—
532C8#88214yescomplemented
533C2×C3—24no—
534C2×C3#284711yescharacteristic
535C2×C3C2×#46511yescharacteristic
536C2×C3C2×#46512yes—
537C2×C3#305012yes—
538C2×C3#305011yescharacteristic
539D3—28no—
540D3—161no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—38no—
546C22—632no—
547C22—64no—
548C22—64no—
549C22—64no—
550C22—64no—
551C22—64no—
552C22—64no—
553C22—241no—
554C22—242no—
555C22—242no—
556C22—241no—
557C22—241no—
558C22#486414yescomplemented
559C22C2×#89511yescharacteristic
560C22D3×#12112yes—
561C22C2×D3×SD1611yescharacteristic
562C22C2×#91611yescharacteristic
563C22D3×#13512yes—
564C4—22no—
565C4—64no—
566C4—82no—
567C4—82no—
568C4—241no—
569C4—241no—
570C4—241no—
571C4C2×#88211yescharacteristic
572C4#552411yescharacteristic
573C4#562712yes—
574C3#3292911yescharacteristic, complemented, Sylow
575C2—24no—
576C2—68no—
577C2—481no—
578C2#7404811yescharacteristic
579C2C2×#486411yescharacteristic
580C2#7918011yescharacteristic
581C2D3×#46512yes—
582C2#8065612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1807C2
socleC23×C3C22×D4
prime radical#74048C2
hypercenter#1807D3
hypocenterC3#32929
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C23×D3×C8C2semidirect
2. C2×#5114C2semidirect
3. C2×#4864C2semidirect
4. #74049C2semidirect
6. C3×#1807C2semidirect
7. #70577C2semidirect
8. #69515C2semidirect
9. #69514C2semidirect
10. #68883C2semidirect
11. #68882C2semidirect
18. C22×#204C22semidirect
26. C2×C3×#106C22semidirect
30. C2×#821C22semidirect
31. C2×#817C22semidirect
32. C2×#816C22semidirect
35. C2×#810C22semidirect
36. C2×#809C22semidirect
43. C3×#340C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
56. #1807D3semidirect
71. C22×C3×C8D4semidirect
103. C2×D3×C8D4semidirect
135. C2×#106C2×D3semidirect
151. #340C2×D3semidirect
158. C23×D3SD16semidirect
256. C22×C8D12semidirect
355. C3×C8#120semidirect
439. C22×C3#465semidirect
457. C2×D3#465semidirect
532. C8#882semidirect
558. C22#4864semidirect
574. C3#32929semidirect

Maximal subgroups

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

subgroupindex
C23×D3×C82
C2×#51142
C2×#48642
#740492
#740482
C3×#18072
#705772
#695152
#695142
#688832
#688822
#329293