Finite Group Databasebeta

← #10823188

Subgroups — #10823188

Order 768.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#949C211yesmaximal, characteristic
2C2×C3×#417C211yesmaximal, characteristic, complemented
3#84086C211yesmaximal, characteristic, complemented
4C2×#5131C211yesmaximal, characteristic, complemented
5C2×#5118C211yesmaximal, characteristic, complemented
6C2×#4988C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#70280C214yesmaximal, complemented
9#70212C214yesmaximal, complemented
10#23546—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#949—24no—
16C2×#949—24no—
17C2×#949—24no—
18C22×#215C2212yes—
19C22×#213C2212yes—
20C22×(C2×D3)⋊C4—22no—
21C2×#926C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#417—24no—
25C2×C3×#103C2212yescomplemented
26C2×C3×#102C2212yescomplemented
27#5229C2212yescomplemented
28#5221C2212yescomplemented
29#5220—22no—
30C4×#217—21no—
31C4×#217—21no—
32#5131—24no—
33#5118—24no—
34C2×#821C2214yes—
35C2×Dic3⋊C8C2214yescomplemented
36#4988—21no—
37#4988—21no—
38#4988C2214yes—
39#4529C2214yescomplemented
40C22×AGL(1,ℤ/8)—31no—
41C2×#423—31no—
42C2×#417D311yescharacteristic, complemented
43C2×#418—31no—
44#2297—31no—
45C2×#390—31no—
46#2055—34no—
47#2054—34no—
48#1736—31no—
49C24×D3—21no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41D414yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×Dic3D412yes—
63C22×D12—22no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—21no—
68C22×D12—22no—
69C22×D12D414yes—
70C22×D12C2312yes—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C22×C3×C8C2312yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2312yes—
79C2×#217—21no—
80C2×#217—22no—
81C2×#217—22no—
82#949—232no—
83C2×#215—24no—
84C2×#215—24no—
85C2×#215—24no—
86C2×#213—24no—
87C2×#213—24no—
88C2×#213—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—48no—
92#926—22no—
93#926—22no—
94C2×C4⋊Dic3C2312yes—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3D411yescharacteristic
99C2×C4×Dic3D411yescharacteristic
100C2×C4×Dic3C2312yes—
101C2×#208—22no—
102C2×#208—22no—
103C2×#208D414yes—
104C22×C3⋊C8C2312yes—
105C3×#103—24no—
106C3×#102—24no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109#821—28no—
110Dic3⋊C8—28no—
111C22×#41C2×D311yescharacteristic
112C23×D4—31no—
113C2×AGL(1,ℤ/8)—64no—
114C2×AGL(1,ℤ/8)—64no—
115C2×AGL(1,ℤ/8)—64no—
116C22×SD16—32no—
117C22×D8—32no—
118C22×M16—31no—
119C22×M16C2×D311yescharacteristic
120C2×#117—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C22⋊C4—62no—
123C22×C42—31no—
124#423—64no—
125C2×C4⋊C8—34no—
126#418—64no—
127#417—24no—
128C2×#103C2×D312yescomplemented
129C2×#102—34no—
130C2×#102C2×D312yescomplemented
131#391—32no—
132#390—32no—
133#390—34no—
134#390—61no—
135#390—61no—
136#386—62no—
137C4×C22⋊C4—61no—
138C4×C22⋊C4—61no—
139#340—34no—
140C24×C3—21no—
141C24×C3C2×D411yescharacteristic
142C23×D3—24no—
143C23×D3—22no—
144C23×D3—44no—
145C23×D3—44no—
146C23×D3—42no—
147C23×D3—42no—
148C3×#41—28no—
149C3×#41—28no—
150C3×#41—44no—
151C3×#41—44no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D412yes—
158C2×C3×D4—24no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4C2×D412yes—
167C22×C3×C4—21no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4—44no—
173C22×C3×C4—42no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4112yes—
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—42no—
186C22×Dic3C2×D412yes—
187C22×Dic3C2×D412yes—
188C2×D12—28no—
189C2×D12—28no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—24no—
195C2×D12—22no—
196C2×D12—42no—
197C2×D12—42no—
198C2×D12—44no—
199C2×D12—44no—
200C2×D12C2×D414yes—
201C3×M16—44no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208#217—44no—
209#217—42no—
210#217—44no—
211#215—216no—
212#213—216no—
213(C2×D3)⋊C4—48no—
214(C2×D3)⋊C4—48no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—42no—
222#208—28no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C2×D414yes—
226C25—61no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41#6714yes—
233C22×Q8C22×D311yescharacteristic
234C22×D4—21no—
235C22×D4—34no—
236C22×D4—32no—
237C22×D4—62no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—61no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4C22×D311yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—32no—
247C23×C4C22×D311yescharacteristic
248AGL(1,ℤ/8)—632no—
249C2×SD16—64no—
250C2×SD16—64no—
251C2×SD16—64no—
252C2×D8—64no—
253C2×D8—64no—
254C2×D8—64no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—34no—
259C2×M16—62no—
260C2×M16—62no—
261C22×C8—32no—
262C22×C8C22×D312yes—
263#117—62no—
264#117—62no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4C22×D312yes—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—128no—
276C2×C42—31no—
277C2×C42—31no—
278C2×C42—32no—
279C2×C42—62no—
280C2×C42—62no—
281C2×C42—61no—
282C4⋊C8—68no—
283#103—24no—
284#102—24no—
285#102—68no—
286#95—62no—
287#95—62no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3#12112yes—
296C23×C3#12112yes—
297C23×C3#12112yes—
298C23×C3C22×D411yescharacteristic
299C22×D3—24no—
300C22×D3—416no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—44no—
304C22×D3—44no—
305C22×D3—48no—
306C22×D3—84no—
307C22×D3—84no—
308C22×D3—88no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—41no—
312C3×Q8—41no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—48no—
316C3×D4—41no—
317C3×D4—41no—
318C2×C3×C4—22no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—48no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—84no—
325C2×C3×C4#12014yes—
326C2×C3×C4#12112yes—
327C2×C3×C4#12112yes—
328C2×C3×C4#12112yes—
329C2×C3×C4#12314yes—
330C2×C3×C4#12314yes—
331C2×C3×C4#12414yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—48no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3AGL(1,ℤ/8)14yescomplemented
348D12—24no—
349D12—24no—
350D12—42no—
351D12—42no—
352D12—48no—
353C3×C8—42no—
354C3⋊C8—24no—
355C24—21no—
356C24—64no—
357C24—62no—
358C24—124no—
359C24—124no—
360C24—122no—
361C24—122no—
362C24D3×D411yescharacteristic
363#41—28no—
364#41—28no—
365#41—44no—
366#41—44no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8C2×#6712yes—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—34no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—68no—
383C2×D4—64no—
384C2×D4—68no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—122no—
391C2×D4—124no—
392C2×D4—124no—
393C2×D4—122no—
394C2×D4C2×#6712yes—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—122no—
408C22×C4#23512yes—
409C22×C4#23512yes—
410C22×C4D3×D412yes—
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D412yes—
413C22×C4#23712yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416SD16—616no—
417D8—616no—
418M16—44no—
419M16—68no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—34no—
424C2×C8—62no—
425C2×C8—62no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—42no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C22⋊C4—128no—
432C22⋊C4—124no—
433C22⋊C4—128no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C42—62no—
437C42—62no—
438C42—62no—
439C42—62no—
440C42—122no—
441C22×C3—24no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3#39414yes—
451C22×C3C2×#12111yescharacteristic
452C22×C3C2×#12111yescharacteristic
453C22×C3C2×#12111yescharacteristic
454C22×C3C2×AGL(1,ℤ/8)12yes—
455C22×C3#57512yes—
456C2×D3—44no—
457C2×D3—48no—
458C2×D3—816no—
459C2×D3—82no—
460C2×D3—82no—
461C2×D3—84no—
462C3×C4—42no—
463C3×C4—82no—
464C3×C4#39414yes—
465C3×C4C2×#12011yescharacteristic
466C3×C4C2×#12311yescharacteristic
467C3×C4C2×#12311yescharacteristic
468C3×C4C2×#12411yescharacteristic
469Dic3—24no—
470Dic3—42no—
471Dic3—82no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—64no—
480C23—1216no—
481C23—128no—
482C23—122no—
483C23—124no—
484C23—124no—
485C23—122no—
486C23—244no—
487C23—248no—
488C23—244no—
489C23#88412yes—
490C23#88412yes—
491C23#93912yes—
492C23C2×D3×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—48no—
500D4—41no—
501D4—41no—
502D4—64no—
503D4—64no—
504D4—128no—
505D4—122no—
506D4—122no—
507C2×C4—22no—
508C2×C4—34no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—84no—
520C2×C4—128no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4#88314yes—
525C2×C4#88412yes—
526C2×C4#88412yes—
527C2×C4#88514yes—
528C2×C4#92914yes—
529C2×C4#93714yes—
530C2×C4#93912yes—
531C2×C4C2×#23511yescharacteristic
532C2×C4C2×#23511yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×#23711yescharacteristic
536C2×C4C22×#6711yescharacteristic
537C8—42no—
538C8—64no—
539C2×C3—24no—
540C2×C3—82no—
541C2×C3C2×#39411yescharacteristic
542C2×C3#303612yes—
543C2×C3#303811yescharacteristic
544C2×C3#304011yescharacteristic
545C2×C3#305512yes—
546D3—84no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22—128no—
557C22—124no—
558C22—2416no—
559C22—242no—
560C22—244no—
561C22—242no—
562C22#495914yes—
563C22C2×#88411yescharacteristic
564C22C2×#88411yescharacteristic
565C22D3×AGL(1,ℤ/8)12yes—
566C22#577712yes—
567C22C2×#93911yescharacteristic
568C4—42no—
569C4—64no—
570C4—82no—
571C4—122no—
572C4—242no—
573C4#495914yes—
574C4C2×#88311yescharacteristic
575C4C2×#88511yescharacteristic
576C4C2×#92911yescharacteristic
577C4C2×#93711yescharacteristic
578C3#2354611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—82no—
581C2—244no—
582C2#7888812yes—
583C2#7891012yes—
584C2C2×#495911yescharacteristic
585C2#8393211yescharacteristic
586C2#8396811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
prime radicalC22×#949C2
hypercenterC2×#417D3
hypocenterC3#23546
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×C3×#417C2semidirect
3. #84086C2semidirect
4. C2×#5131C2semidirect
5. C2×#5118C2semidirect
6. C2×#4988C2semidirect
7. #71558C2semidirect
8. #70280C2semidirect
9. #70212C2semidirect
25. C2×C3×#103C22semidirect
26. C2×C3×#102C22semidirect
27. #5229C22semidirect
28. #5221C22semidirect
35. C2×Dic3⋊C8C22semidirect
39. #4529C22semidirect
42. C2×#417D3semidirect
128. C2×#103C2×D3semidirect
130. C2×#102C2×D3semidirect
347. C2×Dic3AGL(1,ℤ/8)semidirect
578. C3#23546semidirect

Maximal subgroups

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

subgroupindex
C22×#9492
C2×C3×#4172
#840862
C2×#51312
C2×#51182
C2×#49882
#715582
#702802
#702122
#235463