Finite Group Databasebeta

← #10944284

Subgroups — #10944284

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#208C211yesmaximal, characteristic
2C3×#2848C211yesmaximal, characteristic
3#84031C211yesmaximal, characteristic
4C2×#5052C212yesmaximal, complemented
5C2×#5051C211yesmaximal, characteristic, complemented
6#71697C212yesmaximal, complemented
7#71696C211yesmaximal, characteristic, complemented
8#69516C212yesmaximal, complemented
9#69514C211yesmaximal, characteristic, complemented
10#69513C212yesmaximal, complemented
11#69511C211yesmaximal, characteristic, complemented
12#32937—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C22×#216C2211yescharacteristic, complemented
17C2×#924C2211yescharacteristic
18C2×#924C2212yes—
19C22×#208—28no—
20C22×#208—24no—
21C22×#208C2212yes—
22C22×#208C2212yes—
23C23×C3⋊C8C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#385—22no—
26C3×#385—21no—
27C3×#380C2211yescharacteristic
28C3×#380C2212yes—
29C3×#379—21no—
30#5247—21no—
31C2×#835C2212yescomplemented
32C2×#835C2211yescharacteristic, complemented
33#5212C2211yescharacteristic
34#5212C2212yes—
35#5210—22no—
36#5210—21no—
37#5052—28no—
38#5051—24no—
39C2×#810C2211yescharacteristic, complemented
40C2×#810C2212yescomplemented
41C2×#808C2212yescomplemented
42C2×#807C2211yescharacteristic, complemented
43#4529C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#470—32no—
48C2×#467—31no—
49#2848—31no—
50#2848D311yescharacteristic
51#1894—32no—
52#1894—32no—
53#1892—31no—
54#1892—31no—
55#1809—32no—
56#1807—31no—
57C25×C3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4—24no—
60C23×C3×C4—22no—
61C23×C3×C4—24no—
62C23×C3×C4—22no—
63C23×C3×C4D412yes—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2312yes—
67C22×Dic6C2311yescharacteristic
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×#217—22no—
81C2×#217—21no—
82C2×#217—22no—
83C2×#217—21no—
84C2×#216—22no—
85C2×#216—24no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88#924—22no—
89#924—24no—
90#924—22no—
91#924—24no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×#208—232no—
95C2×#208—24no—
96C2×#208—24no—
97C2×#208—24no—
98C2×#208—24no—
99C2×#208—24no—
100C2×#208D418yes—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—24no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8D414yes—
105C22×C3⋊C8C2312yes—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C3×#95—21no—
109C3×#95—22no—
110#835—22no—
111#835—24no—
112C3⋊#95—21no—
113C3⋊#95—22no—
114#810—22no—
115#810—24no—
116#808—24no—
117#807—22no—
118C24×C4C2×D311yescharacteristic
119C22×Q16—31no—
120C22×M16—32no—
121C22×M16—32no—
122C22×M16—64no—
123C22×M16—68no—
124C23×C8—31no—
125C23×C8—31no—
126C2×C22⋊Q8—31no—
127C2×C22⋊Q8—32no—
128C2×C22⋊Q8C2×D311yescharacteristic
129C2×C22⋊Q8C2×D312yes—
130#470—68no—
131#467—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×#103—32no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#103—31no—
138#385—22no—
139#385—21no—
140#385—62no—
141#385—61no—
142#380—31no—
143#380—32no—
144#380C2×D311yescharacteristic
145#380C2×D312yes—
146#379—21no—
147#379—61no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—24no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—42no—
156C24×C3C2×D412yes—
157C24×C3C2×D411yescharacteristic
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C22×C3×C4—28no—
163C22×C3×C4—28no—
164C22×C3×C4—216no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4#4114yes—
184C22×C3×C4#4112yes—
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×Dic6—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—42no—
203#217—44no—
204#217—44no—
205#217—42no—
206#217—42no—
207#216—44no—
208C4⋊Dic3—21no—
209C4⋊Dic3—22no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212Dic3⋊C4—42no—
213Dic3⋊C4—44no—
214#208—216no—
215#208—216no—
216C2×C3⋊C8—28no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—28no—
221C2×C3⋊C8C2×D414yes—
222C2×C3⋊C8C2×D412yes—
223C2×C3⋊C8C2×D412yes—
224C2×C3⋊C8C2×D414yes—
225C25#6711yescharacteristic
226C22×Q8—31no—
227C22×Q8C22×D311yescharacteristic
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4#6712yes—
233C23×C4#6711yescharacteristic
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D312yes—
236C2×Q16—62no—
237C2×Q16—64no—
238C2×M16—38no—
239C2×M16—64no—
240C2×M16—64no—
241C2×M16—64no—
242C2×M16—64no—
243C2×M16—632no—
244C2×M16—64no—
245C22×C8—31no—
246C22×C8—34no—
247C22×C8—31no—
248C22×C8—32no—
249C22×C8—64no—
250C22×C8—64no—
251C22×C8—62no—
252C22⋊Q8—22no—
253C22⋊Q8—24no—
254C22⋊Q8—22no—
255C22⋊Q8—24no—
256C22⋊Q8—62no—
257C22⋊Q8—64no—
258C22⋊Q8—62no—
259C22⋊Q8—64no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276#107—62no—
277#106—64no—
278#103—62no—
279#103—64no—
280#103—62no—
281#103—64no—
282#95—21no—
283#95—22no—
284#95—61no—
285#95—62no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—28no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—48no—
296C23×C3—48no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3#12114yes—
300C23×C3#12112yes—
301C23×C3#13712yes—
302C23×C3#13712yes—
303C23×C3C22×D411yescharacteristic
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C2×C3×C4—28no—
308C2×C3×C4—216no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4#12112yes—
327C2×C3×C4#12114yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D412yes—
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339Dic6—41no—
340Dic6—41no—
341Dic6—42no—
342C3⋊C8—24no—
343C3⋊C8—22no—
344C3⋊C8#12014yes—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—42no—
350C24C2×#6712yes—
351C24C2×#6711yescharacteristic
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—61no—
357C2×Q8—62no—
358C2×Q8—61no—
359C2×Q8—62no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4D3×D412yes—
381C22×C4D3×D414yes—
382C22×C4#23714yes—
383C22×C4#23712yes—
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6712yes—
386C22×C4C23×D311yescharacteristic
387Q16—124no—
388M16—616no—
389M16—616no—
390C2×C8—34no—
391C2×C8—32no—
392C2×C8—32no—
393C2×C8—34no—
394C2×C8—68no—
395C2×C8—68no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—122no—
410C4⋊C4—124no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—122no—
418C22⋊C4—122no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—416no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3#47014yescomplemented
432C22×C3C2×#12111yescharacteristic
433C22×C3C2×#12112yes—
434C22×C3C2×#13711yescharacteristic
435C22×C3C2×#13711yescharacteristic
436C22×C3#57512yes—
437C3×C4—22no—
438C3×C4—24no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4—82no—
442C3×C4C2×#12011yescharacteristic
443C3×C4#52511yescharacteristic
444C3×C4#52512yes—
445Dic3—81no—
446Dic3—81no—
447Dic3—82no—
448C23—22no—
449C23—22no—
450C23—28no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—48no—
458C23—48no—
459C23—44no—
460C23—44no—
461C23#93914yes—
462C23#93912yes—
463C23#94212yes—
464C23#95112yes—
465C23C22×#6711yescharacteristic
466Q8—41no—
467Q8—42no—
468Q8—41no—
469Q8—121no—
470Q8—122no—
471Q8—121no—
472C2×C4—28no—
473C2×C4—216no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4#93912yes—
499C2×C4#93914yes—
500C2×C4C2×D3×D411yescharacteristic
501C2×C4C2×D3×D412yes—
502C2×C4C2×#23711yescharacteristic
503C2×C4C2×#23712yes—
504C8—34no—
505C8—62no—
506C8—64no—
507C2×C3—24no—
508C2×C3—44no—
509C2×C3#284711yescharacteristic
510C2×C3C2×#47011yescharacteristic
511C2×C3#305411yescharacteristic
512C2×C3#305612yes—
513C2×C3#305612yes—
514C22—24no—
515C22—22no—
516C22—22no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—416no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22#505214yescomplemented
527C22C2×#93911yescharacteristic
528C22C2×#93912yes—
529C22C2×#94211yescharacteristic
530C22#582212yes—
531C22C2×#95111yescharacteristic
532C4—22no—
533C4—24no—
534C4—81no—
535C4—81no—
536C4—82no—
537C4—242no—
538C4—241no—
539C4—241no—
540C4#558611yescharacteristic
541C4#558612yes—
542C4D3×#12011yescharacteristic
543C3#3293711yescharacteristic, complemented, Sylow
544C2—24no—
545C2—44no—
546C2#8260111yescharacteristic
547C2C2×#505211yescharacteristic
548C2#8273212yes—
549C2#8292311yescharacteristic
550C2#8292512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2848C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#2848D3
hypocenterC3#32937
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
4. C2×#5052C2semidirect
5. C2×#5051C2semidirect
6. #71697C2semidirect
7. #71696C2semidirect
8. #69516C2semidirect
9. #69514C2semidirect
10. #69513C2semidirect
11. #69511C2semidirect
16. C22×#216C22semidirect
31. C2×#835C22semidirect
32. C2×#835C22semidirect
39. C2×#810C22semidirect
40. C2×#810C22semidirect
41. C2×#808C22semidirect
42. C2×#807C22semidirect
43. #4529C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
431. C22×C3#470semidirect
526. C22#5052semidirect
543. C3#32937semidirect

Maximal subgroups

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

subgroupindex
C23×#2082
C3×#28482
#840312
C2×#50522
C2×#50512
#716972
#716962
#695162
#695142
#695132
#695112
#329373