Finite Group Databasebeta

← #10945046

Subgroups — #10945046

Order 768.

1432 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5521C211yesmaximal, characteristic
2#82263C211yesmaximal, characteristic, complemented
3C2×#4731C211yesmaximal, characteristic
4C2×#4723C211yesmaximal, characteristic, complemented
5C3×#1818C211yesmaximal, characteristic, complemented
6#69959C211yesmaximal, characteristic
7#69926C211yesmaximal, characteristic, complemented
8#68672C212yesmaximal, complemented
9#68672C212yesmaximal, complemented
10#68671C212yesmaximal, complemented
11#68670C212yesmaximal, complemented
12#33018—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×#204C2212yes—
18#5521—24no—
19#5521—24no—
20C2×#893—21no—
21C2×#893—21no—
22C2×D3×C22⋊C4—22no—
23C2×C4×D12C2212yes—
24C2×C4×D12C2212yes—
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#818C2211yescharacteristic, complemented
32C2×#817C2211yescharacteristic, complemented
33C2×#816C2211yescharacteristic, complemented
34C2×#816C2211yescharacteristic, complemented
35#4985—21no—
36#4983—21no—
37#4952—22no—
38#4929C2212yescomplemented
39#4928C2212yescomplemented
40#4731—24no—
41#4723—24no—
42C3×#336C2212yescomplemented
43C3×#336C2212yescomplemented
44#4551C2212yescomplemented
45#4551C2212yescomplemented
46C2×#518—31no—
47C2×#485—31no—
48C2×#450—31no—
49#2625—31no—
50#1877—32no—
51#1877—32no—
52#1864—32no—
53#1862—32no—
54#1825—31no—
55#1818D311yescharacteristic, complemented
56#1786—31no—
57C24×D3—21no—
58C23×C3×C4C2311yescharacteristic
59C22×D12—21no—
60C22×D12—22no—
61C22×D12—24no—
62C22×D12—24no—
63C22×D12D414yes—
64C22×D12C2312yes—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#117—22no—
72C3×#117—22no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42C2312yes—
77C2×#217—21no—
78C2×#217—21no—
79C2×#217—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82#926—22no—
83#926D414yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88#924—22no—
89#924—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—21no—
93C2×#204—24no—
94C2×#204—24no—
95C2×#204—24no—
96#893—28no—
97#893—28no—
98D3×C22⋊C4—216no—
99C4×D12—24no—
100C4×D12—24no—
101C4×D12—24no—
102C4×D12—24no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109#821—24no—
110#821—24no—
111#821—24no—
112#818—22no—
113#817—22no—
114#816—22no—
115#816—22no—
116C23×D4—31no—
117C22×SD16—32no—
118C2×C22⋊Q8—31no—
119#518—64no—
120#518—64no—
121C2×C4×D4—32no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124C2×C4×D4—61no—
125C2×#117—31no—
126C2×#117C2×D311yescharacteristic
127C22×C22⋊C4—62no—
128#485—64no—
129#450—64no—
130C2×#107—31no—
131C2×#106—31no—
132C2×#103—31no—
133C2×#103—31no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×C22⋊C8C2×D311yescharacteristic
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336C2×D312yescomplemented
148#336C2×D312yescomplemented
149C24×C3C2×D411yescharacteristic
150C23×D3—22no—
151C23×D3—28no—
152C23×D3—24no—
153C23×D3—81no—
154C23×D3—82no—
155C23×D3—81no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C2×D12—28no—
170C2×D12—28no—
171C2×D12—28no—
172C2×D12—24no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—216no—
177C2×D12C2×D414yes—
178C2×D12#4114yes—
179C2×D12#4114yes—
180C2×C4×D3—28no—
181C2×C4×D3—28no—
182C2×C4×D3—82no—
183C2×C4×D3—82no—
184C2×C4×D3—82no—
185C2×C4×D3—82no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C22⋊C4—24no—
200C3×C42—22no—
201C3×C42—22no—
202#217—24no—
203#217—42no—
204#217—81no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—28no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3C2×D412yes—
214C4⋊Dic3C2×D412yes—
215Dic3⋊C4—42no—
216Dic3⋊C4—82no—
217C4×Dic3—24no—
218#204—48no—
219C25—61no—
220C22×Q8—31no—
221C22×D4—34no—
222C22×D4—32no—
223C22×D4—64no—
224C22×D4—64no—
225C22×D4—61no—
226C22×D4—62no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4C22×D311yescharacteristic
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—64no—
233C22×C8C22×D312yes—
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D311yescharacteristic
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C4×D4—64no—
239C4×D4—64no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—68no—
243C4×D4—68no—
244#117—22no—
245#117—22no—
246#117—34no—
247#117—62no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4D1211yescharacteristic
255C2×C4⋊C4D1211yescharacteristic
256C2×C22⋊C4—616no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4D1212yes—
263C2×C42—61no—
264C2×C42C22×D312yes—
265#107—62no—
266#106—62no—
267#103—62no—
268#103—62no—
269#102—64no—
270#102—64no—
271#102—64no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274#95—61no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—22no—
281C23×C3—21no—
282C23×C3#12012yes—
283C23×C3C22×D411yescharacteristic
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—216no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—84no—
293C22×D3—84no—
294C22×D3—88no—
295C22×D3—88no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4#12012yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4112yes—
310C2×C3×C4C2×#4112yes—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324D12—216no—
325D12—24no—
326D12—24no—
327D12—24no—
328D12#12318yes—
329C4×D3—88no—
330C4×D3—88no—
331Dic6—42no—
332Dic6—41no—
333Dic6—41no—
334C3×C8—42no—
335C3×C8—41no—
336C3×C8—41no—
337C24—68no—
338C24—62no—
339C24—64no—
340C24—242no—
341C24—241no—
342C24—241no—
343C24C2×D1211yescharacteristic
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×D4—34no—
349C2×D4—34no—
350C2×D4—34no—
351C2×D4—616no—
352C2×D4—68no—
353C2×D4—68no—
354C2×D4—68no—
355C2×D4—64no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—68no—
362C22×C4—68no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—242no—
369C22×C4—242no—
370C22×C4—242no—
371C22×C4—242no—
372C22×C4C2×D1212yes—
373C22×C4C2×D1212yes—
374C22×C4C2×D1211yescharacteristic
375C22×C4D3×D412yes—
376C22×C4D3×D412yes—
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—32no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—122no—
397C4⋊C4—242no—
398C22⋊C4—24no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—64no—
402C22⋊C4—122no—
403C22⋊C4—241no—
404C42—22no—
405C42—22no—
406C42—64no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3C2×#12011yescharacteristic
413C22×C3#52512yes—
414C22×C3#57611yescharacteristic
415C22×C3#57712yes—
416C22×C3#57711yescharacteristic
417C2×D3—28no—
418C2×D3—44no—
419C2×D3—44no—
420C2×D3—44no—
421C2×D3—44no—
422C2×D3—44no—
423C2×D3—44no—
424C2×D3—816no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4C2×#12312yes—
429C3×C4#52511yescharacteristic
430C3×C4#52511yescharacteristic
431Dic3—42no—
432Dic3—42no—
433Dic3—81no—
434Dic3—82no—
435Dic3—81no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—616no—
441C23—64no—
442C23—64no—
443C23—64no—
444C23—122no—
445C23—122no—
446C23—122no—
447C23—122no—
448C23—244no—
449C23—244no—
450C23—248no—
451C23—248no—
452C23#88212yes—
453C23C22×D1211yescharacteristic
454Q8—122no—
455Q8—121no—
456Q8—121no—
457D4—38no—
458D4—616no—
459D4—64no—
460D4—64no—
461D4—64no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—248no—
486C2×C4—248no—
487C2×C4#88212yes—
488C2×C4C2×D3×D411yescharacteristic
489C2×C4C2×D3×D411yescharacteristic
490C2×C4D3×#4112yes—
491C2×C4D3×#4112yes—
492C8—42no—
493C8—41no—
494C8—41no—
495C2×C3—42no—
496C2×C3#284911yescharacteristic
497C2×C3#300911yescharacteristic
498C2×C3#301211yescharacteristic
499C2×C3#311812yes—
500C2×C3#312312yes—
501D3—48no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—42no—
506C22—42no—
507C22—68no—
508C22—124no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—124no—
513C22—124no—
514C22—2416no—
515C22C2×#88211yescharacteristic
516C22#552412yes—
517C22#575112yes—
518C22#575111yescharacteristic
519C22#575211yescharacteristic
520C4—22no—
521C4—42no—
522C4—42no—
523C4—122no—
524C4—122no—
525C4—242no—
526C4—241no—
527C4—241no—
528C4D3×#12312yes—
529C4#565611yescharacteristic
530C4#565611yescharacteristic
531C3#3301811yescharacteristic, complemented, Sylow
532C2—42no—
533C2—128no—
534C2#7414611yescharacteristic
535C2#7867111yescharacteristic
536C2#7867211yescharacteristic
537C2#7938312yes—
538C2#7938712yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1818D3
hypocenterC3#33018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #82263C2semidirect
4. C2×#4723C2semidirect
5. C3×#1818C2semidirect
7. #69926C2semidirect
8. #68672C2semidirect
9. #68672C2semidirect
10. #68671C2semidirect
11. #68670C2semidirect
31. C2×#818C22semidirect
32. C2×#817C22semidirect
33. C2×#816C22semidirect
34. C2×#816C22semidirect
38. #4929C22semidirect
39. #4928C22semidirect
42. C3×#336C22semidirect
43. C3×#336C22semidirect
44. #4551C22semidirect
45. #4551C22semidirect
55. #1818D3semidirect
147. #336C2×D3semidirect
148. #336C2×D3semidirect
531. C3#33018semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
#822632
C2×#47312
C2×#47232
C3×#18182
#699592
#699262
#686722
#686712
#686702
#330183