Finite Group Databasebeta

← #10945045

Subgroups — #10945045

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×#5798C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3C2×#4723C211yesmaximal, characteristic, complemented
4#73951C211yesmaximal, characteristic, complemented
5C3×#1786C211yesmaximal, characteristic
6#70171C212yesmaximal, complemented
7#70171C212yesmaximal, complemented
8#70170C212yesmaximal, complemented
9#70168C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic, complemented
11#69503C211yesmaximal, characteristic, complemented
12#33018—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#217—22no—
15#5798—24no—
16#5798—24no—
17C2×D4×Dic3—21no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20C2×D4×Dic3C2212yes—
21C22×#214C2212yes—
22C2×#926C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5218—24no—
30C2×#832C2212yes—
31C2×#830C2211yescharacteristic
32C2×#818C2211yescharacteristic, complemented
33C2×#817C2211yescharacteristic, complemented
34C2×#816C2211yescharacteristic, complemented
35C2×#816C2211yescharacteristic, complemented
36#4984C2212yescomplemented
37#4973C2212yescomplemented
38#4953—22no—
39#4723—24no—
40#4661—21no—
41#4659—21no—
42#4550C2212yescomplemented
43#4550C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
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#1818—31no—
56#1786D311yescharacteristic
57C25×C3—21no—
58C22×C3×D4—24no—
59C22×C3×D4—24no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4D414yes—
63C22×C3×D4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×Dic3—22no—
66C23×Dic3—22no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—21no—
72C2×#217—216no—
73C2×#217—21no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—21no—
77C2×#217D412yes—
78D4×Dic3—28no—
79D4×Dic3—28no—
80D4×Dic3—24no—
81D4×Dic3—24no—
82D4×Dic3—24no—
83D4×Dic3—24no—
84C2×#214—24no—
85C2×#214—24no—
86C2×#214—24no—
87#926—22no—
88#926—22no—
89#926—22no—
90#926D414yes—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97#924—22no—
98#924—22no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2312yes—
102C22×C3⋊C8C2312yes—
103C3×#102—24no—
104C3×#102—24no—
105C3×C22⋊C8—22no—
106C3×#95—21no—
107#832—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#830—22no—
112#818—22no—
113#817—22no—
114#816—22no—
115#816—22no—
116C23×D4C2×D311yescharacteristic
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×#117—31no—
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×#102C2×D312yes—
136C2×#102C2×D312yes—
137C2×C22⋊C8—31no—
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3—28no—
150C24×C3—22no—
151C24×C3—24no—
152C24×C3—81no—
153C24×C3—81no—
154C24×C3—82no—
155C24×C3C2×D411yescharacteristic
156C2×C3×D4—28no—
157C2×C3×D4—28no—
158C2×C3×D4—28no—
159C2×C3×D4—216no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4C2×D414yes—
165C2×C3×D4#4114yes—
166C2×C3×D4#4114yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—28no—
177C22×Dic3—28no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—82no—
181C22×Dic3—82no—
182C22×Dic3—82no—
183C22×Dic3—82no—
184C22×Dic3C2×D412yes—
185C22×Dic3C2×D412yes—
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—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C22⋊C4—81no—
197#217—28no—
198#217—28no—
199#217—24no—
200#217—24no—
201#217—42no—
202#214—48no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—82no—
210C4⋊Dic3C2×D412yes—
211C4⋊Dic3C2×D412yes—
212Dic3⋊C4—42no—
213C4×Dic3—24no—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C25—21no—
220C22×Q8—31no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4D1214yes—
226C22×D4C22×D312yes—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4C22×D311yescharacteristic
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—64no—
233C22×C8—32no—
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D311yescharacteristic
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C4×D4—64no—
239C4×D4—64no—
240C4×D4—68no—
241C4×D4—68no—
242C4×D4—64no—
243C4×D4—64no—
244#117—34no—
245#117—62no—
246#117—62no—
247#117—62no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—616no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—61no—
263C2×C42—32no—
264C2×C42—61no—
265#107—62no—
266#106—62no—
267#103—62no—
268#103—62no—
269#102—24no—
270#102—24no—
271#102—64no—
272C22⋊C8—22no—
273C22⋊C8—62no—
274#95—21no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3—216no—
279C23×C3—24no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—22no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—84no—
291C23×C3—84no—
292C23×C3—88no—
293C23×C3—88no—
294C23×C3#12012yes—
295C23×C3C22×D411yescharacteristic
296C3×D4—216no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—24no—
300C3×D4#12318yes—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C4C2×#4112yes—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—88no—
330C2×Dic3—88no—
331Dic6—42no—
332Dic6—41no—
333Dic6—41no—
334C3×C8—41no—
335C3×C8—41no—
336C3⋊C8—42no—
337C24—28no—
338C24—22no—
339C24—24no—
340C24—81no—
341C24—81no—
342C24—82no—
343C24D3×D411yescharacteristic
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—216no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4C2×D1214yes—
357C2×D4#23714yes—
358C2×D4#23714yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—68no—
367C22×C4—68no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—242no—
371C22×C4—242no—
372C22×C4—242no—
373C22×C4—242no—
374C22×C4C2×D1212yes—
375C22×C4D3×D411yescharacteristic
376C22×C4D3×D412yes—
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—32no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—122no—
397C4⋊C4—242no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—81no—
403C22⋊C4—122no—
404C42—64no—
405C42—62no—
406C42—62no—
407C22×C3—22no—
408C22×C3—28no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—816no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#52512yes—
422C22×C3#57611yescharacteristic
423C22×C3#57711yescharacteristic
424C22×C3#57712yes—
425C3×C4—22no—
426C3×C4—82no—
427C3×C4C2×#12312yes—
428C3×C4#52511yescharacteristic
429C3×C4#52511yescharacteristic
430Dic3—42no—
431Dic3—42no—
432Dic3—42no—
433Dic3—42no—
434Dic3—81no—
435Dic3—81no—
436C23—216no—
437C23—24no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—84no—
450C23—88no—
451C23—88no—
452C23#88212yes—
453C23C2×D3×D411yescharacteristic
454Q8—122no—
455Q8—121no—
456Q8—121no—
457D4—216no—
458D4—24no—
459D4—24no—
460D4—24no—
461D4#88618yes—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—122no—
485C2×C4—248no—
486C2×C4—248no—
487C2×C4#88212yes—
488C2×C4C22×D1211yescharacteristic
489C2×C4C2×D3×D411yescharacteristic
490C2×C4C2×#23712yes—
491C2×C4C2×#23712yes—
492C8—41no—
493C8—41no—
494C8—122no—
495C2×C3—42no—
496C2×C3—48no—
497C2×C3#284911yescharacteristic
498C2×C3#300911yescharacteristic
499C2×C3#301211yescharacteristic
500C2×C3#311812yes—
501C2×C3#312312yes—
502C22—22no—
503C22—22no—
504C22—28no—
505C22—22no—
506C22—42no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—42no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—816no—
515C22C2×#88211yescharacteristic
516C22#558612yes—
517C22#575711yescharacteristic
518C22#576211yescharacteristic
519C22#577512yes—
520C4—22no—
521C4—82no—
522C4—122no—
523C4—122no—
524C4—122no—
525C4—122no—
526C4—241no—
527C4—241no—
528C4C2×#88612yes—
529C4#558611yescharacteristic
530C4#558611yescharacteristic
531C3#3301811yescharacteristic, complemented, Sylow
532C2—42no—
533C2—48no—
534C2#7384711yescharacteristic
535C2#7874711yescharacteristic
536C2#7892911yescharacteristic
537C2#7999212yes—
538C2#8013712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenter#1786D3
hypocenterC3#33018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
3. C2×#4723C2semidirect
4. #73951C2semidirect
6. #70171C2semidirect
7. #70171C2semidirect
8. #70170C2semidirect
9. #70168C2semidirect
10. #69911C2semidirect
11. #69503C2semidirect
32. C2×#818C22semidirect
33. C2×#817C22semidirect
34. C2×#816C22semidirect
35. C2×#816C22semidirect
36. #4984C22semidirect
37. #4973C22semidirect
42. #4550C22semidirect
43. #4550C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
531. C3#33018semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
C2×#52182
C2×#47232
#739512
C3×#17862
#701712
#701702
#701682
#699112
#695032
#330183