Finite Group Databasebeta

← #10945043

Subgroups — #10945043

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×C3×#518C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3#82263C211yesmaximal, characteristic, complemented
4C2×#4996C211yesmaximal, characteristic, complemented
5#72128C211yesmaximal, characteristic
6#71612C212yesmaximal, complemented
7#71612C212yesmaximal, complemented
8#71607C212yesmaximal, complemented
9#71606C212yesmaximal, complemented
10#71584C211yesmaximal, characteristic, complemented
11#71575C211yesmaximal, characteristic, complemented
12#33018—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C3×#518—24no—
15C3×#518—24no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2212yes—
20C2×C3×#117C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C22×C3×C22⋊C4—22no—
23C22×#214C2212yes—
24C2×#924C2211yescharacteristic
25#5218—24no—
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32#4996—24no—
33#4985—21no—
34#4983—21no—
35C2×#810C2211yescharacteristic, complemented
36C2×#810C2211yescharacteristic, complemented
37C2×#808C2211yescharacteristic, complemented
38C2×#807C2211yescharacteristic, complemented
39#4952—22no—
40#4929C2212yescomplemented
41#4928C2212yescomplemented
42#4529C2212yescomplemented
43#4529C2212yescomplemented
44#4524C2212yescomplemented
45#4524C2212yescomplemented
46C2×#518D311yescharacteristic
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#1786—31no—
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×C4—22no—
65C23×C3×C4—22no—
66C23×C3×C4C2311yescharacteristic
67C22×Dic6C2311yescharacteristic
68C3×C4×D4—28no—
69C3×C4×D4—28no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×#117—22no—
75C3×#117—22no—
76C3×#117—22no—
77C3×#117D414yes—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—216no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—21no—
88C2×C3×C42C2312yes—
89C2×#217—21no—
90C2×#217—21no—
91C2×#214—24no—
92C2×#214—24no—
93C2×#214—24no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96#924—22no—
97#924—22no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—22no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103#832—24no—
104#832—24no—
105#832—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—21no—
110#830—22no—
111#830—22no—
112#810—22no—
113#810—22no—
114#808—22no—
115#807—22no—
116C23×D4C2×D311yescharacteristic
117C22×SD16—32no—
118C2×C22⋊Q8—31no—
119#518—24no—
120#518—24no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2×D312yes—
124C2×C4×D4C2×D312yes—
125C2×#117C2×D311yescharacteristic
126C2×#117C2×D311yescharacteristic
127C22×C22⋊C4—22no—
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⋊C8—31no—
138C2×C22⋊C8—31no—
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#336—32no—
148#336—32no—
149C24×C3—28no—
150C24×C3—22no—
151C24×C3—24no—
152C24×C3—81no—
153C24×C3—82no—
154C24×C3—81no—
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—21no—
170C22×C3×C4—21no—
171C22×C3×C4—28no—
172C22×C3×C4—28no—
173C22×C3×C4—82no—
174C22×C3×C4—82no—
175C22×C3×C4—82no—
176C22×C3×C4—82no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—24no—
200C3×C42—24no—
201C3×C42—22no—
202C3×C42—22no—
203#217—42no—
204#217—81no—
205#214—48no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210Dic3⋊C4—42no—
211Dic3⋊C4—82no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—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×D4#6714yes—
226C22×D4C22×D312yes—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4C22×D311yescharacteristic
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—64no—
233C22×C8—32no—
234C22×C8—31no—
235C22×C8—31no—
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117#6714yes—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4#6711yescharacteristic
253C2×C4⋊C4#6711yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C22⋊C4—216no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4#6712yes—
263C2×C42—21no—
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—62no—
273C22⋊C8—62no—
274#95—61no—
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—22no—
285C23×C3—21no—
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—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—88no—
317C2×C3×C4—88no—
318C2×C3×C4#12012yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4112yes—
322C2×C3×C4C2×#4112yes—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331Dic6—42no—
332Dic6—41no—
333Dic6—41no—
334C3⋊C8—42no—
335C3⋊C8—41no—
336C3⋊C8—41no—
337C24—28no—
338C24—22no—
339C24—24no—
340C24—81no—
341C24—81no—
342C24—82no—
343C24C2×#6711yescharacteristic
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—61no—
347C2×Q8—61no—
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×D4#23514yes—
357C2×D4#23514yes—
358C2×D4C2×#6714yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4C2×#6712yes—
373C22×C4C2×#6712yes—
374C22×C4C2×#6711yescharacteristic
375C22×C4C2×#6712yes—
376C22×C4C2×#6712yes—
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—61no—
385C2×C8—61no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—122no—
395C4⋊C4—242no—
396C4⋊C4C2×#6712yes—
397C4⋊C4C2×#6712yes—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—122no—
403C22⋊C4—241no—
404C42—24no—
405C42—22no—
406C42—22no—
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#57712yes—
424C22×C3#57711yescharacteristic
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4C2×#12312yes—
431C3×C4#52511yescharacteristic
432C3×C4#52511yescharacteristic
433Dic3—81no—
434Dic3—82no—
435Dic3—81no—
436C23—216no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—21no—
441C23—21no—
442C23—22no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—84no—
450C23—88no—
451C23—88no—
452C23#95312yes—
453C23C22×#6711yescharacteristic
454Q8—122no—
455Q8—121no—
456Q8—121no—
457D4—216no—
458D4—24no—
459D4—24no—
460D4—24no—
461D4#92918yes—
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—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—88no—
478C2×C4—88no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4#95312yes—
488C2×C4C2×#23512yes—
489C2×C4C2×#23512yes—
490C2×C4C22×#6711yescharacteristic
491C2×C4C22×#6711yescharacteristic
492C8—122no—
493C8—121no—
494C8—121no—
495C2×C3—42no—
496C2×C3—48no—
497C2×C3#284911yescharacteristic
498C2×C3#300911yescharacteristic
499C2×C3#301211yescharacteristic
500C2×C3#311812yes—
501C2×C3#312312yes—
502C22—22no—
503C22—28no—
504C22—22no—
505C22—22no—
506C22—42no—
507C22—44no—
508C22—44no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—816no—
515C22#579112yes—
516C22#583511yescharacteristic
517C22#583612yes—
518C22#583611yescharacteristic
519C22C2×#95311yescharacteristic
520C4—22no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—241no—
526C4—242no—
527C4—241no—
528C4C2×#92912yes—
529C4#579111yescharacteristic
530C4#579111yescharacteristic
531C3#3301811yescharacteristic, complemented, Sylow
532C2—42no—
533C2—48no—
534C2#8254112yes—
535C2#8254512yes—
536C2#8403311yescharacteristic
537C2#8410211yescharacteristic
538C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#518C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#518D3
hypocenterC3#33018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
3. #82263C2semidirect
4. C2×#4996C2semidirect
6. #71612C2semidirect
7. #71612C2semidirect
8. #71607C2semidirect
9. #71606C2semidirect
10. #71584C2semidirect
11. #71575C2semidirect
35. C2×#810C22semidirect
36. C2×#810C22semidirect
37. C2×#808C22semidirect
38. C2×#807C22semidirect
40. #4929C22semidirect
41. #4928C22semidirect
42. #4529C22semidirect
43. #4529C22semidirect
44. #4524C22semidirect
45. #4524C22semidirect
531. C3#33018semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×#52182
#822632
C2×#49962
#721282
#716122
#716072
#716062
#715842
#715752
#330183