Finite Group Databasebeta

← #10944098

Subgroups — #10944098

Order 768.

1546 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5521C211yesmaximal, characteristic
2#82786C211yesmaximal, characteristic, complemented
3C2×#5046C211yesmaximal, characteristic
4C2×#4818C212yesmaximal, complemented
5C2×#4731C211yesmaximal, characteristic
6C3×#1873C211yesmaximal, characteristic, complemented
7#71585C211yesmaximal, characteristic, complemented
8#70276C212yesmaximal, complemented
9#69909C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68893C212yesmaximal, complemented
12#32914—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#944C2211yescharacteristic, complemented
17C22×#215C2212yes—
18C2×#926C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×#204C2212yes—
21#5521—28no—
22C2×#893C2212yes—
23C2×#893C2212yes—
24C2×D3×C22⋊C4—22no—
25C2×C4×D12—22no—
26C2×C3×#103C2212yescomplemented
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2211yescharacteristic, complemented
29C2×#830C2211yescharacteristic
30C2×#821C2212yes—
31C2×Dic3⋊C8C2212yescomplemented
32#5046—24no—
33#4982—22no—
34#4977C2211yescharacteristic, complemented
35#4973C2211yescharacteristic, complemented
36C2×#810C2212yescomplemented
37C2×#809C2212yes—
38#4952—22no—
39#4818—28no—
40#4731—24no—
41C3×#340C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#518—31no—
45C2×#461—32no—
46C2×#450—31no—
47C2×#450—31no—
48#2621—31no—
49#1877—32no—
50#1877—32no—
51#1873—31no—
52#1873D311yescharacteristic, complemented
53#1804—32no—
54#1798—31no—
55C24×D3—21no—
56C22×C3×Q8C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×D12—21no—
59C22×D12—22no—
60C22×D12—24no—
61C22×D12—24no—
62C22×D12D414yes—
63C22×D12C2312yes—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×#117—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—21no—
77C2×#217—21no—
78C2×#217D412yes—
79#944—24no—
80C2×#215—24no—
81C2×#215—24no—
82C2×#215—24no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85#926—24no—
86C2×C4⋊Dic3D412yes—
87C2×C4⋊Dic3C2311yescharacteristic
88#924—22no—
89#924—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95C2×#204—24no—
96C2×#204—24no—
97C2×#204—24no—
98#893—24no—
99#893—24no—
100#893—24no—
101#893D418yes—
102D3×C22⋊C4—216no—
103C4×D12—216no—
104C3×#103—24no—
105C3×C22⋊C8—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109#830—22no—
110#821—24no—
111Dic3⋊C8—24no—
112#810—24no—
113#809—24no—
114C23×D4—31no—
115C22×SD16—32no—
116C22×SD16—32no—
117C2×C4⋊Q8—31no—
118C2×C22⋊Q8—31no—
119C2×C22⋊Q8C2×D311yescharacteristic
120#518—68no—
121C2×C4×D4—32no—
122C2×C4×D4—32no—
123C2×C4×D4—62no—
124C2×#117—31no—
125C2×#117C2×D311yescharacteristic
126C22×C22⋊C4—62no—
127#461—68no—
128#450—64no—
129#450—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103C2×D312yescomplemented
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#398—62no—
138#397—62no—
139#391—31no—
140#391—31no—
141#382—31no—
142#340—32no—
143#340C2×D312yescomplemented
144#336—32no—
145C24×C3C2×D411yescharacteristic
146C23×D3—22no—
147C23×D3—24no—
148C23×D3—24no—
149C23×D3—24no—
150C23×D3—82no—
151C23×D3—82no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4#4112yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×D12—24no—
170C2×D12—24no—
171C2×D12—216no—
172C2×D12—24no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—216no—
177C2×D12C2×D414yes—
178C2×D12C2×D414yes—
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—22no—
187C2×Dic6—21no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4C2×D414yes—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—42no—
200C3×C42—24no—
201#217—24no—
202#217—42no—
203#215—48no—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—28no—
206C4⋊Dic3—24no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209Dic3⋊C4—42no—
210Dic3⋊C4—44no—
211C4×Dic3—22no—
212C4×Dic3C2×D414yes—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216#204—48no—
217C25—61no—
218C22×Q8—31no—
219C22×Q8C22×D311yescharacteristic
220C22×D4—34no—
221C22×D4—32no—
222C22×D4—64no—
223C22×D4—64no—
224C22×D4—62no—
225C22×D4—61no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4C22×D311yescharacteristic
229C2×SD16—64no—
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—64no—
234C2×SD16—64no—
235C22×C8—32no—
236C22×C8C22×D312yes—
237C4⋊Q8—64no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—62no—
241C22⋊Q8—62no—
242C4×D4—38no—
243C4×D4—616no—
244C4×D4—64no—
245C4×D4—64no—
246C4×D4—64no—
247#117—24no—
248#117—64no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—616no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C42—21no—
264C2×C42—31no—
265C2×C42—31no—
266C4⋊C8—64no—
267#103—24no—
268#103—64no—
269#102—64no—
270#102—64no—
271C22⋊C8—22no—
272C22⋊C8—62no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3#12112yes—
281C23×C3C22×D411yescharacteristic
282C22×D3—24no—
283C22×D3—24no—
284C22×D3—24no—
285C22×D3—216no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—84no—
291C22×D3—84no—
292C22×D3—88no—
293C22×D3—88no—
294C3×Q8—42no—
295C3×Q8—42no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4#12112yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D412yes—
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4112yes—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3#12014yes—
324D12—216no—
325D12—24no—
326D12—24no—
327D12—24no—
328D12#12118yes—
329C4×D3—88no—
330C4×D3—88no—
331Dic6—42no—
332Dic6—42no—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—64no—
336C24—64no—
337C24—62no—
338C24—64no—
339C24—242no—
340C24—242no—
341C24D3×D411yescharacteristic
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—61no—
347C2×Q8—62no—
348C2×Q8—61no—
349C2×Q8—62no—
350C2×D4—34no—
351C2×D4—34no—
352C2×D4—34no—
353C2×D4—616no—
354C2×D4—64no—
355C2×D4—64no—
356C2×D4—616no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—68no—
369C22×C4—68no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—242no—
373C22×C4—242no—
374C22×C4—242no—
375C22×C4—242no—
376C22×C4#23512yes—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D412yes—
379C22×C4C23×D311yescharacteristic
380SD16—128no—
381SD16—128no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—42no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—124no—
396C4⋊C4—122no—
397C4⋊C4D3×D414yes—
398C22⋊C4—24no—
399C22⋊C4—42no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C22⋊C4—64no—
403C22⋊C4—122no—
404C42—24no—
405C42—34no—
406C42—62no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3C2×#12011yescharacteristic
413C22×C3C2×#12111yescharacteristic
414C22×C3#52511yescharacteristic
415C22×C3#57712yes—
416C22×C3#57712yes—
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—44no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4C2×#12112yes—
430C3×C4#52512yes—
431Dic3—24no—
432Dic3—42no—
433Dic3—81no—
434Dic3—81no—
435C23—21no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—616no—
440C23—64no—
441C23—64no—
442C23—64no—
443C23—122no—
444C23—122no—
445C23—122no—
446C23—122no—
447C23—244no—
448C23—244no—
449C23—248no—
450C23—248no—
451C23#88412yes—
452C23C2×D3×D411yescharacteristic
453Q8—42no—
454Q8—42no—
455Q8—122no—
456Q8—122no—
457D4—38no—
458D4—616no—
459D4—64no—
460D4—64no—
461D4—64no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—34no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—248no—
486C2×C4—248no—
487C2×C4#88412yes—
488C2×C4C2×#23511yescharacteristic
489C2×C4C2×D3×D412yes—
490C2×C4C2×D3×D411yescharacteristic
491C2×C4D3×#4112yes—
492C8—42no—
493C8—122no—
494C2×C3—42no—
495C2×C3#284711yescharacteristic
496C2×C3#300912yes—
497C2×C3#304211yescharacteristic
498C2×C3#304211yescharacteristic
499C2×C3#312312yes—
500D3—48no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—42no—
505C22—42no—
506C22—68no—
507C22—124no—
508C22—124no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—124no—
513C22—2416no—
514C22C2×#88411yescharacteristic
515C22D3×#12011yescharacteristic
516C22#565611yescharacteristic
517C22#577012yes—
518C22#577812yes—
519C4—22no—
520C4—44no—
521C4—64no—
522C4—81no—
523C4—81no—
524C4—122no—
525C4—241no—
526C4—241no—
527C4#554712yes—
528C4D3×#12112yes—
529C3#3291411yescharacteristic, complemented, Sylow
530C2—42no—
531C2—128no—
532C2#7493511yescharacteristic
533C2#7886311yescharacteristic
534C2#7897611yescharacteristic
535C2#8022912yes—
536C2#8149812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1873D3
hypocenterC3#32914
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
2. #82786C2semidirect
4. C2×#4818C2semidirect
6. C3×#1873C2semidirect
7. #71585C2semidirect
8. #70276C2semidirect
9. #69909C2semidirect
10. #68896C2semidirect
11. #68893C2semidirect
16. C2×#944C22semidirect
26. C2×C3×#103C22semidirect
28. #5229C22semidirect
31. C2×Dic3⋊C8C22semidirect
34. #4977C22semidirect
35. #4973C22semidirect
36. C2×#810C22semidirect
41. C3×#340C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
52. #1873D3semidirect
132. C2×#103C2×D3semidirect
143. #340C2×D3semidirect
529. C3#32914semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
#827862
C2×#50462
C2×#48182
C2×#47312
C3×#18732
#715852
#702762
#699092
#688962
#688932
#329143