Finite Group Databasebeta

← #10944123

Subgroups — #10944123

Order 768.

1588 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5798C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic
3C2×#5218C211yesmaximal, characteristic, complemented
4C2×#5153C212yesmaximal, complemented
5#82677C211yesmaximal, characteristic, complemented
6#72037C212yesmaximal, complemented
7#71678C211yesmaximal, characteristic, complemented
8#70171C212yesmaximal, complemented
9#70153C212yesmaximal, complemented
10#69931C211yesmaximal, characteristic, complemented
11#69909C211yesmaximal, characteristic, complemented
12#32917—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#121C2211yescharacteristic
16C22×#217—22no—
17#5798—28no—
18C2×#936C2211yescharacteristic, complemented
19C2×D4×Dic3—22no—
20C2×D4×Dic3C2212yes—
21C2×D4×Dic3C2212yes—
22C22×#214C2212yes—
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C3×#447—24no—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29#5222—22no—
30#5221C2211yescharacteristic, complemented
31#5218—24no—
32C2×#832C2212yes—
33C2×#832C2212yescomplemented
34C2×#830C2211yescharacteristic
35#5153—28no—
36C2×#816C2212yescomplemented
37C2×Dic3⋊C8C2212yescomplemented
38#4975C2211yescharacteristic, complemented
39#4973C2211yescharacteristic, complemented
40#4948—22no—
41#4551C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#518—31no—
45C2×#463—32no—
46C2×#450—31no—
47C2×#447D311yescharacteristic
48#2620—31no—
49#1879—32no—
50#1877—32no—
51#1875—31no—
52#1873—31no—
53#1802—32no—
54#1798—31no—
55C25×C3—21no—
56C22×C3×D4—24no—
57C22×C3×D4—24no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4D414yes—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×Dic3—22no—
66C23×Dic3—22no—
67C22×Dic6C2311yescharacteristic
68C2×C3×D8—24no—
69C2×C3×D8—24no—
70C2×C3×D8—24no—
71C22×C3×C8C2312yes—
72C3×#121—22no—
73C3×#121—22no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×#217—216no—
77C2×#217—21no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—21no—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217D412yes—
84#936—24no—
85D4×Dic3—216no—
86D4×Dic3—24no—
87D4×Dic3—24no—
88D4×Dic3—24no—
89D4×Dic3D418yescomplemented
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—24no—
93#926—24no—
94#926—24no—
95C2×C4⋊Dic3D412yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2312yes—
98#924—22no—
99#924—22no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—22no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C3×#102—24no—
107C3×C22⋊C8—22no—
108#832—24no—
109#832—24no—
110C3⋊#95—22no—
111#830—22no—
112#816—24no—
113Dic3⋊C8—24no—
114C23×D4C2×D311yescharacteristic
115C22×SD16—32no—
116C22×D8C2×D312yes—
117C2×#124—31no—
118C2×C22⋊Q8—31no—
119C2×#121C2×D311yescharacteristic
120#518—68no—
121C2×C4×D4—32no—
122C2×C4×D4—32no—
123C2×C4×D4—62no—
124C2×#117—31no—
125C2×#117—31no—
126C22×C22⋊C4—62no—
127#463—68no—
128#450—64no—
129#447—24no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3—24no—
146C24×C3—24no—
147C24×C3—22no—
148C24×C3—24no—
149C24×C3—21no—
150C24×C3—82no—
151C24×C3—82no—
152C24×C3C2×D411yescharacteristic
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—216no—
156C2×C3×D4—216no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—82no—
167C2×C3×D4C2×D414yes—
168C2×C3×D4C2×D414yes—
169C2×C3×D4#4114yes—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C22×Dic3—22no—
178C22×Dic3—28no—
179C22×Dic3—28no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—82no—
183C22×Dic3—82no—
184C22×Dic3—82no—
185C22×Dic3—82no—
186C22×Dic3C2×D412yes—
187C22×Dic3C2×D412yes—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—22no—
192C3×D8—48no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C22⋊C4—42no—
199#217—24no—
200#217—28no—
201#217—24no—
202#217—28no—
203#217—42no—
204#217—44no—
205#217—82no—
206#214—48no—
207C4⋊Dic3—22no—
208C4⋊Dic3—24no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3C2×D414yes—
212Dic3⋊C4—42no—
213Dic3⋊C4—82no—
214C4×Dic3—24no—
215C4×Dic3—22no—
216C4×Dic3C2×D414yes—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C25—21no—
221C22×Q8—31no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4#6714yes—
228C22×D4C22×D311yescharacteristic
229C22×D4C22×D312yes—
230C23×C4—62no—
231C23×C4—62no—
232C23×C4C22×D311yescharacteristic
233C2×SD16—64no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×D8—24no—
237C2×D8—24no—
238C2×D8—24no—
239C22×C8—32no—
240C22×C8C22×D312yes—
241#124—64no—
242C22⋊Q8—62no—
243C22⋊Q8—62no—
244#121—22no—
245#121—22no—
246C4×D4—38no—
247C4×D4—64no—
248C4×D4—64no—
249C4×D4—616no—
250C4×D4—64no—
251#117—64no—
252#117—64no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—32no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—616no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—62no—
268C2×C42—31no—
269C2×C42—31no—
270C2×C42—61no—
271C4⋊C8—64no—
272#103—64no—
273#102—24no—
274#102—64no—
275#102—64no—
276C22⋊C8—22no—
277C22⋊C8—62no—
278#95—62no—
279C23×C3—216no—
280C23×C3—21no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—84no—
297C23×C3—84no—
298C23×C3—88no—
299C23×C3—88no—
300C23×C3#12112yes—
301C23×C3C22×D411yescharacteristic
302C3×D4—216no—
303C3×D4—24no—
304C3×D4—24no—
305C3×D4—24no—
306C3×D4—42no—
307C3×D4—42no—
308C3×D4—84no—
309C3×D4#12118yes—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4#12112yes—
317C2×C3×C4C22×D412yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×C3×C4C2×#4111yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—88no—
335C2×Dic3—88no—
336C2×Dic3#12014yes—
337Dic6—42no—
338Dic6—42no—
339C3×C8—42no—
340C3⋊C8—42no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—21no—
346C24—82no—
347C24—82no—
348C24D3×D411yescharacteristic
349C2×Q8—62no—
350C2×Q8—61no—
351C2×Q8—62no—
352C2×Q8—61no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4D3×D414yes—
368C2×D4#23714yes—
369C2×D4C2×#6714yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—68no—
376C22×C4—62no—
377C22×C4—68no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—242no—
381C22×C4—242no—
382C22×C4—242no—
383C22×C4—242no—
384C22×C4D3×D411yescharacteristic
385C22×C4#23712yes—
386C22×C4C2×#6712yes—
387C22×C4C23×D311yescharacteristic
388SD16—128no—
389D8—48no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—34no—
399C4⋊C4—64no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—122no—
404C4⋊C4—242no—
405C22⋊C4—42no—
406C22⋊C4—64no—
407C22⋊C4—68no—
408C22⋊C4—64no—
409C22⋊C4—68no—
410C22⋊C4—124no—
411C22⋊C4—122no—
412C22⋊C4—242no—
413C42—34no—
414C42—64no—
415C42—62no—
416C22×C3—22no—
417C22×C3—28no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—816no—
429C22×C3—81no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×#12011yescharacteristic
435C22×C3C2×#12111yescharacteristic
436C22×C3#52511yescharacteristic
437C22×C3#57612yes—
438C22×C3#57712yes—
439C3×C4—22no—
440C3×C4—81no—
441C3×C4C2×#12112yes—
442C3×C4#52512yes—
443Dic3—24no—
444Dic3—42no—
445Dic3—44no—
446Dic3—81no—
447Dic3—81no—
448C23—216no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—21no—
453C23—24no—
454C23—21no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—84no—
466C23—84no—
467C23—88no—
468C23—88no—
469C23#93912yes—
470C23C2×D3×D411yescharacteristic
471Q8—122no—
472Q8—122no—
473D4—216no—
474D4—24no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—84no—
480D4#93918yes—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—34no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—248no—
502C2×C4—248no—
503C2×C4#93912yes—
504C2×C4C2×D3×D412yes—
505C2×C4C2×#23712yes—
506C2×C4C2×#23711yescharacteristic
507C2×C4C22×#6711yescharacteristic
508C8—42no—
509C8—122no—
510C2×C3—42no—
511C2×C3—48no—
512C2×C3—161no—
513C2×C3#284711yescharacteristic
514C2×C3#301212yes—
515C2×C3#304111yescharacteristic
516C2×C3#304211yescharacteristic
517C2×C3#311812yes—
518C22—22no—
519C22—22no—
520C22—28no—
521C22—22no—
522C22—44no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—44no—
530C22—816no—
531C22—81no—
532C22—82no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22#558611yescharacteristic
537C22D3×#12011yescharacteristic
538C22#575712yes—
539C22#577512yes—
540C22C2×#93911yescharacteristic
541C4—22no—
542C4—64no—
543C4—81no—
544C4—122no—
545C4—124no—
546C4—241no—
547C4—241no—
548C4#558612yes—
549C4C2×#93912yes—
550C3#3291711yescharacteristic, complemented, Sylow
551C2—42no—
552C2—48no—
553C2—161no—
554C2#7999212yes—
555C2#8000512yes—
556C2#8260111yescharacteristic
557C2#8382611yescharacteristic
558C2#8393511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#447C2
socleC23×C3C22×D4
hypercenterC2×#447D3
hypocenterC3#32917
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#5798C2semidirect
3. C2×#5218C2semidirect
4. C2×#5153C2semidirect
5. #82677C2semidirect
6. #72037C2semidirect
7. #71678C2semidirect
8. #70171C2semidirect
9. #70153C2semidirect
10. #69931C2semidirect
11. #69909C2semidirect
18. C2×#936C22semidirect
30. #5221C22semidirect
33. C2×#832C22semidirect
36. C2×#816C22semidirect
37. C2×Dic3⋊C8C22semidirect
38. #4975C22semidirect
39. #4973C22semidirect
41. #4551C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
89. D4×Dic3D4semidirect
550. C3#32917semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
C2×C3×#4472
C2×#52182
C2×#51532
#826772
#720372
#716782
#701712
#701532
#699312
#699092
#329173