Finite Group Databasebeta

← #10944128

Subgroups — #10944128

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×#5521C211yesmaximal, characteristic, complemented
2#82287C211yesmaximal, characteristic, complemented
3C2×#4844C212yesmaximal, complemented
4C2×#4731C211yesmaximal, characteristic, complemented
5C2×#4732C211yesmaximal, characteristic
6C3×#1798C211yesmaximal, characteristic, complemented
7#70479C212yesmaximal, complemented
8#69933C211yesmaximal, characteristic, complemented
9#69931C211yesmaximal, characteristic, complemented
10#68673C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#32917—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C2×#924C2211yescharacteristic
18C22×D24C2212yes—
19C22×#204C2212yes—
20#5521—28no—
21C2×#893—22no—
22C2×D3×C22⋊C4—22no—
23C2×#875C2211yescharacteristic, complemented
24C2×C4×D12C2212yes—
25C2×C4×D12C2212yes—
26C2×C3×C4⋊C8C2212yescomplemented
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2212yescomplemented
32C2×#816C2212yescomplemented
33#4994—22no—
34#4961—22no—
35#4939C2211yescharacteristic, complemented
36#4935C2211yescharacteristic, complemented
37#4929C2211yescharacteristic, complemented
38#4844—28no—
39#4732—24no—
40#4731—24no—
41C3×#336C2212yescomplemented
42#4551C2212yescomplemented
43#4551C2212yescomplemented
44C2×#518—31no—
45C2×#463—32no—
46C2×#450—31no—
47C2×#447—31no—
48#2620—31no—
49#1879—32no—
50#1877—32no—
51#1875—31no—
52#1873—31no—
53#1802—32no—
54#1798D311yescharacteristic, complemented
55C24×D3—21no—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C22×D12—21no—
59C22×D12—22no—
60C22×D12—24no—
61C22×D12—24no—
62C22×D12D414yes—
63C22×D12C2311yescharacteristic
64C22×D12C2312yes—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#117—24no—
71C2×C3×C4⋊C4D412yes—
72C2×C3×C22⋊C4D412yes—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77#930—22no—
78#930—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84#926—24no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89#924—22no—
90#924—22no—
91C2×Dic3⋊C4—21no—
92C2×C4×Dic3—21no—
93C2×D24—24no—
94C2×D24—24no—
95C2×D24—24no—
96C2×#204—24no—
97C2×#204—24no—
98C2×#204—24no—
99#893—216no—
100D3×C22⋊C4—216no—
101#875—24no—
102C4×D12—24no—
103C4×D12—24no—
104C4×D12—24no—
105C4×D12D418yescomplemented
106C3×C4⋊C8—24no—
107C3×C22⋊C8—22no—
108C3×C22⋊C8—22no—
109C3⋊#95—22no—
110#821—24no—
111#821—24no—
112#821—24no—
113#816—24no—
114C23×D4—31no—
115C22×SD16—32no—
116C22×D8—32no—
117C2×#124—31no—
118C2×C22⋊Q8—31no—
119C2×#121—31no—
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#463—68no—
128#450—64no—
129#447—64no—
130C2×C4⋊C8C2×D312yescomplemented
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8C2×D311yescharacteristic
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#336C2×D312yescomplemented
145C24×C3C2×D411yescharacteristic
146C23×D3—22no—
147C23×D3—24no—
148C23×D3—24no—
149C23×D3—24no—
150C23×D3—21no—
151C23×D3—82no—
152C23×D3—82no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C2×#67—42no—
162C2×#67—82no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C2×D12—24no—
168C2×D12—24no—
169C2×D12—216no—
170C2×D12—24no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—216no—
175C2×D12—22no—
176C2×D12—21no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12C2×D414yes—
180C2×D12C2×D414yes—
181C2×D12#4114yes—
182C2×C4×D3—28no—
183C2×C4×D3—28no—
184C2×C4×D3—82no—
185C2×C4×D3—82no—
186C2×C4×D3—82no—
187C2×C4×D3—82no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—24no—
199C3×C22⋊C4—24no—
200C3×C42—22no—
201C3×C42C2×D414yes—
202#217—24no—
203#217—42no—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—42no—
208(C2×D3)⋊C4—82no—
209C4⋊Dic3—22no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—82no—
215C4⋊Dic3C2×D414yes—
216Dic3⋊C4—42no—
217C4×Dic3—24no—
218D24—48no—
219#204—48no—
220C25—61no—
221C22×Q8—31no—
222C22×D4—34no—
223C22×D4—32no—
224C22×D4—31no—
225C22×D4—64no—
226C22×D4—64no—
227C22×D4—61no—
228C22×D4—61no—
229C22×D4—62no—
230C23×C4—62no—
231C23×C4—62no—
232C23×C4C22×D311yescharacteristic
233C2×SD16—64no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×D8—64no—
237C2×D8—64no—
238C2×D8—64no—
239C22×C8C22×D312yes—
240C22×C8C22×D312yes—
241#124—64no—
242C22⋊Q8—62no—
243C22⋊Q8—62no—
244#121—62no—
245#121—62no—
246C4×D4—38no—
247C4×D4—64no—
248C4×D4—64no—
249C4×D4—616no—
250C4×D4—64no—
251#117—24no—
252#117—64no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4D1212yes—
259C2×C22⋊C4—616no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4D1212yes—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270C2×C42C22×D311yescharacteristic
271C4⋊C8—24no—
272#103—64no—
273#102—64no—
274#102—64no—
275#102—64no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278#95—62no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3#12112yes—
284C23×C3C22×D411yescharacteristic
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—24no—
288C22×D3—216no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—84no—
299C22×D3—84no—
300C22×D3—88no—
301C22×D3—88no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4#12014yes—
312C2×C3×C4#12112yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D412yes—
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4112yes—
317#67—84no—
318C2×Dic3—24no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328D12—216no—
329D12—24no—
330D12—24no—
331D12—24no—
332D12—42no—
333D12—42no—
334D12#12118yes—
335C4×D3—88no—
336C4×D3—88no—
337Dic6—42no—
338Dic6—42no—
339C3×C8—42no—
340C3×C8—42no—
341C24—64no—
342C24—64no—
343C24—62no—
344C24—64no—
345C24—61no—
346C24—242no—
347C24—242no—
348C24C2×D1211yescharacteristic
349C2×Q8—62no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—62no—
353C2×D4—34no—
354C2×D4—34no—
355C2×D4—34no—
356C2×D4—616no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—616no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—61no—
364C2×D4—62no—
365C2×D4—61no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4—242no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—68no—
373C22×C4—62no—
374C22×C4—68no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—242no—
379C22×C4—242no—
380C22×C4—242no—
381C22×C4—242no—
382C22×C4C2×D1212yes—
383C22×C4C2×D1212yes—
384C22×C4C2×D1211yescharacteristic
385C22×C4D3×D412yes—
386C22×C4#23912yes—
387C22×C4C23×D311yescharacteristic
388SD16—128no—
389D8—128no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C4⋊C4—24no—
397C4⋊C4—34no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—122no—
404C4⋊C4—242no—
405C22⋊C4—24no—
406C22⋊C4—68no—
407C22⋊C4—64no—
408C22⋊C4—68no—
409C22⋊C4—122no—
410C22⋊C4—124no—
411C22⋊C4—122no—
412C22⋊C4—242no—
413C42—22no—
414C42—64no—
415C42D3×D414yes—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3C2×#12011yescharacteristic
422C22×C3C2×#12111yescharacteristic
423C22×C3#52511yescharacteristic
424C22×C3#57612yes—
425C22×C3#57712yes—
426C2×D3—28no—
427C2×D3—44no—
428C2×D3—44no—
429C2×D3—44no—
430C2×D3—44no—
431C2×D3—44no—
432C2×D3—44no—
433C2×D3—816no—
434C2×D3—81no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—81no—
438C2×D3—81no—
439C3×C4—22no—
440C3×C4—24no—
441C3×C4—42no—
442C3×C4C2×#12112yes—
443C3×C4#52512yes—
444Dic3—44no—
445Dic3—81no—
446Dic3—81no—
447Dic3—81no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—21no—
452C23—616no—
453C23—64no—
454C23—64no—
455C23—64no—
456C23—122no—
457C23—122no—
458C23—122no—
459C23—122no—
460C23—121no—
461C23—122no—
462C23—122no—
463C23—121no—
464C23—121no—
465C23—244no—
466C23—244no—
467C23—248no—
468C23—248no—
469C23#89512yes—
470C23C22×D1211yescharacteristic
471Q8—122no—
472Q8—122no—
473D4—38no—
474D4—616no—
475D4—64no—
476D4—64no—
477D4—64no—
478D4—122no—
479D4—122no—
480D4—244no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—64no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—248no—
501C2×C4—248no—
502C2×C4#88214yes—
503C2×C4#89512yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×D3×D412yes—
506C2×C4C2×#23911yescharacteristic
507C2×C4D3×#4112yes—
508C8—42no—
509C8—42no—
510C2×C3—42no—
511C2×C3#284711yescharacteristic
512C2×C3#301212yes—
513C2×C3#304111yescharacteristic
514C2×C3#304211yescharacteristic
515C2×C3#311812yes—
516D3—48no—
517D3—161no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—68no—
524C22—124no—
525C22—124no—
526C22—124no—
527C22—124no—
528C22—124no—
529C22—124no—
530C22—2416no—
531C22—242no—
532C22—241no—
533C22—242no—
534C22—241no—
535C22—241no—
536C22C2×#88211yescharacteristic
537C22C2×#89511yescharacteristic
538C22#552411yescharacteristic
539C22#575112yes—
540C22#575212yes—
541C4—22no—
542C4—24no—
543C4—42no—
544C4—124no—
545C4—241no—
546C4—241no—
547C4—241no—
548C4D3×#12112yes—
549C4#562712yes—
550C3#3291711yescharacteristic, complemented, Sylow
551C2—42no—
552C2—128no—
553C2—481no—
554C2#7404811yescharacteristic
555C2#7867212yes—
556C2#7908611yescharacteristic
557C2#7908711yescharacteristic
558C2#7938312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1798C2
socleC23×C3C22×D4
prime radicalC2×#4732C2
hypercenter#1798D3
hypocenterC3#32917
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
1. C2×#5521C2semidirect
2. #82287C2semidirect
3. C2×#4844C2semidirect
4. C2×#4731C2semidirect
6. C3×#1798C2semidirect
7. #70479C2semidirect
8. #69933C2semidirect
9. #69931C2semidirect
10. #68673C2semidirect
11. #68672C2semidirect
23. C2×#875C22semidirect
26. C2×C3×C4⋊C8C22semidirect
31. C2×#821C22semidirect
32. C2×#816C22semidirect
35. #4939C22semidirect
36. #4935C22semidirect
37. #4929C22semidirect
41. C3×#336C22semidirect
42. #4551C22semidirect
43. #4551C22semidirect
54. #1798D3semidirect
105. C4×D12D4semidirect
130. C2×C4⋊C8C2×D3semidirect
144. #336C2×D3semidirect
550. C3#32917semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
#822872
C2×#48442
C2×#47312
C2×#47322
C3×#17982
#704792
#699332
#699312
#686732
#686722
#329173