Finite Group Databasebeta

← #10822433

Subgroups — #10822433

Order 768.

2258 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85735C211yesmaximal, characteristic, complemented
2C2×#5524C212yesmaximal, complemented
3#82267C211yesmaximal, characteristic, complemented
4C2×#5006C212yesmaximal, complemented
5C3×#1740C211yesmaximal, characteristic, complemented
6#68966C214yesmaximal, complemented
7#68965C214yesmaximal, complemented
8#23443—31nomaximal, Sylow
9C22×#235C2212yes—
10C23×C4×D3C2211yescharacteristic
11C2×C3×#117C2212yes—
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14#5526—22no—
15#5525—22no—
16#5524—28no—
17#5524—24no—
18#5524—28no—
19#5510—22no—
20C2×#897—22no—
21C2×#895—22no—
22#5493C2214yes—
23C2×#886C2214yes—
24C2×#882C2214yes—
25#5477—22no—
26C2×C4×D12—24no—
27#5006—28no—
28#4987—22no—
29#4955C2212yes—
30C2×#806C2218yes—
31#4943C2212yes—
32C4×(C2×D3)⋊C4—22no—
33C3×#342C2214yes—
34#3449—31no—
35C2×#525—32no—
36C2×#410—32no—
37#2280—31no—
38#2022—34no—
39#2021—34no—
40#1740D311yescharacteristic, complemented
41C24×D3C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67C2314yes—
44C23×Dic3C2311yescharacteristic
45C2×#235—22no—
46C2×#235—22no—
47C2×#235—24no—
48C2×#235—24no—
49C2×#235—22no—
50C2×#235—24no—
51C2×#235—22no—
52C2×#235—24no—
53C22×D12—24no—
54C22×D12D412yescomplemented
55C22×C4×D3—21no—
56C22×C4×D3—22no—
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C22×C4×D3—24no—
60C22×C4×D3—22no—
61C22×C4×D3D414yescomplemented
62C22×C4×D3D414yescomplemented
63C22×Dic6D412yescomplemented
64C3×#117—24no—
65C3×#117—22no—
66C3×#117—22no—
67C3×#117D418yescomplemented
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4D412yescomplemented
70C2×C3×C22⋊C4C2312yes—
71C2×C3×C22⋊C4C2314yes—
72C2×C3×C42—22no—
73C2×C3×C42D412yescomplemented
74C2×#217C2312yes—
75#930—28no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79#926—24no—
80C2×C4⋊Dic3—24no—
81C2×C4⋊Dic3—21no—
82C2×C4×Dic3—21no—
83#897—216no—
84#897—28no—
85#895—216no—
86#895—28no—
87#886—28no—
88#886—28no—
89#886—28no—
90#882—28no—
91#882—28no—
92#882—28no—
93#875—24no—
94#874—22no—
95C4×D12—232no—
96C4×D12—28no—
97#869—22no—
98C3⋊#95—24no—
99#806—216no—
100C22×#41—32no—
101C24×C4—31no—
102#537—62no—
103#536—62no—
104#535—62no—
105#534—34no—
106#532—62no—
107#525—68no—
108#525—64no—
109#525—68no—
110C2×#123—34no—
111C2×C22⋊Q8—62no—
112C2×#121—62no—
113C2×#120—34no—
114C2×C4×D4—64no—
115C2×#117—31no—
116C2×#117C2×D312yes—
117C2×#117C2×D311yescharacteristic
118#410—68no—
119C2×#99—38no—
120#381—32no—
121#380—62no—
122#379—32no—
123C4×C22⋊C4—62no—
124#342C2×D314yes—
125C24×C3C2411yescharacteristic
126C23×D3—21no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—21no—
130C23×D3—44no—
131C23×D3C2×D412yes—
132C23×D3C2×D414yes—
133C22×C3×C4—24no—
134C22×C3×C4C2×D414yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D414yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4#4112yes—
142C2×#67—24no—
143C2×#67—24no—
144C2×#67—24no—
145C2×#67—24no—
146C2×#67—24no—
147C2×#67—24no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—21no—
152C22×Dic3—22no—
153C22×Dic3C2×D412yes—
154C22×Dic3C2×D414yes—
155#235—44no—
156#235—44no—
157#235—44no—
158#235—44no—
159#235—44no—
160#235—44no—
161#235—44no—
162#235—44no—
163C2×D12—216no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—24no—
167C2×D12—24no—
168C2×D12—28no—
169C2×D12—416no—
170C2×C4×D3—28no—
171C2×C4×D3—28no—
172C2×C4×D3—24no—
173C2×C4×D3—28no—
174C2×C4×D3—24no—
175C2×C4×D3—24no—
176C2×C4×D3—24no—
177C2×C4×D3—28no—
178C2×C4×D3—41no—
179C2×C4×D3—42no—
180C2×C4×D3—41no—
181C2×C4×D3—44no—
182C2×C4×D3—44no—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—24no—
188C2×Dic6—24no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4C2×D418yes—
194C3×C42—22no—
195C3×C42—28no—
196C3×C42—22no—
197#217—24no—
198(C2×D3)⋊C4—216no—
199(C2×D3)⋊C4—216no—
200(C2×D3)⋊C4—216no—
201(C2×D3)⋊C4—48no—
202C4⋊Dic3—28no—
203C4⋊Dic3—216no—
204C4⋊Dic3—42no—
205C4×Dic3—42no—
206C25—31no—
207C2×#41—64no—
208C2×#41—62no—
209C2×#41—62no—
210C2×#41—64no—
211C2×#41—64no—
212C2×#41—62no—
213C2×#41—62no—
214C2×#41—64no—
215C22×Q8—32no—
216C22×D4—34no—
217C22×D4—32no—
218C22×D4—64no—
219C23×C4—31no—
220C23×C4—34no—
221C23×C4—34no—
222C23×C4—64no—
223C23×C4—61no—
224C23×C4—61no—
225C23×C4—62no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4C22×D311yescharacteristic
229C4⋊Q8—62no—
230#127—62no—
231#124—64no—
232#123—68no—
233#123—68no—
234#123—68no—
235C22⋊Q8—616no—
236C22⋊Q8—68no—
237#121—616no—
238#121—68no—
239#121—68no—
240#120—68no—
241#120—68no—
242#120—68no—
243C4×D4—632no—
244C4×D4—68no—
245#117—24no—
246#117—22no—
247#117—22no—
248#117—64no—
249#117D1218yescomplemented
250C2×C4⋊C4—22no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4D1212yescomplemented
254C2×C22⋊C4—32no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4C22×D312yes—
259C2×C22⋊C4C22×D314yes—
260C2×C42—22no—
261C2×C42—61no—
262C2×C42D1212yescomplemented
263#99—616no—
264#95—64no—
265C23×C3—42no—
266C23×C3C22×D411yescharacteristic
267C23×C3C22×D412yes—
268C23×C3C22×D412yes—
269C23×C3C22×D411yescharacteristic
270C23×C3C2×#4111yescharacteristic
271C22×D3—24no—
272C22×D3—28no—
273C22×D3—24no—
274C22×D3—22no—
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—24no—
278C22×D3—44no—
279C22×D3—48no—
280C22×D3—44no—
281C22×D3—42no—
282C22×D3—44no—
283C22×D3—41no—
284C22×D3—42no—
285C22×D3—41no—
286C22×D3—48no—
287C2×C3×C4—24no—
288C2×C3×C4—28no—
289C2×C3×C4—48no—
290C2×C3×C4—44no—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12014yes—
293C2×C3×C4#12012yes—
294C2×C3×C4#12014yes—
295C2×C3×C4#12011yescharacteristic
296C2×C3×C4#12114yes—
297C2×C3×C4#12114yes—
298C2×C3×C4#12112yes—
299C2×C3×C4#12112yes—
300C2×C3×C4#12712yes—
301C2×C3×C4#12712yes—
302#67—44no—
303#67—44no—
304#67—44no—
305#67—44no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—22no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314C2×Dic3—44no—
315C2×Dic3—42no—
316D12—432no—
317D12—44no—
318D12—44no—
319D12—88no—
320C4×D3—44no—
321C4×D3—48no—
322C4×D3—44no—
323C4×D3—48no—
324C4×D3—48no—
325Dic6—44no—
326Dic6—44no—
327C24—34no—
328C24—32no—
329C24—61no—
330C24—62no—
331C24—62no—
332C24—61no—
333C24—124no—
334C24C23×D311yescharacteristic
335#41—124no—
336#41—124no—
337#41—124no—
338#41—124no—
339#41—124no—
340#41—124no—
341#41—124no—
342#41—124no—
343C2×Q8—62no—
344C2×Q8—64no—
345C2×Q8—62no—
346C2×Q8—64no—
347C2×D4—616no—
348C2×D4—68no—
349C2×D4—64no—
350C2×D4—64no—
351C2×D4—64no—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—62no—
356C2×D4—64no—
357C2×D4—62no—
358C2×D4—64no—
359C2×D4—1216no—
360C22×C4—24no—
361C22×C4—34no—
362C22×C4—32no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—68no—
368C22×C4—68no—
369C22×C4—68no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—68no—
375C22×C4—62no—
376C22×C4—122no—
377C22×C4—122no—
378C22×C4—121no—
379C22×C4—122no—
380C22×C4—121no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4C2×D1214yes—
384C22×C4C2×D1211yescharacteristic
385C22×C4C2×D1214yes—
386C22×C4C2×D1211yescharacteristic
387C22×C4D3×D412yes—
388C22×C4D3×D412yes—
389C22×C4D3×D412yes—
390C22×C4#23912yes—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—616no—
394C4⋊C4—68no—
395C4⋊C4—122no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—616no—
399C22⋊C4—616no—
400C22⋊C4—616no—
401C22⋊C4—64no—
402C22⋊C4—128no—
403C22⋊C4C2×D1218yes—
404C42—28no—
405C42—22no—
406C42—22no—
407C42—122no—
408C22×C3—22no—
409C22×C3—24no—
410C22×C3—42no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3C2×#12012yes—
414C22×C3C2×#12011yescharacteristic
415C22×C3C2×#12112yes—
416C22×C3C2×#12111yescharacteristic
417C22×C3C2×#12711yescharacteristic
418C2×D3—28no—
419C2×D3—24no—
420C2×D3—44no—
421C2×D3—44no—
422C2×D3—42no—
423C2×D3—48no—
424C2×D3—42no—
425C2×D3—44no—
426C2×D3—48no—
427C2×D3—44no—
428C2×D3—48no—
429C2×D3—84no—
430C3×C4—48no—
431C3×C4—44no—
432C3×C4#39212yes—
433C3×C4#39214yes—
434C3×C4#39212yes—
435Dic3—44no—
436Dic3—42no—
437Dic3—82no—
438C23—42no—
439C23—64no—
440C23—68no—
441C23—64no—
442C23—62no—
443C23—64no—
444C23—64no—
445C23—64no—
446C23—122no—
447C23—124no—
448C23—128no—
449C23—124no—
450C23—128no—
451C23—124no—
452C23—121no—
453C23—121no—
454C23—122no—
455C23C22×D1211yescharacteristic
456C23C22×D1212yes—
457C23C2×D3×D412yes—
458C23C2×D3×D411yescharacteristic
459C23C2×#23911yescharacteristic
460Q8—124no—
461Q8—124no—
462D4—1232no—
463D4—124no—
464D4—124no—
465D4—124no—
466D4—124no—
467D4—124no—
468D4—124no—
469D4—248no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—64no—
475C2×C4—64no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—64no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4—124no—
483C2×C4—128no—
484C2×C4—124no—
485C2×C4—128no—
486C2×C4—128no—
487C2×C4—124no—
488C2×C4—122no—
489C2×C4#87412yes—
490C2×C4#87412yes—
491C2×C4#88214yes—
492C2×C4#88211yescharacteristic
493C2×C4#88212yes—
494C2×C4#88214yes—
495C2×C4#88211yescharacteristic
496C2×C4#89514yes—
497C2×C4#89512yes—
498C2×C4#89512yes—
499C2×C4#89514yes—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3C2×#39211yescharacteristic
505C2×C3#302512yes—
506D3—44no—
507D3—84no—
508C22—22no—
509C22—24no—
510C22—42no—
511C22—41no—
512C22—41no—
513C22—68no—
514C22—64no—
515C22—124no—
516C22—124no—
517C22—122no—
518C22—128no—
519C22—128no—
520C22—124no—
521C22—122no—
522C22—128no—
523C22—124no—
524C22—244no—
525C22C2×#87411yescharacteristic
526C22C2×#88212yes—
527C22C2×#88211yescharacteristic
528C22C2×#89512yes—
529C22C2×#89511yescharacteristic
530C4—48no—
531C4—44no—
532C4—122no—
533C4—124no—
534C4—242no—
535C4#467112yes—
536C4#467114yes—
537C4#467112yes—
538C3#2344311yescharacteristic, complemented, Sylow
539C2—22no—
540C2—21no—
541C2—21no—
542C2—22no—
543C2—124no—
544C2—244no—
545C2C2×#467111yescharacteristic
546C2#7869012yes—
subgroupG/H
centerC2×C4#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#1740C2
socleC22×C3C2×#120
hypercenter#1740D3
hypocenterC3#23443
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
1. #85735C2semidirect
2. C2×#5524C2semidirect
3. #82267C2semidirect
4. C2×#5006C2semidirect
5. C3×#1740C2semidirect
6. #68966C2semidirect
7. #68965C2semidirect
40. #1740D3semidirect
54. C22×D12D4semidirect
61. C22×C4×D3D4semidirect
62. C22×C4×D3D4semidirect
63. C22×Dic6D4semidirect
67. C3×#117D4semidirect
69. C2×C3×C4⋊C4D4semidirect
73. C2×C3×C42D4semidirect
249. #117D12semidirect
253. C2×C4⋊C4D12semidirect
262. C2×C42D12semidirect
538. C3#23443semidirect

Maximal subgroups

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

subgroupindex
#857352
C2×#55242
#822672
C2×#50062
C3×#17402
#689662
#689652
#234433