Finite Group Databasebeta

← #10796821

Subgroups — #10796821

Order 768.

2126 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic
2C2×#5798C211yesmaximal, characteristic, complemented
3#84043C212yesmaximal, complemented
4C22×#834C211yesmaximal, characteristic, complemented
5#71704C214yesmaximal, complemented
6#71669C214yesmaximal, complemented
7C2×#4527C212yesmaximal, complemented
8#20960—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×C3×D4C2211yescharacteristic
11C3×#534—24no—
12C3×#534C418yescomplemented
13C2×C3×#123—21no—
14C2×C3×#123C2212yes—
15C2×C3×#121—22no—
16C2×C3×#120—21no—
17C2×C3×#120C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×#217—21no—
20C22×#217C2212yes—
21C22×#217C2212yes—
22#5798—22no—
23#5798—24no—
24#5798C2214yes—
25C2×D4×Dic3—24no—
26C2×#926—21no—
27#5247C2214yescomplemented
28#5220—24no—
29#5220—24no—
30C2×#834—24no—
31C2×#834C2214yes—
32#5211C2214yescomplemented
33#4527C2218yescomplemented
34C2×#534D311yescharacteristic
35C2×#518—31no—
36C22×#99—31no—
37#2308—32no—
38#1890—34no—
39#1889—34no—
40C2×#342—32no—
41C25×C3D411yescharacteristic
42C25×C3C2311yescharacteristic
43C25×C3C2311yescharacteristic
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—24no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4—24no—
53C22×C3×D4—22no—
54C22×C3×D4C2×C414yes—
55C22×C3×D4C2×C414yescomplemented
56C22×C3×D4D414yes—
57C22×C3×D4D412yes—
58C22×C3×D4C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—22no—
63C23×Dic3—22no—
64C23×Dic3D414yescomplemented
65C3×#123—24no—
66C3×#123—28no—
67C3×#123C2×C418yescomplemented
68C3×#121—216no—
69C3×#120—28no—
70C3×#120—24no—
71C3×#120C2×C418yescomplemented
72C2×C3×C4⋊C4—22no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4C2×C414yescomplemented
78C2×C3×C22⋊C4C2312yes—
79C2×#217—22no—
80C2×#217—24no—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—22no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—24no—
88C2×#217—22no—
89C2×#217—28no—
90C2×#217D414yescomplemented
91C2×#217C2314yes—
92C2×#217C2314yes—
93D4×Dic3—216no—
94D4×Dic3—216no—
95#926—24no—
96#926—24no—
97C2×C4⋊Dic3—22no—
98C2×C4×Dic3—22no—
99#834—28no—
100C3⋊#95—28no—
101C23×D4C2×D311yescharacteristic
102C23×D4C2×D311yescharacteristic
103#534—24no—
104#534Dic318yescomplemented
105C2×#123—21no—
106C2×#123C2×D312yes—
107C2×#121—22no—
108C2×#120—21no—
109C2×#120C2×D312yes—
110#518—34no—
111#518—64no—
112#518—62no—
113C2×C4×D4—64no—
114C2×#117—61no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4—61no—
118C22×C22⋊C4C2×D311yescharacteristic
119C2×#99—34no—
120C2×#99—64no—
121#386—64no—
122#386—64no—
123#381—34no—
124#379—34no—
125#342—38no—
126C24×C3—21no—
127C24×C3—21no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—21no—
131C24×C3—21no—
132C24×C3—22no—
133C24×C3—82no—
134C24×C3—82no—
135C24×C3—82no—
136C24×C3C22⋊C414yescomplemented
137C24×C3C22×C412yes—
138C24×C3C22×C412yescomplemented
139C24×C3C2×D412yes—
140C24×C3C2×D412yes—
141C24×C3C2×D412yes—
142C24×C3C2×D411yescharacteristic
143C24×C3C2×D411yescharacteristic
144C24×C3C2×D412yes—
145C24×C3#4114yes—
146C24×C3C2411yescharacteristic
147C2×C3×D4—28no—
148C2×C3×D4—28no—
149C2×C3×D4—24no—
150C2×C3×D4—28no—
151C2×C3×D4—28no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—28no—
158C2×C3×D4—28no—
159C2×C3×D4—28no—
160C2×C3×D4—41no—
161C2×C3×D4—42no—
162C2×C3×D4—41no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—84no—
168C2×C3×D4C22⋊C418yescomplemented
169C2×C3×D4C22×C414yes—
170C2×C3×D4C2×D414yes—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—82no—
181C22×C3×C4C22⋊C414yescomplemented
182C22×C3×C4C22×C412yescomplemented
183C22×C3×C4C2×D412yes—
184C22×Dic3—24no—
185C22×Dic3—28no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—44no—
189C22×Dic3—84no—
190C22×Dic3—84no—
191C22×Dic3—84no—
192C22×Dic3C2×D418yes—
193C3×C4⋊C4—28no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—28no—
198C3×C22⋊C4C22×C414yescomplemented
199#217—28no—
200#217—24no—
201#217—24no—
202#217—24no—
203#217—24no—
204#217—24no—
205#217—24no—
206#217—44no—
207#217—88no—
208C4⋊Dic3—28no—
209C4×Dic3—28no—
210C25#6711yescharacteristic
211C25C22×D311yescharacteristic
212C25C22×D311yescharacteristic
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4C2×Dic314yes—
224C22×D4C2×Dic314yescomplemented
225C22×D4#6714yes—
226C22×D4#6712yes—
227C22×D4C22×D312yes—
228C23×C4—21no—
229C23×C4—34no—
230C23×C4—62no—
231C23×C4—62no—
232C23×C4#6711yescharacteristic
233C23×C4C22×D311yescharacteristic
234#123—24no—
235#123—28no—
236#123C2×Dic318yescomplemented
237#121—216no—
238#120—28no—
239#120—24no—
240#120C2×Dic318yescomplemented
241C4×D4—616no—
242C4×D4—616no—
243#117—64no—
244#117—64no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—62no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—34no—
252C2×C22⋊C4—34no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—68no—
265C2×C22⋊C4C2×Dic314yescomplemented
266C2×C22⋊C4C22×D312yes—
267C2×C42—62no—
268#99—68no—
269#95—68no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—22no—
275C23×C3—24no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3—48no—
284C23×C3—44no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—82no—
289C23×C3—82no—
290C23×C3—82no—
291C23×C3—82no—
292C23×C3—84no—
293C23×C3—84no—
294C23×C3—84no—
295C23×C3—82no—
296C23×C3—84no—
297C23×C3C2×C22⋊C414yes—
298C23×C3C2×C22⋊C412yes—
299C23×C3C2×C22⋊C412yes—
300C23×C3C4×D414yes—
301C23×C3C4×D414yes—
302C23×C3#12011yescharacteristic
303C23×C3#12012yes—
304C23×C3#12011yescharacteristic
305C23×C3#12114yes—
306C23×C3#12114yes—
307C23×C3#12312yes—
308C23×C3#12312yes—
309C23×C3C23×C411yescharacteristic
310C23×C3C22×D412yes—
311C23×C3C22×D411yescharacteristic
312C23×C3C22×D411yescharacteristic
313C23×C3C2×#4112yes—
314C3×D4—28no—
315C3×D4—28no—
316C3×D4—216no—
317C3×D4—44no—
318C3×D4—44no—
319C3×D4—48no—
320C3×D4—88no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—44no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—84no—
333C2×C3×C4C2×C22⋊C414yes—
334C2×Dic3—28no—
335C2×Dic3—48no—
336C2×Dic3—48no—
337C2×Dic3—48no—
338C2×Dic3—44no—
339C2×Dic3—84no—
340C2×Dic3—88no—
341C2×Dic3—88no—
342C24—21no—
343C24—22no—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—21no—
349C24—82no—
350C24—82no—
351C24—82no—
352C24#21714yescomplemented
353C24D3×D412yes—
354C24D3×D412yes—
355C24#23714yes—
356C24C22×Dic312yes—
357C24C22×Dic312yescomplemented
358C24C2×#6712yes—
359C24C2×#6711yescharacteristic
360C24C2×#6711yescharacteristic
361C24C2×#6712yes—
362C24C23×D311yescharacteristic
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—42no—
380C2×D4—41no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—84no—
384C2×D4#21718yescomplemented
385C2×D4C22×Dic314yes—
386C2×D4C2×#6714yes—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—38no—
397C22×C4—64no—
398C22×C4—68no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—82no—
402C22×C4—124no—
403C22×C4—244no—
404C22×C4—244no—
405C22×C4—244no—
406C22×C4#21714yescomplemented
407C22×C4C22×Dic312yescomplemented
408C22×C4C2×#6712yes—
409C4⋊C4—28no—
410C4⋊C4—68no—
411C22⋊C4—24no—
412C22⋊C4—28no—
413C22⋊C4—22no—
414C22⋊C4—28no—
415C22⋊C4—68no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—124no—
423C22⋊C4—248no—
424C22⋊C4C22×Dic314yescomplemented
425C42—68no—
426C22×C3—22no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—42no—
443C22×C3—44no—
444C22×C3—48no—
445C22×C3—44no—
446C22×C3—48no—
447C22×C3—44no—
448C22×C3—42no—
449C22×C3—82no—
450C22×C3—84no—
451C22×C3—84no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3#38612yes—
455C22×C3#38612yes—
456C22×C3#38614yes—
457C22×C3C22×C22⋊C411yescharacteristic
458C22×C3C2×C4×D412yes—
459C22×C3C2×#12011yescharacteristic
460C22×C3C2×#12112yes—
461C22×C3C2×#12311yescharacteristic
462C3×C4—24no—
463C3×C4—44no—
464C3×C4—42no—
465Dic3—48no—
466Dic3—84no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—48no—
481C23—44no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—82no—
486C23—84no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—82no—
494C23D4×Dic314yes—
495C23D4×Dic314yes—
496C23#93512yes—
497C23#93512yes—
498C23#93711yescharacteristic
499C23#93712yes—
500C23#93711yescharacteristic
501C23#93914yes—
502C23#93914yes—
503C23C2×#21714yes—
504C23C2×#21712yes—
505C23C2×#21712yes—
506C23C2×D3×D412yes—
507C23C2×#23712yes—
508C23C23×Dic311yescharacteristic
509C23C22×#6711yescharacteristic
510C23C22×#6711yescharacteristic
511D4—216no—
512D4—28no—
513D4—28no—
514D4—44no—
515D4—44no—
516D4—48no—
517D4—88no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—68no—
530C2×C4—84no—
531C2×C4—128no—
532C2×C4—128no—
533C2×C4—128no—
534C2×C4—124no—
535C2×C4—244no—
536C2×C4—248no—
537C2×C4—248no—
538C2×C4C2×#21714yes—
539C2×C3—22no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—22no—
545C2×C3—24no—
546C2×C3—44no—
547C2×C3—82no—
548C2×C3C2×#38611yescharacteristic
549C2×C3#302612yes—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—22no—
555C22—24no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—48no—
569C22—44no—
570C22—48no—
571C22—44no—
572C22—42no—
573C22—82no—
574C22—84no—
575C22—84no—
576C22—82no—
577C22—82no—
578C22#522012yes—
579C22#522012yes—
580C22#522014yes—
581C22C2×D4×Dic312yes—
582C22C2×#93511yescharacteristic
583C22C2×#93711yescharacteristic
584C22C2×#93912yes—
585C22C22×#21711yescharacteristic
586C4—24no—
587C4—44no—
588C4—42no—
589C4—128no—
590C4—244no—
591C3#2096011yescharacteristic, complemented, Sylow
592C2—21no—
593C2—22no—
594C2—22no—
595C2—22no—
596C2—21no—
597C2—22no—
598C2—24no—
599C2—44no—
600C2—82no—
601C2#8337512yes—
602C2C2×#522011yescharacteristic
subgroupG/H
centerC22C2×#937
commutatorC23×C3C23×C4
FrattiniC24C23×D3
Fitting, prime radicalC2×C3×#534C2
socleC22×C3C2×#120
cosocleC24×C3C24
hypercenterC2×#534D3
hypocenterC3#20960
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
2. C2×#5798C2semidirect
3. #84043C2semidirect
4. C22×#834C2semidirect
5. #71704C2semidirect
6. #71669C2semidirect
7. C2×#4527C2semidirect
12. C3×#534C4semidirect
27. #5247C22semidirect
32. #5211C22semidirect
33. #4527C22semidirect
55. C22×C3×D4C2×C4semidirect
64. C23×Dic3D4semidirect
67. C3×#123C2×C4semidirect
71. C3×#120C2×C4semidirect
77. C2×C3×C22⋊C4C2×C4semidirect
90. C2×#217D4semidirect
104. #534Dic3semidirect
136. C24×C3C22⋊C4semidirect
138. C24×C3C22×C4semidirect
168. C2×C3×D4C22⋊C4semidirect
181. C22×C3×C4C22⋊C4semidirect
182. C22×C3×C4C22×C4semidirect
198. C3×C22⋊C4C22×C4semidirect
224. C22×D4C2×Dic3semidirect
236. #123C2×Dic3semidirect
240. #120C2×Dic3semidirect
265. C2×C22⋊C4C2×Dic3semidirect
352. C24#217semidirect
357. C24C22×Dic3semidirect
384. C2×D4#217semidirect
406. C22×C4#217semidirect
407. C22×C4C22×Dic3semidirect
424. C22⋊C4C22×Dic3semidirect
591. C3#20960semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C2×#57982
#840432
C22×#8342
#717042
#716692
C2×#45272
#209603