Finite Group Databasebeta

← #10796817

Subgroups — #10796817

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×#5804C211yesmaximal, characteristic
2C2×#5798C211yesmaximal, characteristic, complemented
3C22×C3×#99C211yesmaximal, characteristic, complemented
4#82121C212yesmaximal, complemented
5C2×#4527C212yesmaximal, complemented
6#70023C214yesmaximal, complemented
7#70022C214yesmaximal, complemented
8#20960—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C2×#953—21no—
13C22×#217—21no—
14C22×#217C2212yes—
15C22×#217C2211yescharacteristic
16#5804—24no—
17#5804C418yescomplemented
18C2×#939—22no—
19C2×#937C2212yes—
20#5798—22no—
21#5798—24no—
22#5798C2214yes—
23C2×#935C2212yes—
24C2×D4×Dic3—24no—
25C2×#929—21no—
26C2×#926—21no—
27C2×C3×#99—24no—
28C2×C3×#99C2214yes—
29#5213—24no—
30#4958—24no—
31#4955C2214yescomplemented
32#4943C2214yescomplemented
33#4527C2218yescomplemented
34C2×#534—31no—
35C2×#518—31no—
36C22×#99D311yescharacteristic, complemented
37#2308—32no—
38#1890—34no—
39#1889—34no—
40C2×#342—32no—
41C25×C3D411yescharacteristic
42C25×C3C2311yescharacteristic
43C24×D3C2311yescharacteristic
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—24no—
47C22×C3×D4C2×C414yes—
48C22×C3×D4D412yes—
49C22×C3×D4D414yes—
50C22×C3×D4C2312yes—
51C23×C3×C4—22no—
52C23×C3×C4D411yescharacteristic
53C22×#67—22no—
54C22×#67—22no—
55C22×#67—21no—
56C22×#67—21no—
57C22×#67—22no—
58C22×#67—24no—
59C22×#67—22no—
60C22×#67C2×C414yescomplemented
61C23×Dic3—22no—
62C23×Dic3—21no—
63C23×Dic3D414yescomplemented
64C23×Dic3C2311yescharacteristic
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4C2314yes—
69#953—28no—
70C2×#217—22no—
71C2×#217—24no—
72C2×#217—22no—
73C2×#217—21no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—24no—
79C2×#217—22no—
80C2×#217C2×C414yescomplemented
81C2×#217D414yescomplemented
82C2×#217C2314yes—
83C2×#217C2312yes—
84#939—216no—
85#937—24no—
86#937C2×C418yescomplemented
87#935—24no—
88#935C2×C418yescomplemented
89D4×Dic3—216no—
90D4×Dic3—216no—
91#929—28no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—28no—
94#926—24no—
95#926—24no—
96C2×C4⋊Dic3—22no—
97C2×Dic3⋊C4—22no—
98C2×C4×Dic3—22no—
99C3×#99—28no—
100C3⋊#95—28no—
101C23×D4—31no—
102C23×D4C2×D311yescharacteristic
103#534—38no—
104#534—64no—
105C2×#123—32no—
106C2×#123—61no—
107C2×#121—62no—
108C2×#120—32no—
109C2×#120—61no—
110#518—34no—
111#518—64no—
112#518—62no—
113C2×C4×D4—64no—
114C2×#117—61no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4—61no—
118C22×C22⋊C4C2×D312yes—
119C2×#99—24no—
120C2×#99C2×D314yes—
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—82no—
130C24×C3—82no—
131C24×C3C22⋊C414yescomplemented
132C24×C3C22×C412yes—
133C24×C3C2×D412yes—
134C24×C3C2×D411yescharacteristic
135C24×C3C2×D411yescharacteristic
136C24×C3C2×D412yes—
137C24×C3C2×D412yes—
138C24×C3C2×D412yes—
139C24×C3#4114yes—
140C24×C3C2411yescharacteristic
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—22no—
145C23×D3—82no—
146C23×D3C22×C412yescomplemented
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×D4C22⋊C418yescomplemented
156C2×C3×D4C22×C414yes—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—44no—
163C22×C3×C4—84no—
164C22×C3×C4C22⋊C414yescomplemented
165C22×C3×C4C2×D412yes—
166C2×#67—24no—
167C2×#67—24no—
168C2×#67—28no—
169C2×#67—28no—
170C2×#67—28no—
171C2×#67—41no—
172C2×#67—42no—
173C2×#67—41no—
174C2×#67—44no—
175C2×#67—44no—
176C2×#67—42no—
177C2×#67—42no—
178C2×#67—84no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—28no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—84no—
189C22×Dic3—84no—
190C22×Dic3—82no—
191C22×Dic3C22×C412yescomplemented
192C22×Dic3C2×D418yes—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—44no—
196#217—28no—
197#217—24no—
198#217—28no—
199#217—24no—
200#217—24no—
201#217—24no—
202#217—24no—
203#217—22no—
204#217C22×C414yescomplemented
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—88no—
207C4⋊Dic3—28no—
208Dic3⋊C4—28no—
209C4×Dic3—28no—
210C25—31no—
211C25D1211yescharacteristic
212C25C22×D311yescharacteristic
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—34no—
217C22×D4—61no—
218C22×D4—61no—
219C22×D4—62no—
220C22×D4—62no—
221C22×D4—64no—
222C22×D4—62no—
223C22×D4—62no—
224C22×D4C4×D314yes—
225C22×D4D1212yes—
226C22×D4#6714yes—
227C22×D4C22×D312yes—
228C23×C4—22no—
229C23×C4—31no—
230C23×C4—34no—
231C23×C4—62no—
232C23×C4—61no—
233C23×C4D1211yescharacteristic
234#123—38no—
235#123—64no—
236#123—68no—
237#121—616no—
238#120—38no—
239#120—68no—
240#120—64no—
241C4×D4—616no—
242C4×D4—616no—
243#117—64no—
244#117—64no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—34no—
251C2×C22⋊C4—32no—
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—61no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—68no—
266C2×C22⋊C4C22×D314yes—
267C2×C42—62no—
268#99—28no—
269#95—68no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—42no—
275C23×C3—44no—
276C23×C3—44no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3—82no—
282C23×C3—82no—
283C23×C3—82no—
284C23×C3—84no—
285C23×C3—84no—
286C23×C3—84no—
287C23×C3C2×C22⋊C414yes—
288C23×C3C2×C22⋊C412yes—
289C23×C3C2×C22⋊C412yes—
290C23×C3C4×D414yes—
291C23×C3C4×D414yes—
292C23×C3#12011yescharacteristic
293C23×C3#12012yes—
294C23×C3#12011yescharacteristic
295C23×C3#12114yes—
296C23×C3#12114yes—
297C23×C3#12312yes—
298C23×C3#12312yes—
299C23×C3C23×C411yescharacteristic
300C23×C3C22×D411yescharacteristic
301C23×C3C22×D412yes—
302C23×C3C22×D411yescharacteristic
303C23×C3C2×#4112yes—
304C22×D3—22no—
305C22×D3—24no—
306C22×D3—48no—
307C22×D3—44no—
308C22×D3—41no—
309C22×D3—42no—
310C22×D3—41no—
311C22×D3—82no—
312C22×D3—82no—
313C22×D3—84no—
314C3×D4—28no—
315C3×D4—28no—
316C3×D4—216no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—48no—
322C2×C3×C4—44no—
323C2×C3×C4—88no—
324C2×C3×C4C2×C22⋊C414yes—
325#67—44no—
326#67—44no—
327#67—48no—
328#67—88no—
329C2×Dic3—22no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—28no—
333C2×Dic3—48no—
334C2×Dic3—44no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—48no—
339C2×Dic3—84no—
340C2×Dic3—84no—
341C2×Dic3—88no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—32no—
346C24—61no—
347C24—62no—
348C24—62no—
349C24—61no—
350C24—82no—
351C24—82no—
352C24—242no—
353C24(C2×D3)⋊C414yescomplemented
354C24C2×C4×D312yes—
355C24C2×D1212yes—
356C24C2×D1211yescharacteristic
357C24C2×D1211yescharacteristic
358C24D3×D412yes—
359C24D3×D412yes—
360C24#23714yes—
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—68no—
372C2×D4—68no—
373C2×D4—64no—
374C2×D4—64no—
375C2×D4—68no—
376C2×D4—121no—
377C2×D4—122no—
378C2×D4—121no—
379C2×D4—122no—
380C2×D4—122no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—244no—
384C2×D4(C2×D3)⋊C418yescomplemented
385C2×D4C2×C4×D314yes—
386C2×D4C2×#6714yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—32no—
392C22×C4—38no—
393C22×C4—44no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—68no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—84no—
404C22×C4—244no—
405C22×C4—244no—
406C22×C4—242no—
407C22×C4(C2×D3)⋊C414yescomplemented
408C22×C4C2×D1212yes—
409C4⋊C4—68no—
410C4⋊C4—68no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—34no—
414C22⋊C4—44no—
415C22⋊C4—68no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—68no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—62no—
423C22⋊C4—68no—
424C22⋊C4—248no—
425C42—68no—
426C22×C3—22no—
427C22×C3—24no—
428C22×C3—22no—
429C22×C3—24no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—42no—
433C22×C3—44no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—42no—
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—82no—
447C22×C3—84no—
448C22×C3#38612yes—
449C22×C3#38612yes—
450C22×C3#38614yes—
451C22×C3C22×C22⋊C411yescharacteristic
452C22×C3C2×C4×D412yes—
453C22×C3C2×#12011yescharacteristic
454C22×C3C2×#12112yes—
455C22×C3C2×#12311yescharacteristic
456C2×D3—44no—
457C2×D3—48no—
458C2×D3—42no—
459C2×D3—84no—
460C2×D3—82no—
461C2×D3—82no—
462C3×C4—24no—
463C3×C4—84no—
464Dic3—44no—
465Dic3—48no—
466Dic3—42no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—64no—
479C23—62no—
480C23—82no—
481C23—82no—
482C23—84no—
483C23—82no—
484C23—84no—
485C23—84no—
486C23—128no—
487C23—124no—
488C23—121no—
489C23—121no—
490C23—122no—
491C23—242no—
492C23—244no—
493C23—242no—
494C23#67⋊C414yes—
495C23#67⋊C414yes—
496C23#88211yescharacteristic
497C23#88212yes—
498C23#88211yescharacteristic
499C23#88612yes—
500C23#88612yes—
501C23C2×(C2×D3)⋊C414yes—
502C23C2×(C2×D3)⋊C412yes—
503C23C2×(C2×D3)⋊C412yes—
504C23#93914yes—
505C23#93914yes—
506C23C22×C4×D311yescharacteristic
507C23C22×D1211yescharacteristic
508C23C2×D3×D412yes—
509C23C2×#23712yes—
510C23C22×#6711yescharacteristic
511D4—216no—
512D4—28no—
513D4—28no—
514D4—124no—
515D4—124no—
516D4—128no—
517D4—248no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—44no—
524C2×C4—64no—
525C2×C4—68no—
526C2×C4—64no—
527C2×C4—62no—
528C2×C4—88no—
529C2×C4—128no—
530C2×C4—124no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—128no—
535C2×C4—244no—
536C2×C4—244no—
537C2×C4—248no—
538C2×C4C2×(C2×D3)⋊C414yes—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—24no—
546C2×C3—44no—
547C2×C3C2×#38611yescharacteristic
548C2×C3#302612yes—
549D3—82no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—22no—
555C22—24no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—41no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—48no—
569C22—44no—
570C22—82no—
571C22—84no—
572C22—124no—
573C22—128no—
574C22—122no—
575C22—244no—
576C22—242no—
577C22—242no—
578C22#495812yes—
579C22#495812yes—
580C22#495814yes—
581C22C2×#67⋊C412yes—
582C22C2×#88211yescharacteristic
583C22C2×#88611yescharacteristic
584C22C22×(C2×D3)⋊C411yescharacteristic
585C22C2×#93912yes—
586C4—24no—
587C4—84no—
588C4—124no—
589C4—128no—
590C4—122no—
591C3#2096011yescharacteristic, complemented, Sylow
592C2—22no—
593C2—21no—
594C2—22no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—24no—
599C2—44no—
600C2—242no—
601C2#7868612yes—
602C2C2×#495811yescharacteristic
subgroupG/H
centerC22C2×#882
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×#99C2
socleC22×C3C2×#120
prime radicalC2×#5804C2
cosocleC24×C3C24
hypercenterC22×#99D3
hypocenterC3#20960
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
2. C2×#5798C2semidirect
3. C22×C3×#99C2semidirect
4. #82121C2semidirect
5. C2×#4527C2semidirect
6. #70023C2semidirect
7. #70022C2semidirect
17. #5804C4semidirect
31. #4955C22semidirect
32. #4943C22semidirect
33. #4527C22semidirect
36. C22×#99D3semidirect
60. C22×#67C2×C4semidirect
63. C23×Dic3D4semidirect
80. C2×#217C2×C4semidirect
81. C2×#217D4semidirect
86. #937C2×C4semidirect
88. #935C2×C4semidirect
131. C24×C3C22⋊C4semidirect
146. C23×D3C22×C4semidirect
155. C2×C3×D4C22⋊C4semidirect
164. C22×C3×C4C22⋊C4semidirect
191. C22×Dic3C22×C4semidirect
204. #217C22×C4semidirect
353. C24(C2×D3)⋊C4semidirect
384. C2×D4(C2×D3)⋊C4semidirect
407. C22×C4(C2×D3)⋊C4semidirect
591. C3#20960semidirect

Maximal subgroups

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

subgroupindex
C2×#58042
C2×#57982
C22×C3×#992
#821212
C2×#45272
#700232
#700222
#209603