Finite Group Databasebeta

← #10796820

Subgroups — #10796820

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×#518C211yesmaximal, characteristic, complemented
2C2×#5804C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic, complemented
4#82121C212yesmaximal, complemented
5#71655C214yesmaximal, complemented
6#71654C214yesmaximal, complemented
7C2×#4526C212yesmaximal, complemented
8#20960—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67C2211yescharacteristic
11C3×#518—22no—
12C3×#518—24no—
13C3×#518C2214yes—
14C2×C3×C4×D4—24no—
15C2×C3×#117—21no—
16C22×C3×C22⋊C4—21no—
17C22×C3×C22⋊C4C2212yes—
18C2×#953—21no—
19C22×#217C2212yes—
20C22×#217C2211yescharacteristic
21#5804—24no—
22#5804C418yescomplemented
23C2×#939—22no—
24C2×#937C2212yes—
25C2×#935C2212yes—
26C2×#929—21no—
27C2×#834—24no—
28C2×#834C2214yes—
29#5213—24no—
30#4958—24no—
31#4955C2214yescomplemented
32#4943C2214yescomplemented
33#4526C2218yescomplemented
34C2×#534—31no—
35C2×#518D311yescharacteristic, complemented
36C22×#99—31no—
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×D4D414yes—
49C22×C3×D4D412yes—
50C22×C3×D4C2312yes—
51C23×C3×C4—22no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D414yescomplemented
54C22×#67—22no—
55C22×#67—22no—
56C22×#67—21no—
57C22×#67—21no—
58C22×#67—22no—
59C22×#67—24no—
60C22×#67—22no—
61C22×#67C2×C414yescomplemented
62C23×Dic3—21no—
63C23×Dic3—22no—
64C23×Dic3C2311yescharacteristic
65C3×C4×D4—216no—
66C3×C4×D4—216no—
67C3×#117—24no—
68C3×#117—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D414yescomplemented
78C2×C3×C22⋊C4C2314yes—
79C2×C3×C42—22no—
80#953—28no—
81C2×#217—21no—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—22no—
87C2×#217C2×C414yescomplemented
88C2×#217C2314yes—
89C2×#217C2312yes—
90#939—216no—
91#937—24no—
92#937C2×C418yescomplemented
93#935—24no—
94#935C2×C418yescomplemented
95#929—28no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—28no—
98C2×Dic3⋊C4—22no—
99#834—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—22no—
111#518—24no—
112#518C2×D314yes—
113C2×C4×D4—24no—
114C2×#117—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4C2×D312yes—
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—82no—
130C24×C3—82no—
131C24×C3C22⋊C414yescomplemented
132C24×C3C22×C412yes—
133C24×C3C2×D412yes—
134C24×C3C2×D412yes—
135C24×C3C2×D411yescharacteristic
136C24×C3C2×D411yescharacteristic
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—28no—
163C22×C3×C4—24no—
164C22×C3×C4—84no—
165C22×C3×C4—84no—
166C22×C3×C4C22⋊C414yescomplemented
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D418yes—
169C2×#67—24no—
170C2×#67—24no—
171C2×#67—28no—
172C2×#67—28no—
173C2×#67—28no—
174C2×#67—41no—
175C2×#67—42no—
176C2×#67—41no—
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—42no—
180C2×#67—42no—
181C2×#67—84no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—44no—
190C22×Dic3—82no—
191C22×Dic3—84no—
192C22×Dic3C22×C412yescomplemented
193C3×C4⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—24no—
199C3×C42—28no—
200#217—24no—
201#217—28no—
202#217—22no—
203#217—24no—
204#217—24no—
205#217—44no—
206#217C22×C414yescomplemented
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—88no—
209Dic3⋊C4—28no—
210C25—31no—
211C25#6711yescharacteristic
212C25C22×D311yescharacteristic
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—34no—
217C22×D4—62no—
218C22×D4—64no—
219C22×D4—61no—
220C22×D4—61no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4—62no—
224C22×D4C4×D314yes—
225C22×D4D1214yes—
226C22×D4#6712yes—
227C22×D4C22×D312yes—
228C23×C4—22no—
229C23×C4—31no—
230C23×C4—61no—
231C23×C4—62no—
232C23×C4#6711yescharacteristic
233C23×C4#6714yescomplemented
234#123—38no—
235#123—64no—
236#123—68no—
237#121—616no—
238#120—38no—
239#120—68no—
240#120—64no—
241C4×D4—216no—
242C4×D4—216no—
243#117—24no—
244#117—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—62no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—34no—
255C2×C22⋊C4—32no—
256C2×C22⋊C4—34no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—68no—
265C2×C22⋊C4#6714yescomplemented
266C2×C22⋊C4C22×D314yes—
267C2×C42—22no—
268#99—68no—
269#95—68no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—44no—
275C23×C3—42no—
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×D411yescharacteristic
302C23×C3C22×D412yes—
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—28no—
322C2×C3×C4—48no—
323C2×C3×C4—48no—
324C2×C3×C4—84no—
325C2×C3×C4—88no—
326C2×C3×C4C2×C22⋊C414yes—
327#67—44no—
328#67—44no—
329#67—48no—
330#67—88no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—44no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—48no—
339C2×Dic3—44no—
340C2×Dic3—84no—
341C2×Dic3—88no—
342C24—21no—
343C24—22no—
344C24—21no—
345C24—32no—
346C24—62no—
347C24—61no—
348C24—61no—
349C24—62no—
350C24—82no—
351C24—82no—
352C24—242no—
353C24(C2×D3)⋊C414yescomplemented
354C24C2×C4×D312yes—
355C24C2×D1212yes—
356C24#23514yes—
357C24C2×#6712yes—
358C24C2×#6711yescharacteristic
359C24C2×#6711yescharacteristic
360C24C2×#6712yes—
361C24C2×#6712yes—
362C24C23×D311yescharacteristic
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—24no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—64no—
372C2×D4—64no—
373C2×D4—68no—
374C2×D4—68no—
375C2×D4—68no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—124no—
379C2×D4—124no—
380C2×D4—121no—
381C2×D4—122no—
382C2×D4—121no—
383C2×D4—244no—
384C2×D4(C2×D3)⋊C418yescomplemented
385C2×D4C2×C4×D314yes—
386C2×D4C2×D1214yes—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—32no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—124no—
404C22×C4—242no—
405C22×C4—244no—
406C22×C4(C2×D3)⋊C414yescomplemented
407C22×C4C2×#6712yes—
408C22×C4C2×#6718yes—
409C4⋊C4—28no—
410C4⋊C4—68no—
411C22⋊C4—28no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—34no—
417C22⋊C4—64no—
418C22⋊C4—68no—
419C22⋊C4—62no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—124no—
424C22⋊C4—248no—
425C42—28no—
426C22×C3—24no—
427C22×C3—22no—
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—48no—
464Dic3—44no—
465Dic3—42no—
466Dic3—84no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—62no—
479C23—64no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—128no—
487C23—124no—
488C23—121no—
489C23—122no—
490C23—121no—
491C23—242no—
492C23—242no—
493C23—244no—
494C23C2×(C2×D3)⋊C414yes—
495C23C2×(C2×D3)⋊C412yes—
496C23C2×(C2×D3)⋊C412yes—
497C23C4×#6714yes—
498C23C4×#6714yes—
499C23#92912yes—
500C23#92912yes—
501C23#93014yes—
502C23#93014yes—
503C23#95311yescharacteristic
504C23#95312yes—
505C23#95311yescharacteristic
506C23C22×C4×D311yescharacteristic
507C23C22×D1211yescharacteristic
508C23C2×#23512yes—
509C23C22×#6711yescharacteristic
510C23C22×#6712yes—
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—28no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—64no—
526C2×C4—62no—
527C2×C4—64no—
528C2×C4—84no—
529C2×C4—88no—
530C2×C4—124no—
531C2×C4—121no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—128no—
535C2×C4—124no—
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—22no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—24no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—42no—
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#521312yes—
579C22#521312yes—
580C22#521314yes—
581C22C22×(C2×D3)⋊C411yescharacteristic
582C22C2×C4×#6712yes—
583C22C2×#92911yescharacteristic
584C22C2×#93012yes—
585C22C2×#95311yescharacteristic
586C4—24no—
587C4—48no—
588C4—124no—
589C4—122no—
590C4—244no—
591C3#2096011yescharacteristic, complemented, Sylow
592C2—21no—
593C2—22no—
594C2—22no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—24no—
599C2—44no—
600C2—242no—
601C2C2×#521311yescharacteristic
602C2#8411012yes—
subgroupG/H
centerC22C2×#953
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC2×C3×#518C2
socleC22×C3C2×#120
prime radicalC2×#5804C2
cosocleC24×C3C24
hypercenterC2×#518D3
hypocenterC3#20960
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
3. C22×#834C2semidirect
4. #82121C2semidirect
5. #71655C2semidirect
6. #71654C2semidirect
7. C2×#4526C2semidirect
22. #5804C4semidirect
31. #4955C22semidirect
32. #4943C22semidirect
33. #4526C22semidirect
35. C2×#518D3semidirect
53. C23×C3×C4D4semidirect
61. C22×#67C2×C4semidirect
77. C2×C3×C22⋊C4D4semidirect
87. C2×#217C2×C4semidirect
92. #937C2×C4semidirect
94. #935C2×C4semidirect
131. C24×C3C22⋊C4semidirect
146. C23×D3C22×C4semidirect
155. C2×C3×D4C22⋊C4semidirect
166. C22×C3×C4C22⋊C4semidirect
192. C22×Dic3C22×C4semidirect
206. #217C22×C4semidirect
233. C23×C4#67semidirect
265. C2×C22⋊C4#67semidirect
353. C24(C2×D3)⋊C4semidirect
384. C2×D4(C2×D3)⋊C4semidirect
406. 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×C3×#5182
C2×#58042
C22×#8342
#821212
#716552
#716542
C2×#45262
#209603