Finite Group Databasebeta

← #10822436

Subgroups — #10822436

Order 768.

2418 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic, complemented
2C2×#5804C212yesmaximal, complemented
3C22×#834C212yesmaximal, complemented
4#82121C211yesmaximal, characteristic, complemented
5#71851C214yesmaximal, complemented
6#71849C214yesmaximal, complemented
7C2×#4526C211yesmaximal, characteristic, complemented
8#23444—31nomaximal, Sylow
9C23×C3×D4C2212yes—
10C23×#67C2211yescharacteristic
11C3×#534—24no—
12C3×#534—22no—
13C3×#534C2214yes—
14C2×C3×#123—22no—
15C2×C3×#121—22no—
16C2×C3×#120—22no—
17C22×C3×C22⋊C4C2211yescharacteristic
18C2×#953—22no—
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21#5804—28no—
22#5804—28no—
23C2×#939—24no—
24C2×#937C2214yescomplemented
25C2×#935C2214yescomplemented
26C2×#929—22no—
27C2×#834—28no—
28C2×#834C2218yescomplemented
29#5213—22no—
30#4958—22no—
31#4955C2212yescomplemented
32#4943C2212yescomplemented
33#4526C2214yescomplemented
34C2×#534—32no—
35C2×#534D311yescharacteristic, complemented
36C22×#99—32no—
37#2308—31no—
38#2022—34no—
39#2021—34no—
40C2×#342—31no—
41C25×C3D412yescomplemented
42C25×C3C2311yescharacteristic
43C24×D3C2311yescharacteristic
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—24no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—24no—
52C22×C3×D4—22no—
53C22×C3×D4D414yescomplemented
54C22×C3×D4C2314yes—
55C23×C3×C4D412yescomplemented
56C23×C3×C4D412yescomplemented
57C22×#67—21no—
58C22×#67—21no—
59C22×#67—22no—
60C22×#67—22no—
61C22×#67—24no—
62C22×#67—22no—
63C22×#67—24no—
64C22×#67D414yescomplemented
65C23×Dic3—22no—
66C23×Dic3—21no—
67C23×Dic3C2311yescharacteristic
68C3×#123—28no—
69C3×#123—28no—
70C3×#121—28no—
71C3×#121—28no—
72C3×#120—28no—
73C3×#120—28no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D412yescomplemented
79C2×C3×C22⋊C4C2312yes—
80#953—216no—
81C2×#217—22no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—21no—
86C2×#217D418yescomplemented
87C2×#217C2312yes—
88C2×#217C2314yes—
89#939—232no—
90#937—28no—
91#937—28no—
92#935—28no—
93#935—28no—
94#929—216no—
95C2×(C2×D3)⋊C4—24no—
96C2×(C2×D3)⋊C4—24no—
97C2×Dic3⋊C4—24no—
98#834—216no—
99C3⋊#95—24no—
100C23×D4—31no—
101C23×D4C2×D312yes—
102#534—24no—
103#534—22no—
104#534—68no—
105#534—68no—
106#534C2×D314yes—
107C2×#123—22no—
108C2×#123—34no—
109C2×#123—62no—
110C2×#121—22no—
111C2×#121—64no—
112C2×#120—22no—
113C2×#120—34no—
114C2×#120—62no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4C2×D311yescharacteristic
118C2×#99—38no—
119C2×#99—68no—
120#386—62no—
121#386—62no—
122#381—32no—
123#379—32no—
124#342—34no—
125C24×C3—22no—
126C24×C3—24no—
127C24×C3—22no—
128C24×C3—24no—
129C24×C3—82no—
130C24×C3—84no—
131C24×C3C2×D412yes—
132C24×C3C2×D414yes—
133C24×C3C2×D414yes—
134C24×C3C2×D412yes—
135C24×C3C2×D412yes—
136C24×C3C2×D411yescharacteristic
137C24×C3C2×D411yescharacteristic
138C24×C3#4112yes—
139C24×C3C2411yescharacteristic
140C23×D3—21no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—21no—
144C23×D3—82no—
145C23×D3C2×D412yescomplemented
146C2×C3×D4—28no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—28no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—28no—
157C2×C3×D4—44no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—84no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D414yes—
175C2×#67—28no—
176C2×#67—28no—
177C2×#67—24no—
178C2×#67—24no—
179C2×#67—216no—
180C2×#67—42no—
181C2×#67—41no—
182C2×#67—41no—
183C2×#67—44no—
184C2×#67—44no—
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—88no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—28no—
194C22×Dic3—22no—
195C22×Dic3—42no—
196C22×Dic3—84no—
197C22×Dic3—82no—
198C22×Dic3C2×D412yescomplemented
199C3×C4⋊C4—28no—
200C3×C22⋊C4—216no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203#217—216no—
204#217—28no—
205#217—24no—
206#217—24no—
207#217—42no—
208#217C2×D418yescomplemented
209(C2×D3)⋊C4—216no—
210(C2×D3)⋊C4—84no—
211Dic3⋊C4—216no—
212C25—31no—
213C25#6712yescomplemented
214C25C22×D311yescharacteristic
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—34no—
225C22×D4—64no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4—62no—
229C22×D4—61no—
230C22×D4—61no—
231C22×D4—64no—
232C22×D4#6714yescomplemented
233C22×D4C22×D314yes—
234C23×C4—31no—
235C23×C4—62no—
236C23×C4—61no—
237C23×C4#6712yescomplemented
238C23×C4#6712yescomplemented
239#123—28no—
240#123—28no—
241#123—68no—
242#123—616no—
243#123—68no—
244#121—28no—
245#121—28no—
246#121—632no—
247#120—28no—
248#120—28no—
249#120—68no—
250#120—68no—
251#120—616no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—64no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—38no—
258C2×C22⋊C4—34no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4#6712yescomplemented
268C2×C22⋊C4C22×D312yes—
269#99—616no—
270#95—64no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—42no—
275C23×C3—44no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—44no—
279C23×C3—48no—
280C23×C3—48no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3—84no—
284C23×C3—82no—
285C23×C3—84no—
286C23×C3—82no—
287C23×C3—88no—
288C23×C3—88no—
289C23×C3#12014yes—
290C23×C3#12014yes—
291C23×C3#12012yes—
292C23×C3#12011yescharacteristic
293C23×C3#12011yescharacteristic
294C23×C3#12112yes—
295C23×C3#12112yes—
296C23×C3#12114yes—
297C23×C3#12114yes—
298C23×C3#12712yes—
299C23×C3#12712yes—
300C23×C3C22×D411yescharacteristic
301C23×C3C22×D412yes—
302C23×C3C22×D412yes—
303C23×C3C22×D411yescharacteristic
304C23×C3C2×#4111yescharacteristic
305C22×D3—24no—
306C22×D3—22no—
307C22×D3—44no—
308C22×D3—48no—
309C22×D3—41no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—82no—
313C22×D3—84no—
314C22×D3—82no—
315C3×D4—44no—
316C3×D4—44no—
317C3×D4—44no—
318C3×D4—44no—
319C3×D4—48no—
320C3×D4—48no—
321C3×D4—88no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4—82no—
331#67—44no—
332#67—48no—
333#67—44no—
334#67—816no—
335C2×Dic3—24no—
336C2×Dic3—28no—
337C2×Dic3—22no—
338C2×Dic3—48no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—44no—
343C2×Dic3—42no—
344C2×Dic3—88no—
345C2×Dic3—84no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—32no—
351C24—61no—
352C24—62no—
353C24—62no—
354C24—61no—
355C24—82no—
356C24—84no—
357C24—242no—
358C24D3×D412yes—
359C24D3×D414yes—
360C24#23712yes—
361C24C2×#6714yes—
362C24C2×#6712yes—
363C24C2×#6712yes—
364C24C2×#6711yescharacteristic
365C24C2×#6711yescharacteristic
366C24C23×D311yescharacteristic
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—28no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—68no—
387C2×D4—68no—
388C2×D4—64no—
389C2×D4—64no—
390C2×D4—616no—
391C2×D4—84no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4—122no—
397C2×D4—121no—
398C2×D4—121no—
399C2×D4—248no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—32no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—68no—
413C22×C4—62no—
414C22×C4—82no—
415C22×C4—122no—
416C22×C4—244no—
417C22×C4—242no—
418C22×C4C2×#6714yes—
419C22×C4C2×#6714yes—
420C4⋊C4—28no—
421C4⋊C4—616no—
422C22⋊C4—216no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—38no—
426C22⋊C4—616no—
427C22⋊C4—616no—
428C22⋊C4—68no—
429C22⋊C4—64no—
430C22⋊C4—64no—
431C22⋊C4—122no—
432C22⋊C4—244no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—41no—
443C22×C3—44no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—44no—
448C22×C3—42no—
449C22×C3—48no—
450C22×C3—44no—
451C22×C3—48no—
452C22×C3—84no—
453C22×C3—88no—
454C22×C3—88no—
455C22×C3#39212yes—
456C22×C3#39214yes—
457C22×C3#39212yes—
458C22×C3C2×#12012yes—
459C22×C3C2×#12011yescharacteristic
460C22×C3C2×#12111yescharacteristic
461C22×C3C2×#12112yes—
462C22×C3C2×#12711yescharacteristic
463C2×D3—48no—
464C2×D3—44no—
465C2×D3—42no—
466C2×D3—82no—
467C2×D3—84no—
468C2×D3—82no—
469C3×C4—44no—
470C3×C4—44no—
471Dic3—48no—
472Dic3—42no—
473Dic3—82no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—48no—
483C23—48no—
484C23—44no—
485C23—44no—
486C23—64no—
487C23—62no—
488C23—84no—
489C23—82no—
490C23—84no—
491C23—82no—
492C23—88no—
493C23—88no—
494C23—124no—
495C23—128no—
496C23—122no—
497C23—121no—
498C23—121no—
499C23—242no—
500C23—244no—
501C23—242no—
502C23#93714yes—
503C23#93714yes—
504C23#93812yes—
505C23#93812yes—
506C23#93914yes—
507C23#93914yes—
508C23#94012yes—
509C23#94012yes—
510C23#95312yes—
511C23#95311yescharacteristic
512C23#95311yescharacteristic
513C23C2×D3×D411yescharacteristic
514C23C2×D3×D412yes—
515C23C2×#23711yescharacteristic
516C23C22×#6712yes—
517C23C22×#6711yescharacteristic
518D4—48no—
519D4—48no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—44no—
524D4—88no—
525D4—124no—
526D4—128no—
527D4—124no—
528D4—2416no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—68no—
538C2×C4—64no—
539C2×C4—62no—
540C2×C4—82no—
541C2×C4—128no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—248no—
548C2×C4—244no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—22no—
554C2×C3—24no—
555C2×C3—22no—
556C2×C3—84no—
557C2×C3C2×#39211yescharacteristic
558C2×C3#302612yes—
559D3—82no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—44no—
566C22—42no—
567C22—44no—
568C22—44no—
569C22—41no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—48no—
577C22—44no—
578C22—48no—
579C22—84no—
580C22—88no—
581C22—88no—
582C22—128no—
583C22—124no—
584C22—122no—
585C22—242no—
586C22—244no—
587C22—242no—
588C22#522312yes—
589C22#522314yes—
590C22#522312yes—
591C22C2×#93712yes—
592C22C2×#93811yescharacteristic
593C22C2×#93912yes—
594C22C2×#94011yescharacteristic
595C22C2×#95311yescharacteristic
596C4—44no—
597C4—44no—
598C4—128no—
599C4—122no—
600C4—242no—
601C3#2344411yescharacteristic, complemented, Sylow
602C2—22no—
603C2—21no—
604C2—22no—
605C2—22no—
606C2—24no—
607C2—21no—
608C2—22no—
609C2—84no—
610C2—242no—
611C2C2×#522311yescharacteristic
612C2#8411012yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#534C2
socleC22×C3C2×#120
hypercenterC2×#534D3
hypocenterC3#23444
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. C2×C3×#534C2semidirect
2. C2×#5804C2semidirect
3. C22×#834C2semidirect
4. #82121C2semidirect
5. #71851C2semidirect
6. #71849C2semidirect
7. C2×#4526C2semidirect
24. C2×#937C22semidirect
25. C2×#935C22semidirect
28. C2×#834C22semidirect
31. #4955C22semidirect
32. #4943C22semidirect
33. #4526C22semidirect
35. C2×#534D3semidirect
41. C25×C3D4semidirect
53. C22×C3×D4D4semidirect
55. C23×C3×C4D4semidirect
56. C23×C3×C4D4semidirect
64. C22×#67D4semidirect
78. C2×C3×C22⋊C4D4semidirect
86. C2×#217D4semidirect
145. C23×D3C2×D4semidirect
198. C22×Dic3C2×D4semidirect
208. #217C2×D4semidirect
213. C25#67semidirect
232. C22×D4#67semidirect
237. C23×C4#67semidirect
238. C23×C4#67semidirect
267. C2×C22⋊C4#67semidirect
601. C3#23444semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C2×#58042
C22×#8342
#821212
#718512
#718492
C2×#45262
#234443