Finite Group Databasebeta

← #10944102

Subgroups — #10944102

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5572C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×#5186C212yesmaximal
4#83261C211yesmaximal, characteristic
5C2×#5090C211yesmaximal, characteristic, complemented
6#72065C212yesmaximal, complemented
7#71627C211yesmaximal, characteristic, complemented
8#70486C211yesmaximal, characteristic, complemented
9#70476C211yesmaximal, characteristic
10#70281C212yesmaximal, complemented
11#70269C212yesmaximal
12#32915—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×#946C2212yes—
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2212yes—
20#5572—24no—
21#5572—24no—
22C2×#892—21no—
23C2×#892C2211yescharacteristic
24C2×#887—21no—
25C2×#887C2211yescharacteristic
26C2×#872—21no—
27C2×#872C2211yescharacteristic
28C2×C4×Dic6—21no—
29C2×C4×Dic6C2211yescharacteristic
30C3×#462—24no—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33#5229C2212yes—
34C2×#835C2212yes—
35C2×#835C2212yes—
36#5186—28no—
37C2×(C2×D3)⋊C8C2212yes—
38C2×#816C2212yes—
39#5090—24no—
40#4992—21no—
41#4992—21no—
42#4983—21no—
43#4983—21no—
44#4936C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#519—31no—
50C2×#462—31no—
51C2×#462D311yescharacteristic, complemented
52C2×#451—32no—
53#2621—31no—
54#1878—32no—
55#1878—32no—
56#1873—32no—
57#1805—31no—
58#1805—31no—
59#1798—31no—
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C2×D3×Q8—22no—
63C2×D3×Q8—22no—
64C2×D3×Q8—22no—
65C2×D3×Q8—22no—
66C2×D3×Q8—22no—
67C2×D3×Q8D414yes—
68C22×C4×D3—21no—
69C22×C4×D3C2311yescharacteristic
70C22×Dic6—21no—
71C22×Dic6C2311yescharacteristic
72C2×C3×Q16—24no—
73C2×C3×Q16—24no—
74C2×C3×Q16—24no—
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8D414yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86#946—24no—
87#946—24no—
88Q8×Dic3—22no—
89Q8×Dic3—22no—
90C2×#216—24no—
91C2×#216—24no—
92C2×#216—24no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111#892—28no—
112#892—22no—
113#892—22no—
114#887—28no—
115#887—22no—
116#887—22no—
117#872—28no—
118#872—22no—
119#872—22no—
120C4×Dic6—28no—
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C3×C4⋊C8—22no—
124C3×#103—24no—
125#835—24no—
126#835—24no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—22no—
130C3⋊#95—22no—
131(C2×D3)⋊C8—24no—
132#816—24no—
133C4⋊(C3⋊C8)—22no—
134C23×Q8—31no—
135C22×Q16—32no—
136C22×Q16C2×D312yes—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×C22⋊Q8—32no—
139#519—64no—
140#519—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×Q8—61no—
146C2×C4×Q8C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—31no—
149C2×#117—61no—
150C2×#117—61no—
151#462—24no—
152#462—64no—
153#451—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#103C2×D312yes—
160C2×C22⋊C8—32no—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382—31no—
167#340—32no—
168#340—32no—
169#336—32no—
170C23×D3C2×D411yescharacteristic
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8#4112yes—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195D3×Q8—28no—
196D3×Q8—28no—
197D3×Q8—28no—
198D3×Q8—28no—
199C2×C4×D3—24no—
200C2×C4×D3—24no—
201C2×C4×D3—24no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3C2×D412yes—
207C2×C4×D3#4112yes—
208C2×Dic6—24no—
209C2×Dic6—24no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—24no—
214C2×Dic6C2×D412yes—
215C2×Dic6C2×D412yes—
216C2×Dic6#4112yes—
217C3×Q16—48no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4C2×D412yes—
226C3×C42—24no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229#216—48no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—44no—
234C4⋊Dic3—24no—
235C4⋊Dic3—21no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3C2×D412yes—
239Dic3⋊C4—24no—
240Dic3⋊C4—22no—
241Dic3⋊C4—24no—
242Dic3⋊C4—22no—
243Dic3⋊C4—44no—
244C4×Dic3—24no—
245C4×Dic3—24no—
246C4×Dic3—24no—
247C4×Dic3—24no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C22×Q8—34no—
254C22×Q8—31no—
255C22×Q8—61no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×Q8C22×D312yes—
262C22×Q8C22×D311yescharacteristic
263C23×C4—31no—
264C23×C4—61no—
265C2×Q16—24no—
266C2×Q16—24no—
267C2×Q16—24no—
268C2×Q16—64no—
269C2×Q16—64no—
270C2×Q16—64no—
271C22×C8—32no—
272C22×C8C22×D312yes—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C22⋊Q8—64no—
276C22⋊Q8—64no—
277C4×Q8—22no—
278C4×Q8—34no—
279C4×Q8—68no—
280C4×Q8—62no—
281C4×Q8—68no—
282C4×Q8—62no—
283C4×Q8—62no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×Q8#6714yes—
287#117—68no—
288#117—62no—
289#117—68no—
290#117—62no—
291#117—62no—
292#117—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—62no—
309C2×C42—21no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42C22×D311yescharacteristic
316C4⋊C8—22no—
317C4⋊C8—62no—
318#103—24no—
319#103—64no—
320#103—64no—
321#103—64no—
322C22⋊C8—64no—
323#95—62no—
324#95—62no—
325#95—61no—
326#95—61no—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3#12112yes—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8#12114yes—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4#12012yes—
352C2×C3×C4#12012yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3#12112yes—
374C4×D3—216no—
375C4×D3—22no—
376C4×D3—22no—
377C4×D3—22no—
378C4×D3—22no—
379Dic6—216no—
380Dic6—22no—
381Dic6—22no—
382Dic6—22no—
383Dic6#12114yes—
384C3×C8—42no—
385C3⋊C8—42no—
386C24—31no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—32no—
397C2×Q8—32no—
398C2×Q8—32no—
399C2×Q8—64no—
400C2×Q8—64no—
401C2×Q8—64no—
402C2×Q8—68no—
403C2×Q8—68no—
404C2×Q8—68no—
405C2×Q8—68no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8D3×D412yes—
410C2×Q8#23712yes—
411C2×Q8C2×#6712yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—31no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4C23×D311yescharacteristic
433Q16—48no—
434Q16—128no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—62no—
439C2×C8—62no—
440C2×C8—62no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—32no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—124no—
455C4⋊C4C2×#6712yes—
456C22⋊C4—64no—
457C22⋊C4—62no—
458C22⋊C4—62no—
459C22⋊C4—124no—
460C42—24no—
461C42—64no—
462C42—64no—
463C42—64no—
464C42—64no—
465C42—61no—
466C42—61no—
467C42D3×D412yes—
468C42C2×#6712yes—
469C22×C3C2×#12011yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3#57812yes—
473C22×C3#57812yes—
474C2×D3—22no—
475C2×D3—24no—
476C2×D3—42no—
477C2×D3—41no—
478C2×D3—41no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—82no—
484C3×C4C2×#12111yescharacteristic
485C3×C4C2×#12111yescharacteristic
486C3×C4#52511yescharacteristic
487C3×C4#52511yescharacteristic
488Dic3—22no—
489Dic3—24no—
490Dic3—41no—
491Dic3—41no—
492Dic3—42no—
493Dic3—82no—
494C23—32no—
495C23—62no—
496C23—62no—
497C23—61no—
498C23—61no—
499C23C2×D3×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—34no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—616no—
509Q8—62no—
510Q8—62no—
511Q8—62no—
512Q8#93814yes—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—32no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—616no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4#93712yes—
544C2×C4#93712yes—
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×#23711yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4C22×#6711yescharacteristic
551C8—42no—
552C8—122no—
553C2×C3#284711yescharacteristic
554C2×C3#301111yescharacteristic
555C2×C3#301111yescharacteristic
556C2×C3#304412yes—
557C2×C3#312712yes—
558D3—42no—
559C22—62no—
560C22—64no—
561C22—121no—
562C22—122no—
563C22—121no—
564C22D3×#12111yescharacteristic
565C22#560611yescharacteristic
566C22#576612yes—
567C22#577912yes—
568C22C2×#93711yescharacteristic
569C4—22no—
570C4—24no—
571C4—22no—
572C4—42no—
573C4—62no—
574C4—64no—
575C4—82no—
576C4—121no—
577C4—121no—
578C4—122no—
579C4—242no—
580C4D3×#12111yescharacteristic
581C4#560611yescharacteristic
582C4C2×#93811yescharacteristic
583C4#582811yescharacteristic
584C3#3291511yescharacteristic, complemented, Sylow
585C2—122no—
586C2#8046112yes—
587C2#8047812yes—
588C2#8272311yescharacteristic
589C2#8388411yescharacteristic
590C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#462C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#462D3
hypocenterC3#32915
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×C3×#462C2semidirect
5. C2×#5090C2semidirect
6. #72065C2semidirect
7. #71627C2semidirect
8. #70486C2semidirect
10. #70281C2semidirect
51. C2×#462D3semidirect
584. C3#32915semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C2×C3×#4622
C2×#51862
#832612
C2×#50902
#720652
#716272
#704862
#704762
#702812
#702692
#329153