Finite Group Databasebeta

← #10700635

Subgroups — #10700635

Order 768.

1498 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5798C211yesmaximal, characteristic
2C22×C3×#99C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4C2×#4527C212yesmaximal, complemented
5C2×#4526C212yesmaximal, complemented
6#10768—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4C2212yes—
9C22×#217C412yescomplemented
10C22×#217C2212yes—
11C22×#217C2211yescharacteristic
12C22×#217C2211yescharacteristic
13#5798—24no—
14#5798—24no—
15C2×D4×Dic3—22no—
16C2×D4×Dic3—22no—
17C2×#926C412yescomplemented
18C2×C3×#99—22no—
19C2×C3×#99—22no—
20C2×C3×#99C414yescomplemented
21C2×#834—22no—
22C2×#834—22no—
23C2×#834C414yescomplemented
24#4527—24no—
25#4526—24no—
26C2×#518—31no—
27C22×#99—31no—
28C22×#99D311yescharacteristic
29C2×#342—32no—
30C2×#342—32no—
31C25×C3C2×C411yescharacteristic
32C25×C3C2311yescharacteristic
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4C2×C412yes—
38C22×C3×D4C2×C412yes—
39C22×C3×D4D412yes—
40C22×C3×D4D412yes—
41C22×C3×D4D412yes—
42C22×C3×D4Q812yes—
43C23×C3×C4—22no—
44C23×C3×C4C2×C411yescharacteristic
45C23×Dic3—22no—
46C23×Dic3—21no—
47C23×Dic3—21no—
48C23×Dic3—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4C2×C414yescomplemented
55C2×#217—22no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—24no—
61C2×#217—24no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—22no—
65C2×#217—22no—
66C2×#217—22no—
67C2×#217C2×C412yes—
68C2×#217C2×C412yes—
69C2×#217C2×C414yescomplemented
70C2×#217D414yes—
71C2×#217D414yes—
72D4×Dic3—216no—
73D4×Dic3—216no—
74#926—24no—
75#926—24no—
76C2×C4⋊Dic3—22no—
77C2×C4×Dic3—22no—
78C3×#99—28no—
79#834—28no—
80C23×D4C2×D311yescharacteristic
81#518—64no—
82#518—64no—
83C2×C4×D4—62no—
84C2×C4×D4—62no—
85C2×#117—32no—
86C22×C22⋊C4—31no—
87C22×C22⋊C4—31no—
88C22×C22⋊C4—32no—
89C22×C22⋊C4—32no—
90C22×C22⋊C4C2×D312yes—
91C2×#99—22no—
92C2×#99—22no—
93C2×#99—34no—
94C2×#99—62no—
95C2×#99—62no—
96C2×#99Dic314yescomplemented
97#342—64no—
98#342—64no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—21no—
102C24×C3—21no—
103C24×C3—82no—
104C24×C3—82no—
105C24×C3C4212yes—
106C24×C3C22⋊C412yes—
107C24×C3C4⋊C412yes—
108C24×C3C22×C411yescharacteristic
109C24×C3C22×C411yescharacteristic
110C24×C3C22×C411yescharacteristic
111C24×C3C2×D411yescharacteristic
112C24×C3C2×D412yes—
113C24×C3C2×D412yes—
114C24×C3C2×D411yescharacteristic
115C24×C3C2×D411yescharacteristic
116C24×C3C2×Q811yescharacteristic
117C24×C3#4112yes—
118C24×C3#4112yes—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4C22⋊C414yes—
132C2×C3×D4C22⋊C414yes—
133C2×C3×D4C4⋊C414yes—
134C2×C3×D4C4⋊C414yes—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—24no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—84no—
143C22×C3×C4C4212yes—
144C22×C3×C4C22⋊C412yes—
145C22×C3×C4C4⋊C412yes—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—28no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—82no—
155C22×Dic3—82no—
156C22×Dic3—84no—
157C22×Dic3—84no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—24no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—42no—
162#217—28no—
163#217—22no—
164#217—22no—
165#217—24no—
166#217—24no—
167#217—24no—
168#217—24no—
169#217—42no—
170#217—42no—
171#217C22⋊C414yes—
172#217C22⋊C414yes—
173C4⋊Dic3—28no—
174C4×Dic3—28no—
175C25C4×D311yescharacteristic
176C25C22×D311yescharacteristic
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4Dic612yes—
182C22×D4C4×D312yes—
183C22×D4D1212yes—
184C22×D4C2×Dic312yes—
185C22×D4#6712yes—
186C22×D4#6712yes—
187C23×C4—22no—
188C23×C4—62no—
189C23×C4—61no—
190C23×C4—61no—
191C23×C4—62no—
192C23×C4C4×D311yescharacteristic
193C4×D4—616no—
194C4×D4—616no—
195#117—64no—
196#117—64no—
197C2×C4⋊C4—62no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—32no—
205C2×C22⋊C4—34no—
206C2×C22⋊C4—34no—
207C2×C22⋊C4—34no—
208C2×C22⋊C4—64no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4C2×Dic314yescomplemented
221C2×C42—62no—
222#99—28no—
223#99—68no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—44no—
233C23×C3—44no—
234C23×C3—42no—
235C23×C3—44no—
236C23×C3—82no—
237C23×C3—82no—
238C23×C3—81no—
239C23×C3—81no—
240C23×C3—82no—
241C23×C3—82no—
242C23×C3—81no—
243C23×C3—81no—
244C23×C3—82no—
245C23×C3—82no—
246C23×C3—82no—
247C23×C3—82no—
248C23×C3#9514yes—
249C23×C3C2×C4211yescharacteristic
250C23×C3C2×C22⋊C411yescharacteristic
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3C2×C22⋊C411yescharacteristic
253C23×C3C2×C22⋊C411yescharacteristic
254C23×C3C2×C22⋊C411yescharacteristic
255C23×C3C2×C4⋊C411yescharacteristic
256C23×C3C2×C4⋊C411yescharacteristic
257C23×C3C2×C4⋊C411yescharacteristic
258C23×C3#11711yescharacteristic
259C23×C3#11711yescharacteristic
260C23×C3C4×D412yes—
261C23×C3C4×D412yes—
262C23×C3C4×D412yes—
263C23×C3C4×D412yes—
264C23×C3#12012yes—
265C23×C3#12012yes—
266C23×C3#12112yes—
267C23×C3#12112yes—
268C23×C3C22⋊Q812yes—
269C23×C3C22⋊Q812yes—
270C23×C3#12312yes—
271C23×C3#12312yes—
272C3×D4—28no—
273C3×D4—28no—
274C3×D4—28no—
275C3×D4—28no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—44no—
283C2×C3×C4—84no—
284C2×C3×C4—84no—
285C2×C3×C4—84no—
286C2×C3×C4#9514yes—
287C2×Dic3—24no—
288C2×Dic3—24no—
289C2×Dic3—44no—
290C2×Dic3—44no—
291C2×Dic3—44no—
292C2×Dic3—84no—
293C2×Dic3—84no—
294C2×Dic3—88no—
295C2×Dic3—84no—
296C2×Dic3—84no—
297C2×Dic3—84no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—82no—
303C24—82no—
304C24C4×Dic312yes—
305C24Dic3⋊C412yes—
306C24(C2×D3)⋊C412yes—
307C24C2×Dic611yescharacteristic
308C24C2×C4×D311yescharacteristic
309C24C2×C4×D311yescharacteristic
310C24C2×D1211yescharacteristic
311C24D3×D412yes—
312C24D3×D412yes—
313C24#23712yes—
314C24#23712yes—
315C24C22×Dic311yescharacteristic
316C24C2×#6711yescharacteristic
317C24C2×#6711yescharacteristic
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4Dic3⋊C414yes—
331C2×D4C4⋊Dic314yes—
332C2×D4(C2×D3)⋊C414yes—
333C2×D4#21714yes—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—68no—
344C22×C4—64no—
345C22×C4—64no—
346C22×C4—64no—
347C22×C4—84no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4—242no—
351C22×C4—242no—
352C22×C4—244no—
353C22×C4—244no—
354C22×C4C4×Dic312yes—
355C22×C4Dic3⋊C412yes—
356C22×C4(C2×D3)⋊C412yes—
357C4⋊C4—68no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—34no—
361C22⋊C4—34no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—68no—
365C22⋊C4—62no—
366C22⋊C4—62no—
367C22⋊C4—64no—
368C22⋊C4—64no—
369C22⋊C4—64no—
370C22⋊C4—64no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C42—68no—
374C22×C3—22no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—24no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3C2×#9511yescharacteristic
406C22×C3C4×C22⋊C411yescharacteristic
407C22×C3C4×C22⋊C411yescharacteristic
408C22×C3#37911yescharacteristic
409C22×C3#38011yescharacteristic
410C22×C3#38011yescharacteristic
411C22×C3#38111yescharacteristic
412C22×C3#38512yes—
413C22×C3#38512yes—
414C22×C3#38612yes—
415C22×C3#38612yes—
416C3×C4—24no—
417C3×C4—84no—
418Dic3—44no—
419Dic3—44no—
420Dic3—84no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—82no—
434C23—82no—
435C23—81no—
436C23—81no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23C3⋊#9514yes—
446C23#87711yescharacteristic
447C23#87711yescharacteristic
448C23#87812yes—
449C23#87812yes—
450C23D3×C22⋊C411yescharacteristic
451C23D3×C22⋊C411yescharacteristic
452C23#67⋊C412yes—
453C23#67⋊C412yes—
454C23#88212yes—
455C23#88612yes—
456C23C2×C4×Dic311yescharacteristic
457C23C2×Dic3⋊C411yescharacteristic
458C23C2×Dic3⋊C411yescharacteristic
459C23C2×C4⋊Dic311yescharacteristic
460C23C2×(C2×D3)⋊C411yescharacteristic
461C23C2×(C2×D3)⋊C411yescharacteristic
462C23D4×Dic312yes—
463C23D4×Dic312yes—
464C23#93512yes—
465C23#93712yes—
466C23#93912yes—
467C23#93912yes—
468C23C2×#21711yescharacteristic
469D4—28no—
470D4—28no—
471D4—28no—
472D4—28no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—44no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—124no—
486C2×C4—124no—
487C2×C4—124no—
488C2×C4—244no—
489C2×C4—244no—
490C2×C4—248no—
491C2×C4—244no—
492C2×C4—244no—
493C2×C4—244no—
494C2×C4C3⋊#9514yes—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—22no—
505C2×C3—44no—
506C2×C3C22⋊#9511yescharacteristic
507C2×C3#182812yes—
508C22—24no—
509C22—22no—
510C22—24no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22Dic3×C22⋊C411yescharacteristic
540C22Dic3×C22⋊C411yescharacteristic
541C22#494211yescharacteristic
542C22#494211yescharacteristic
543C22#494311yescharacteristic
544C22#494612yes—
545C22#495012yes—
546C22#495511yescharacteristic
547C22#495812yes—
548C22C2×C3⋊#9511yescharacteristic
549C22#522012yes—
550C4—24no—
551C4—84no—
552C4—124no—
553C4—124no—
554C4—244no—
555C3#1076811yescharacteristic, complemented, Sylow
556C2—22no—
557C2—21no—
558C2—22no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—22no—
565C2—22no—
566C2—44no—
567C2#7002012yes—
568C2#7154411yescharacteristic
subgroupG/H
centerC22#4943
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC22×C3×#99C2
socleC22×C3#381
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenterC22×#99D3
hypocenterC3#10768
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
4. C2×#4527C2semidirect
5. C2×#4526C2semidirect
9. C22×#217C4semidirect
17. C2×#926C4semidirect
20. C2×C3×#99C4semidirect
23. C2×#834C4semidirect
54. C2×C3×C22⋊C4C2×C4semidirect
69. C2×#217C2×C4semidirect
96. C2×#99Dic3semidirect
220. C2×C22⋊C4C2×Dic3semidirect
555. C3#10768semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
C22×C3×#992
C22×#8342
C2×#45272
C2×#45262
#107683