Finite Group Databasebeta

← #10796782

Subgroups — #10796782

Order 768.

1710 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5675C211yesmaximal, characteristic
2#85856C211yesmaximal, characteristic
3D3×#419C212yesmaximal, complemented
4C2×#4745C211yesmaximal, characteristic
5C3×#1885C211yesmaximal, characteristic, complemented
6#71723C211yesmaximal, characteristic, complemented
7#71649C211yesmaximal, characteristic, complemented
8#70345C212yesmaximal, complemented
9#70069C211yesmaximal, characteristic, complemented
10#68765C212yesmaximal, complemented
11#68753C212yesmaximal, complemented
12#20955—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×#237C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C3×#518C2211yescharacteristic
17C2×#944C2211yescharacteristic
18#5798C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×D3×M16C2212yes—
21#5675—24no—
22#5675C418yescomplemented
23#5546—24no—
24C4×#237—24no—
25D3×#117C2212yes—
26#5516C2212yes—
27C2×#883—22no—
28#5486—22no—
29C2×#878—22no—
30C2×#877—22no—
31C2×#872C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C3×#419C2212yes—
34C2×C3×#100C2211yescharacteristic
35#5225C2212yes—
36C2×#833C2211yescharacteristic
37C2×#824C2212yes—
38#5120C2212yes—
39#5004C2212yes—
40#4802—24no—
41D3×#103—24no—
42#4745—24no—
43C3×#339C2212yes—
44#4555C2212yes—
45#4532C2212yes—
46#4523C2212yes—
47C2×#536—31no—
48#3428—31no—
49C2×#419—32no—
50C2×#413—31no—
51#1886—32no—
52#1886—32no—
53#1885—31no—
54#1885—31no—
55#1885—31no—
56#1885D311yescharacteristic, complemented
57#1733—32no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C22×#67D412yes—
61C23×Dic3D412yes—
62C2×D3×Q8—22no—
63C2×D3×Q8—22no—
64C2×D3×Q8—22no—
65C2×D3×Q8—24no—
66C2×D3×Q8C2×C414yescomplemented
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—24no—
70C2×#237C2×C414yes—
71C2×#237D414yes—
72C22×C4×D3—21no—
73C22×C4×D3C2311yescharacteristic
74C22×Dic6—21no—
75C22×Dic6C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C2×C3×M16C2312yes—
78C3×#124—22no—
79C3×#124C2×C414yescomplemented
80C3×C4×D4—24no—
81C3×#117C2312yes—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—22no—
86C2×#217—22no—
87#944—22no—
88#944C2×C414yescomplemented
89#936—22no—
90#936C2×C414yescomplemented
91D4×Dic3—24no—
92C4×#67—28no—
93C2×(C2×D3)⋊C4—21no—
94#926C2312yes—
95C2×C4⋊Dic3—22no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—24no—
100C2×C4×Dic3C2311yescharacteristic
101C2×#208C2312yes—
102D3×M16—24no—
103D3×M16—24no—
104C2×ΓL(1,𝔽25)—22no—
105C2×D3×C8—22no—
106#892D414yes—
107D3×C4⋊C4D414yes—
108#887—24no—
109#883—216no—
110#67⋊C4—28no—
111D3×C22⋊C4—22no—
112#878—216no—
113#877—24no—
114#877—24no—
115#877—22no—
116#877—24no—
117#872—22no—
118#872—22no—
119#872C2×C414yescomplemented
120C42×D3—22no—
121#869—22no—
122#869C2×C414yescomplemented
123C4×Dic6—24no—
124C3×#103—24no—
125C3×#100—22no—
126#835—24no—
127#833—22no—
128#824—24no—
129#816—24no—
130#810—24no—
131C22×#41—31no—
132C23×Q8—31no—
133C22×M16—32no—
134#536—38no—
135#536—64no—
136C2×C4⋊Q8—31no—
137C2×C4⋊Q8—31no—
138C2×#124—31no—
139C2×#124C2×D311yescharacteristic
140C2×#123—62no—
141C2×C22⋊Q8—62no—
142#520—64no—
143#519—32no—
144#518—31no—
145#518—62no—
146#518C2×D311yescharacteristic
147C4×#41—64no—
148C2×#117—31no—
149C2×#117—32no—
150C2×#117—62no—
151#419—32no—
152#419—32no—
153#419—32no—
154#419—64no—
155#419C2×D312yes—
156C2×#103—64no—
157#413—64no—
158C2×#101—32no—
159C2×#100—31no—
160C2×#100C2×D311yescharacteristic
161#339—32no—
162#339—32no—
163#339—32no—
164#339C2×D312yes—
165C24×C3C2×D412yes—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8C22×C412yes—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4C22×C412yes—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—24no—
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—24no—
181C2×#67C22⋊C418yes—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—24no—
188C22×Dic3C22⋊C418yes—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D411yescharacteristic
191D3×Q8—28no—
192D3×Q8—28no—
193D3×Q8—44no—
194D3×Q8—44no—
195#237—28no—
196#237—28no—
197#237—28no—
198#237—28no—
199C2×C4×D3—24no—
200C2×C4×D3—24no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C4×D3C22×C412yes—
204C2×C4×D3C2×D412yes—
205C2×C4×D3C2×D412yes—
206C2×C4×D3C2×D412yes—
207C2×C4×D3#4114yes—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—21no—
213C2×Dic6—24no—
214C2×Dic6—41no—
215C2×Dic6—41no—
216C2×Dic6—42no—
217C2×Dic6C22×C412yes—
218C2×Dic6C22×C412yes—
219C2×Dic6C2×D412yes—
220C3×M16—22no—
221C3×M16—22no—
222C2×C3×C8—22no—
223C3×C4⋊C4C2×D414yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—28no—
226C3×C22⋊C4—22no—
227C3×C42—22no—
228C3×C42—21no—
229C3×C42C22×C412yes—
230#217—24no—
231#217—28no—
232#217—22no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—24no—
235C4⋊Dic3—28no—
236C4⋊Dic3C2×D414yes—
237Dic3⋊C4—28no—
238Dic3⋊C4—24no—
239Dic3⋊C4—28no—
240Dic3⋊C4—24no—
241C4×Dic3—28no—
242C4×Dic3—22no—
243C4×Dic3—21no—
244C4×Dic3—24no—
245C4×Dic3C22×C412yes—
246#208—22no—
247#208—22no—
248C2×C3⋊C8—22no—
249ΓL(1,𝔽25)—44no—
250D3×C8—44no—
251C2×#41—34no—
252C2×#41—34no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—64no—
256C22×Q8—31no—
257C22×Q8—34no—
258C22×Q8—31no—
259C22×Q8—61no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8—62no—
263C22×Q8—64no—
264C22×Q8C22×D311yescharacteristic
265C22×D4—32no—
266C22×D4C22×D311yescharacteristic
267C23×C4—31no—
268C23×C4—32no—
269C23×C4—61no—
270C2×M16—32no—
271C2×M16—62no—
272C2×M16—64no—
273C2×M16—64no—
274C2×M16C22×D312yes—
275C22×C8—62no—
276C4⋊Q8—34no—
277C4⋊Q8—34no—
278C4⋊Q8—62no—
279C4⋊Q8—62no—
280#124—22no—
281#124—34no—
282#124—62no—
283#124C4×D314yescomplemented
284#123—616no—
285C22⋊Q8—616no—
286C4×Q8—64no—
287C4×Q8—64no—
288C4×D4—24no—
289C4×D4—68no—
290C4×D4—64no—
291C4×D4—68no—
292#117—34no—
293#117—32no—
294#117—34no—
295#117—62no—
296#117—62no—
297#117—62no—
298#117—64no—
299#117—64no—
300#117—64no—
301#117C22×D312yes—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—64no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—62no—
312C2×C22⋊C4—62no—
313C2×C42—31no—
314C2×C42—62no—
315C2×C42—64no—
316C2×C42C22×D311yescharacteristic
317#103—24no—
318#103—64no—
319#103—64no—
320#103—64no—
321#101—64no—
322#100—22no—
323#100—62no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—44no—
327C23×C3C2×C22⋊C414yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—42no—
330C22×D3C2×C22⋊C412yes—
331C22×D3#12012yes—
332C22×D3#12312yes—
333C3×Q8—24no—
334C3×Q8—42no—
335C3×D4—24no—
336C3×D4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—84no—
344C2×C3×C4C23×C411yescharacteristic
345C2×C3×C4C22×D412yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348#67—216no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—28no—
352C2×Dic3—28no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—84no—
360C2×Dic3C2×C22⋊C414yes—
361C2×Dic3C2×C22⋊C412yes—
362C2×Dic3#12011yescharacteristic
363C2×Dic3#12011yescharacteristic
364C2×Dic3#12312yes—
365C4×D3—48no—
366C4×D3—84no—
367C4×D3C4×D414yes—
368C4×D3C4×D414yes—
369C4×D3#12114yes—
370C4×D3#12114yes—
371Dic6—24no—
372Dic6—24no—
373Dic6—24no—
374Dic6—42no—
375Dic6—44no—
376Dic6—44no—
377C3×C8—42no—
378C3⋊C8—42no—
379C24—31no—
380C24D3×D412yes—
381#41—68no—
382#41—68no—
383#41—68no—
384#41—68no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—32no—
389C2×Q8—32no—
390C2×Q8—32no—
391C2×Q8—64no—
392C2×Q8—68no—
393C2×Q8—68no—
394C2×Q8—62no—
395C2×Q8—61no—
396C2×Q8—62no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8—122no—
400C2×Q8—121no—
401C2×Q8—121no—
402C2×Q8—124no—
403C2×Q8—124no—
404C2×Q8C2×C4×D312yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—38no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4C2×C4×D312yes—
411C2×D4D3×D412yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—31no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—34no—
421C22×C4—38no—
422C22×C4—32no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—64no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—64no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4C23×D311yescharacteristic
434M16—22no—
435M16—22no—
436M16—62no—
437M16—62no—
438M16—124no—
439C2×C8—22no—
440C2×C8—62no—
441C2×C8—124no—
442C4⋊C4—34no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—68no—
446C4⋊C4—68no—
447C4⋊C4—68no—
448C4⋊C4D3×D414yes—
449C22⋊C4—24no—
450C22⋊C4—28no—
451C22⋊C4—22no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—68no—
456C22⋊C4—62no—
457C42—22no—
458C42—21no—
459C42—32no—
460C42—64no—
461C42—68no—
462C42—61no—
463C42—62no—
464C42C2×C4×D312yes—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—82no—
470C22×C3C22×C22⋊C411yescharacteristic
471C22×C3C2×#12011yescharacteristic
472C22×C3C2×#12311yescharacteristic
473C2×D3—24no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3#38614yes—
477C3×C4—42no—
478C3×C4—44no—
479C3×C4—42no—
480C3×C4C2×C4×D412yes—
481C3×C4C2×#12112yes—
482Dic3—24no—
483Dic3—42no—
484Dic3—44no—
485Dic3—42no—
486Dic3#38612yes—
487Dic3#38612yes—
488C23—22no—
489C23—22no—
490C23—32no—
491C23—32no—
492C23—32no—
493C23—44no—
494C23—122no—
495C23D3×C22⋊C414yes—
496C23C2×D3×D411yescharacteristic
497Q8—24no—
498Q8—42no—
499Q8—64no—
500Q8—64no—
501Q8—64no—
502Q8—124no—
503Q8—124no—
504Q8—122no—
505D4—24no—
506D4—24no—
507D4—616no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—32no—
512C2×C4—32no—
513C2×C4—31no—
514C2×C4—31no—
515C2×C4—34no—
516C2×C4—34no—
517C2×C4—34no—
518C2×C4—34no—
519C2×C4—34no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—68no—
524C2×C4—68no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—64no—
530C2×C4—84no—
531C2×C4—128no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—244no—
536C2×C4—244no—
537C2×C4C22×C4×D311yescharacteristic
538C2×C4C2×D3×D412yes—
539C2×C4C2×D3×D411yescharacteristic
540C2×C4D3×#4112yes—
541C8—42no—
542C8—122no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—44no—
546C2×C3C2×#38611yescharacteristic
547C2×C3#302212yes—
548D3—24no—
549C22—22no—
550C22—24no—
551C22—34no—
552C22—44no—
553C22—44no—
554C22—64no—
555C22—82no—
556C22—122no—
557C22—122no—
558C22C2×D3×C22⋊C411yescharacteristic
559C22D3×#12011yescharacteristic
560C22D3×#12311yescharacteristic
561C4—32no—
562C4—32no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—64no—
567C4—122no—
568C4—122no—
569C4—124no—
570C4C4×D3×D412yes—
571C4D3×#12112yes—
572C3#2095511yescharacteristic, complemented, Sylow
573C2—21no—
574C2—21no—
575C2—44no—
576C2—64no—
577C2D3×#38611yescharacteristic
578C2#8029312yes—
subgroupG/H
centerC22D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1885C2
socleC22×C3C2×#120
prime radicalC22×#237C22
cosocleC22×C3×C4C24
hypercenter#1885D3
hypocenterC3#20955
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1885

Complements

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

NH ≅ G/Ntype
3. D3×#419C2semidirect
5. C3×#1885C2semidirect
6. #71723C2semidirect
7. #71649C2semidirect
8. #70345C2semidirect
9. #70069C2semidirect
10. #68765C2semidirect
11. #68753C2semidirect
22. #5675C4semidirect
56. #1885D3semidirect
66. C2×D3×Q8C2×C4semidirect
79. C3×#124C2×C4semidirect
88. #944C2×C4semidirect
90. #936C2×C4semidirect
119. #872C2×C4semidirect
122. #869C2×C4semidirect
283. #124C4×D3semidirect
572. C3#20955semidirect

Maximal subgroups

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

subgroupindex
C2×#56752
#858562
D3×#4192
C2×#47452
C3×#18852
#717232
#716492
#703452
#700692
#687652
#687532
#209553