Finite Group Databasebeta

← #10943821

Subgroups — #10943821

Order 768.

3876 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic, complemented
3C2×#5170C214yesmaximal, complemented
4#82723C211yesmaximal, characteristic
5C2×#5040C211yesmaximal, characteristic, complemented
6#72132C211yesmaximal, characteristic, complemented
7#70266C214yesmaximal, complemented
8#69963C211yesmaximal, characteristic, complemented
9#69955C211yesmaximal, characteristic, complemented
10#32879—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×D3×Q8—24no—
13C22×D3×Q8—216no—
14C22×D3×Q8—22no—
15C22×D3×Q8—24no—
16C22×D3×Q8C2214yes—
17C23×C4×D3—21no—
18C23×C4×D3C2211yescharacteristic
19C23×Dic6—21no—
20C23×Dic6C2211yescharacteristic
21C22×C3×SD16C2212yescomplemented
22C2×C3×#121C2211yescharacteristic
23C2×#938C2212yes—
24C2×#937—21no—
25C22×#214C2212yescomplemented
26C3×#448—24no—
27C2×C3×#103C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#835C2214yescomplemented
30#5220—21no—
31#5212C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#5170—216no—
34C2×(C2×D3)⋊C8C2214yes—
35C2×#816C2214yescomplemented
36#5040—24no—
37#4987C2212yes—
38C2×#810C2212yescomplemented
39#4956—22no—
40C24×Q8—31no—
41C2×#448—34no—
42C2×#448—31no—
43C2×#448D311yescharacteristic, complemented
44#2847—31no—
45#1787—34no—
46#1787—31no—
47#1787—31no—
48#1779—31no—
49C24×D3D411yescharacteristic
50C22×C3×Q8—28no—
51C22×C3×Q8—22no—
52C22×C3×Q8—21no—
53C22×C3×Q8D414yescomplemented
54C22×C3×Q8C2312yes—
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60C23×Dic3—21no—
61C23×Dic3D411yescharacteristic
62C2×D3×Q8—264no—
63C2×D3×Q8—216no—
64C2×D3×Q8—216no—
65C2×D3×Q8—264no—
66C2×D3×Q8—216no—
67C2×D3×Q8—28no—
68C2×D3×Q8—28no—
69C2×D3×Q8—28no—
70C2×D3×Q8—28no—
71C2×D3×Q8—28no—
72C2×D3×Q8D4116yescomplemented
73C22×C4×D3—28no—
74C22×C4×D3—216no—
75C22×C4×D3—24no—
76C22×C4×D3—22no—
77C22×C4×D3—22no—
78C22×C4×D3—28no—
79C22×C4×D3D412yes—
80C22×C4×D3C2312yes—
81C22×Dic6—28no—
82C22×Dic6—216no—
83C22×Dic6—21no—
84C22×Dic6—22no—
85C22×Dic6—24no—
86C22×Dic6—28no—
87C22×Dic6D414yescomplemented
88C22×Dic6C2312yes—
89C2×C3×SD16—28no—
90C2×C3×SD16—24no—
91C22×C3×C8C2312yes—
92C3×#121—22no—
93C3×#121—22no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217—22no—
98#938—28no—
99#937—44no—
100C2×#214—28no—
101C2×#214—24no—
102C2×(C2×D3)⋊C4—22no—
103C2×C4⋊Dic3C2312yes—
104C22×C3⋊C8C2312yes—
105C3×#103—24no—
106C3×C22⋊C8—22no—
107#835—28no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#830—22no—
111(C2×D3)⋊C8—28no—
112#816—28no—
113#810—24no—
114C23×Q8—34no—
115C23×Q8—31no—
116C23×Q8—61no—
117C23×Q8—62no—
118C23×Q8—616no—
119C23×Q8—64no—
120C23×Q8—64no—
121C23×Q8C2×D311yescharacteristic
122C24×C4—31no—
123C24×C4—61no—
124C22×SD16—32no—
125C22×SD16C2×D312yescomplemented
126C2×#121—32no—
127C2×#121C2×D311yescharacteristic
128C2×#120—61no—
129#448—24no—
130#448—64no—
131#448—616no—
132C2×#103—34no—
133C2×#103—34no—
134C2×#103—32no—
135C2×#103C2×D312yescomplemented
136C2×C22⋊C8—31no—
137C2×C22⋊C8—34no—
138C2×C22⋊C8C2×D311yescharacteristic
139#386—62no—
140#386—61no—
141#380—31no—
142#380—32no—
143C24×C3—21no—
144C24×C3C2×D411yescharacteristic
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—24no—
148C23×D3—24no—
149C23×D3SD1614yescomplemented
150C23×D3C2×D412yes—
151C2×C3×Q8—216no—
152C2×C3×Q8—28no—
153C2×C3×Q8—216no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—24no—
157C2×C3×Q8C2×D418yes—
158C2×C3×Q8C2×D414yes—
159C2×C3×D4—21no—
160C2×C3×D4—24no—
161C2×C3×D4—21no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C22×C3×C4—28no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2411yescharacteristic
174C2×#67—44no—
175C2×#67—44no—
176C2×#67—44no—
177C22×Dic3—28no—
178C22×Dic3—22no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3—22no—
183C22×Dic3C2×D412yes—
184D3×Q8—2128no—
185D3×Q8—232no—
186D3×Q8—232no—
187D3×Q8—232no—
188D3×Q8—232no—
189C2×C4×D3—264no—
190C2×C4×D3—216no—
191C2×C4×D3—216no—
192C2×C4×D3—216no—
193C2×C4×D3—216no—
194C2×C4×D3—216no—
195C2×C4×D3—24no—
196C2×C4×D3—24no—
197C2×C4×D3—22no—
198C2×C4×D3—24no—
199C2×C4×D3—22no—
200C2×C4×D3—24no—
201C2×C4×D3C2×D418yes—
202C2×Dic6—264no—
203C2×Dic6—216no—
204C2×Dic6—216no—
205C2×Dic6—28no—
206C2×Dic6—216no—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C2×Dic6—22no—
210C2×Dic6—216no—
211C2×Dic6—216no—
212C2×Dic6C2×D414yes—
213C2×Dic6C2×D418yes—
214C3×SD16—48no—
215C2×C3×C8—24no—
216C2×C3×C8—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C22⋊C4—42no—
220#217—42no—
221#217—44no—
222#214—48no—
223(C2×D3)⋊C4—44no—
224C4⋊Dic3—24no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—24no—
227C25—31no—
228C22×Q8—28no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—316no—
232C22×Q8—34no—
233C22×Q8—32no—
234C22×Q8—64no—
235C22×Q8—68no—
236C22×Q8—616no—
237C22×Q8—68no—
238C22×Q8—68no—
239C22×Q8—68no—
240C22×Q8—68no—
241C22×Q8—616no—
242C22×Q8—616no—
243C22×Q8—664no—
244C22×Q8—664no—
245C22×Q8—616no—
246C22×Q8—68no—
247C22×Q8—68no—
248C22×Q8—61no—
249C22×Q8—62no—
250C22×Q8#6714yescomplemented
251C22×Q8C22×D312yes—
252C22×D4—21no—
253C22×D4—62no—
254C22×D4C22×D311yescharacteristic
255C23×C4—21no—
256C23×C4—32no—
257C23×C4—31no—
258C23×C4—32no—
259C23×C4—68no—
260C23×C4—64no—
261C23×C4—616no—
262C23×C4—68no—
263C23×C4—61no—
264C23×C4—62no—
265C23×C4—62no—
266C23×C4C22×D311yescharacteristic
267C2×SD16—28no—
268C2×SD16—24no—
269C2×SD16—64no—
270C2×SD16—68no—
271C22×C8—32no—
272C22×C8C22×D312yes—
273#121—22no—
274#121—22no—
275#121—68no—
276#120—124no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—61no—
283#103—24no—
284#103—64no—
285#103—68no—
286#103—68no—
287C22⋊C8—22no—
288C22⋊C8—68no—
289C22⋊C8—62no—
290#95—62no—
291#95—61no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3#12012yes—
300C23×C3C2×SD1612yes—
301C23×C3#13712yes—
302C23×C3C22×D411yescharacteristic
303C22×D3—28no—
304C22×D3—216no—
305C22×D3—28no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—28no—
311C22×D3—216no—
312C22×D3#12014yes—
313C22×D3C2×SD1614yes—
314C22×D3#13714yes—
315C3×Q8—216no—
316C3×Q8—24no—
317C3×Q8—28no—
318C3×Q8#12018yescomplemented
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—84no—
322C2×C3×C4—216no—
323C2×C3×C4—24no—
324C2×C3×C4—28no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4#12012yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D414yes—
334C2×C3×C4C22×D411yescharacteristic
335#67—88no—
336C2×Dic3—28no—
337C2×Dic3—216no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—28no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3#12014yes—
346C4×D3—216no—
347C4×D3—264no—
348C4×D3—28no—
349C4×D3—28no—
350Dic6—264no—
351Dic6—216no—
352Dic6—28no—
353Dic6—24no—
354Dic6#12018yescomplemented
355C3×C8—42no—
356C3⋊C8—42no—
357C24—21no—
358C24—32no—
359C24—34no—
360C24—64no—
361C24—64no—
362C24—62no—
363C24—62no—
364C24D3×D411yescharacteristic
365C2×Q8—216no—
366C2×Q8—28no—
367C2×Q8—216no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—24no—
371C2×Q8—34no—
372C2×Q8—38no—
373C2×Q8—616no—
374C2×Q8—616no—
375C2×Q8—664no—
376C2×Q8—616no—
377C2×Q8—616no—
378C2×Q8—68no—
379C2×Q8—616no—
380C2×Q8—632no—
381C2×Q8—6128no—
382C2×Q8—632no—
383C2×Q8—632no—
384C2×Q8—632no—
385C2×Q8—64no—
386C2×Q8—64no—
387C2×Q8—62no—
388C2×Q8D3×D418yes—
389C2×Q8C2×#6714yes—
390C2×D4—21no—
391C2×D4—24no—
392C2×D4—21no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—124no—
397C2×D4—124no—
398C2×D4—124no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—32no—
405C22×C4—38no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—616no—
409C22×C4—664no—
410C22×C4—616no—
411C22×C4—616no—
412C22×C4—616no—
413C22×C4—616no—
414C22×C4—64no—
415C22×C4—68no—
416C22×C4—62no—
417C22×C4—64no—
418C22×C4—62no—
419C22×C4—64no—
420C22×C4—62no—
421C22×C4—64no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D412yes—
426C22×C4C2×#6712yes—
427C22×C4C23×D311yescharacteristic
428SD16—48no—
429SD16—128no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—64no—
433C2×C8—62no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—64no—
437C22⋊C4—42no—
438C22⋊C4—124no—
439C22⋊C4—124no—
440C22⋊C4—122no—
441C22×C3—22no—
442C22×C3—24no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3—81no—
446C22×C3—82no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—82no—
450C22×C3#44814yescomplemented
451C22×C3C2×#12011yescharacteristic
452C22×C3C2×#12012yes—
453C22×C3C22×SD1611yescharacteristic
454C22×C3C2×#13711yescharacteristic
455C22×C3C2×#13712yes—
456C2×D3—232no—
457C2×D3—24no—
458C2×D3—24no—
459C2×D3—24no—
460C2×D3—24no—
461C2×D3—24no—
462C2×D3—24no—
463C2×D3#44818yescomplemented
464C3×C4—28no—
465C3×C4—22no—
466C3×C4—81no—
467C3×C4C2×#12012yes—
468C3×C4C2×#12012yes—
469Dic3—24no—
470Dic3—28no—
471Dic3—82no—
472C23—22no—
473C23—22no—
474C23—34no—
475C23—34no—
476C23—34no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—616no—
483C23—68no—
484C23—68no—
485C23—616no—
486C23—68no—
487C23—62no—
488C23—62no—
489C23—62no—
490C23—62no—
491C23D3×SD1612yes—
492C23#91512yes—
493C23#93712yes—
494C23C2×D3×D411yescharacteristic
495Q8—216no—
496Q8—24no—
497Q8—28no—
498Q8—38no—
499Q8—664no—
500Q8—616no—
501Q8—68no—
502Q8—64no—
503Q8#93718yescomplemented
504D4—42no—
505D4—42no—
506D4—84no—
507D4—248no—
508C2×C4—216no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—34no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—664no—
519C2×C4—616no—
520C2×C4—68no—
521C2×C4—616no—
522C2×C4—62no—
523C2×C4—64no—
524C2×C4—62no—
525C2×C4—68no—
526C2×C4—68no—
527C2×C4—68no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#93712yes—
532C2×C4C2×D3×D414yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C8—42no—
536C8—122no—
537C2×C3—24no—
538C2×C3—161no—
539C2×C3#284611yescharacteristic
540C2×C3C2×#44811yescharacteristic
541C2×C3C2×#44812yes—
542C2×C3#300411yescharacteristic
543C2×C3#300412yes—
544D3—28no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—22no—
549C22—38no—
550C22—632no—
551C22—64no—
552C22—64no—
553C22—64no—
554C22—64no—
555C22—64no—
556C22—64no—
557C22—81no—
558C22—81no—
559C22—82no—
560C22—82no—
561C22—81no—
562C22#517014yescomplemented
563C22D3×#12012yes—
564C22C2×D3×SD1611yescharacteristic
565C22C2×#91511yescharacteristic
566C22D3×#13712yes—
567C22C2×#93711yescharacteristic
568C4—28no—
569C4—22no—
570C4—64no—
571C4—68no—
572C4—81no—
573C4—242no—
574C4D3×#12012yes—
575C4C2×#93712yes—
576C3#3287911yescharacteristic, complemented, Sylow
577C2—24no—
578C2—68no—
579C2—161no—
580C2D3×#44812yes—
581C2#8003912yes—
582C2#8261211yescharacteristic
583C2C2×#517011yescharacteristic
584C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radical#82723C2
hypercenterC2×#448D3
hypocenterC3#32879
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C23×D3×Q8C2semidirect
2. C2×C3×#448C2semidirect
3. C2×#5170C2semidirect
5. C2×#5040C2semidirect
6. #72132C2semidirect
7. #70266C2semidirect
8. #69963C2semidirect
9. #69955C2semidirect
21. C22×C3×SD16C22semidirect
25. C22×#214C22semidirect
27. C2×C3×#103C22semidirect
29. C2×#835C22semidirect
35. C2×#816C22semidirect
38. C2×#810C22semidirect
43. C2×#448D3semidirect
53. C22×C3×Q8D4semidirect
72. C2×D3×Q8D4semidirect
87. C22×Dic6D4semidirect
125. C22×SD16C2×D3semidirect
135. C2×#103C2×D3semidirect
149. C23×D3SD16semidirect
250. C22×Q8#67semidirect
318. C3×Q8#120semidirect
354. Dic6#120semidirect
450. C22×C3#448semidirect
463. C2×D3#448semidirect
503. Q8#937semidirect
562. C22#5170semidirect
576. C3#32879semidirect

Maximal subgroups

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

subgroupindex
C23×D3×Q82
C2×C3×#4482
C2×#51702
#827232
C2×#50402
#721322
#702662
#699632
#699552
#328793