Finite Group Databasebeta

← #10943824

Subgroups — #10943824

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×#5227C211yesmaximal, characteristic, complemented
3C2×#4806C214yesmaximal, complemented
4C2×#4738C211yesmaximal, characteristic, complemented
5#74048C211yesmaximal, characteristic
6C3×#1787C211yesmaximal, characteristic, complemented
7#70266C214yesmaximal, complemented
8#69955C211yesmaximal, characteristic, complemented
9#69525C211yesmaximal, 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×#215C2212yescomplemented
22C2×#930C2211yescharacteristic
23C22×#204C2212yescomplemented
24C2×#895C2212yes—
25C2×#882—21no—
26C2×C3×#103C2214yescomplemented
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5227—24no—
30C2×#835C2212yescomplemented
31C2×#830C2211yescharacteristic
32C2×(C2×D3)⋊C8C2214yes—
33C2×#816C2212yescomplemented
34#4987C2212yes—
35C2×#810C2214yescomplemented
36#4958—21no—
37#4806—216no—
38#4738—24no—
39#4665—22no—
40C24×Q8—31no—
41C2×#448—34no—
42C2×#448—31no—
43C2×#448—31no—
44#2847—31no—
45#1787—34no—
46#1787—31no—
47#1787D311yescharacteristic, complemented
48#1779—31no—
49C24×D3D411yescharacteristic
50C22×C3×Q8—28no—
51C22×C3×Q8—22no—
52C22×C3×Q8—21no—
53C22×C3×Q8D414yescomplemented
54C22×C3×Q8C2312yes—
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C2×D3×Q8—264no—
61C2×D3×Q8—216no—
62C2×D3×Q8—216no—
63C2×D3×Q8—264no—
64C2×D3×Q8—216no—
65C2×D3×Q8—28no—
66C2×D3×Q8—28no—
67C2×D3×Q8—28no—
68C2×D3×Q8—28no—
69C2×D3×Q8—28no—
70C2×D3×Q8D4116yescomplemented
71C22×D12—22no—
72C22×D12C2311yescharacteristic
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—22no—
84C22×Dic6—21no—
85C22×Dic6—24no—
86C22×Dic6—28no—
87C22×Dic6D414yescomplemented
88C22×Dic6C2312yes—
89C22×C3×C8C2312yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—21no—
92C2×#215—24no—
93C2×#215—28no—
94#930—22no—
95#930—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C2×#204—24no—
102C2×#204—28no—
103#895—28no—
104#882—44no—
105C3×#103—28no—
106C3×C22⋊C8—22no—
107C3×#95—21no—
108#835—24no—
109C3⋊#95—22no—
110#830—22no—
111(C2×D3)⋊C8—28no—
112#816—24no—
113#810—28no—
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×SD16—32no—
126C2×#121—31no—
127C2×#121—32no—
128C2×#120—61no—
129#448—64no—
130#448—616no—
131#448—64no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—34no—
135C2×#103C2×D314yescomplemented
136C2×C22⋊C8—31no—
137C2×C22⋊C8—34no—
138C2×C22⋊C8C2×D311yescharacteristic
139#386—62no—
140#386—61no—
141#380—32no—
142#380C2×D311yescharacteristic
143C24×C3C2×D411yescharacteristic
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—24no—
147C23×D3—24no—
148C23×D3—21no—
149C23×D3SD1614yescomplemented
150C23×D3C2×D412yes—
151C2×C3×Q8—216no—
152C2×C3×Q8—28no—
153C2×C3×Q8—216no—
154C2×C3×Q8—24no—
155C2×C3×Q8—24no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D414yes—
158C2×C3×Q8C2×D418yes—
159C22×C3×C4—28no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2411yescharacteristic
168C2×#67—42no—
169C2×#67—42no—
170C2×#67—42no—
171C22×Dic3—28no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—22no—
175C22×Dic3—24no—
176C22×Dic3—21no—
177C22×Dic3C2×D412yes—
178D3×Q8—2128no—
179D3×Q8—232no—
180D3×Q8—232no—
181D3×Q8—232no—
182D3×Q8—232no—
183C2×D12—24no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—44no—
187C2×D12—44no—
188C2×D12—44no—
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—22no—
209C2×Dic6—24no—
210C2×Dic6—216no—
211C2×Dic6—216no—
212C2×Dic6C2×D418yes—
213C2×Dic6C2×D414yes—
214C2×C3×C8—22no—
215C2×C3×C8—24no—
216C3×C4⋊C4—24no—
217C3×C22⋊C4—42no—
218#215—48no—
219(C2×D3)⋊C4—44no—
220(C2×D3)⋊C4—42no—
221(C2×D3)⋊C4—44no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C2×C3⋊C8—24no—
225C2×C3⋊C8—22no—
226#204—48no—
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—62no—
249C22×Q8—61no—
250C22×Q8D1214yescomplemented
251C22×Q8C22×D312yes—
252C22×D4—31no—
253C22×D4—62no—
254C22×D4—61no—
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—64no—
268C2×SD16—68no—
269C2×SD16—64no—
270C2×SD16—68no—
271C22×C8—32no—
272C22×C8C22×D312yes—
273#121—68no—
274#121—62no—
275#121—62no—
276#120—124no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4C22×D312yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—61no—
283#103—28no—
284#103—68no—
285#103—64no—
286#103—64no—
287C22⋊C8—22no—
288C22⋊C8—68no—
289C22⋊C8—62no—
290#95—21no—
291#95—62no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#12012yes—
295C23×C3C2×SD1612yes—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C22×D3—28no—
299C22×D3—216no—
300C22×D3—28no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—28no—
306C22×D3—216no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3#12014yes—
313C22×D3C2×SD1614yes—
314C22×D3#13714yes—
315C3×Q8—216no—
316C3×Q8—24no—
317C3×Q8—28no—
318C3×Q8#12018yescomplemented
319C2×C3×C4—216no—
320C2×C3×C4—24no—
321C2×C3×C4—28no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D414yes—
331C2×C3×C4C22×D411yescharacteristic
332#67—84no—
333C2×Dic3—28no—
334C2×Dic3—216no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—28no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3#12014yes—
343D12—42no—
344D12—42no—
345D12—88no—
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—32no—
358C24—34no—
359C24—64no—
360C24—64no—
361C24—62no—
362C24—62no—
363C24—61no—
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—62no—
387C2×Q8—64no—
388C2×Q8C2×D1214yes—
389C2×Q8D3×D418yes—
390C2×D4—61no—
391C2×D4—64no—
392C2×D4—61no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—124no—
396C2×D4—122no—
397C2×D4—122no—
398C2×D4—122no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
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—62no—
422C22×C4—64no—
423C22×C4—61no—
424C22×C4C2×D1212yes—
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D412yes—
427C22×C4C23×D311yescharacteristic
428SD16—128no—
429SD16—128no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—62no—
433C2×C8—64no—
434C4⋊C4—24no—
435C4⋊C4—61no—
436C4⋊C4—61no—
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#44814yescomplemented
446C22×C3C2×#12011yescharacteristic
447C22×C3C2×#12012yes—
448C22×C3C22×SD1611yescharacteristic
449C22×C3C2×#13711yescharacteristic
450C22×C3C2×#13712yes—
451C2×D3—232no—
452C2×D3—24no—
453C2×D3—24no—
454C2×D3—24no—
455C2×D3—24no—
456C2×D3—24no—
457C2×D3—24no—
458C2×D3—81no—
459C2×D3—82no—
460C2×D3—81no—
461C2×D3—82no—
462C2×D3—81no—
463C2×D3#44818yescomplemented
464C3×C4—28no—
465C3×C4—22no—
466C3×C4—82no—
467C3×C4C2×#12012yes—
468C3×C4C2×#12012yes—
469Dic3—24no—
470Dic3—28no—
471Dic3—81no—
472C23—22no—
473C23—22no—
474C23—34no—
475C23—34no—
476C23—34no—
477C23—616no—
478C23—68no—
479C23—616no—
480C23—68no—
481C23—62no—
482C23—62no—
483C23—62no—
484C23—68no—
485C23—62no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—122no—
490C23—122no—
491C23#88212yes—
492C23D3×SD1612yes—
493C23#91812yes—
494C23C2×D3×D411yescharacteristic
495Q8—216no—
496Q8—24no—
497Q8—28no—
498Q8—38no—
499Q8—664no—
500Q8—616no—
501Q8—68no—
502Q8—64no—
503Q8#88218yescomplemented
504D4—122no—
505D4—122no—
506D4—248no—
507D4—244no—
508C2×C4—216no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—34no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
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—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4#88212yes—
532C2×C4C22×D1211yescharacteristic
533C2×C4C2×D3×D414yes—
534C2×C4C2×D3×D411yescharacteristic
535C8—42no—
536C8—122no—
537C2×C3—24no—
538C2×C3#284611yescharacteristic
539C2×C3C2×#44812yes—
540C2×C3C2×#44811yescharacteristic
541C2×C3#300412yes—
542C2×C3#300411yescharacteristic
543D3—28no—
544D3—161no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—38no—
550C22—632no—
551C22—64no—
552C22—64no—
553C22—64no—
554C22—64no—
555C22—64no—
556C22—64no—
557C22—242no—
558C22—241no—
559C22—241no—
560C22—242no—
561C22—241no—
562C22#480614yescomplemented
563C22C2×#88211yescharacteristic
564C22D3×#12012yes—
565C22C2×D3×SD1611yescharacteristic
566C22C2×#91811yescharacteristic
567C22D3×#13712yes—
568C4—28no—
569C4—22no—
570C4—64no—
571C4—68no—
572C4—82no—
573C4—241no—
574C4C2×#88212yes—
575C4D3×#12012yes—
576C3#3287911yescharacteristic, complemented, Sylow
577C2—24no—
578C2—68no—
579C2—481no—
580C2#7386111yescharacteristic
581C2C2×#480611yescharacteristic
582C2#7895811yescharacteristic
583C2D3×#44812yes—
584C2#8018312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radical#74048C2
hypercenter#1787D3
hypocenterC3#32879
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C23×D3×Q8C2semidirect
2. C2×#5227C2semidirect
3. C2×#4806C2semidirect
4. C2×#4738C2semidirect
6. C3×#1787C2semidirect
7. #70266C2semidirect
8. #69955C2semidirect
9. #69525C2semidirect
21. C22×#215C22semidirect
23. C22×#204C22semidirect
26. C2×C3×#103C22semidirect
30. C2×#835C22semidirect
33. C2×#816C22semidirect
35. C2×#810C22semidirect
47. #1787D3semidirect
53. C22×C3×Q8D4semidirect
70. C2×D3×Q8D4semidirect
87. C22×Dic6D4semidirect
135. C2×#103C2×D3semidirect
149. C23×D3SD16semidirect
250. C22×Q8D12semidirect
318. C3×Q8#120semidirect
354. Dic6#120semidirect
445. C22×C3#448semidirect
463. C2×D3#448semidirect
503. Q8#882semidirect
562. C22#4806semidirect
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×#52272
C2×#48062
C2×#47382
#740482
C3×#17872
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