Finite Group Databasebeta

← #10822686

Subgroups — #10822686

Order 768.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3C2×#5225C211yesmaximal, characteristic, complemented
4C2×#5126C211yesmaximal, characteristic
5#82267C211yesmaximal, characteristic
6C2×#4995C211yesmaximal, characteristic, complemented
7#71985C211yesmaximal, characteristic, complemented
8#70275C214yesmaximal, complemented
9#70274C214yesmaximal, complemented
10#23481—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#946—24no—
16C2×#946—22no—
17C2×#946—24no—
18C2×#946C2214yes—
19C22×(C2×D3)⋊C4—21no—
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4—21no—
23C22×#208C2211yescharacteristic
24C3×#419—24no—
25C2×C3×#103C2212yescomplemented
26C2×C3×#103C2212yescomplemented
27#5225—24no—
28C2×#835C2214yescomplemented
29#5126—24no—
30C2×(C2×D3)⋊C8C2214yes—
31#4995—24no—
32#4987C2212yes—
33#4987C2212yes—
34C2×#808C2212yescomplemented
35C2×#807C2212yescomplemented
36#4955—21no—
37#4943—21no—
38C4×(C2×D3)⋊C4—22no—
39#4551C2214yescomplemented
40C22×C22⋊Q8—31no—
41C2×#428—31no—
42C2×#419—31no—
43C2×#419D311yescharacteristic, complemented
44C2×#407—31no—
45#2280—31no—
46#2036—34no—
47#2035—34no—
48#1736—31no—
49C24×D3D411yescharacteristic
50C22×C3×Q8—22no—
51C22×C3×Q8—21no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8D414yes—
57C22×C3×Q8C2312yes—
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C23×Dic3—21no—
61C23×Dic3D411yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3D412yes—
69C22×C4×D3C2312yes—
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117—22no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—21no—
80C2×#217—21no—
81#946—216no—
82#946—28no—
83#946—28no—
84C2×(C2×D3)⋊C4—28no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88#926—22no—
89#926—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3D414yes—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4—28no—
96C2×Dic3⋊C4—24no—
97C2×C4×Dic3—21no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208D414yes—
101C22×C3⋊C8C2312yes—
102C3×#103—24no—
103C3×#103—24no—
104#835—28no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107(C2×D3)⋊C8—28no—
108#808—24no—
109#807—24no—
110C23×Q8C2×D311yescharacteristic
111C24×C4—31no—
112C22×M16—31no—
113C22×M16C2×D311yescharacteristic
114C2×C22⋊Q8—34no—
115C2×C22⋊Q8—64no—
116C2×C22⋊Q8—62no—
117C2×C22⋊Q8—64no—
118C2×#117—31no—
119C2×#117C2×D311yescharacteristic
120C22×C4⋊C4—31no—
121C22×C4⋊C4—61no—
122C22×C22⋊C4—61no—
123#428—64no—
124C2×#107—32no—
125C2×#106—32no—
126#419—24no—
127#419—64no—
128C2×#103—34no—
129C2×#103C2×D312yescomplemented
130C2×#103C2×D312yescomplemented
131#407—64no—
132C2×C22⋊C8—34no—
133#381—61no—
134#380—32no—
135#380—32no—
136#379—61no—
137C4×C22⋊C4—62no—
138#340—34no—
139C24×C3C2×D411yescharacteristic
140C23×D3—22no—
141C23×D3—24no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—42no—
145C23×D3C2×D412yes—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—28no—
149C2×C3×Q8—28no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—42no—
155C2×C3×Q8—44no—
156C2×C3×Q8—42no—
157C2×C3×Q8—44no—
158C2×C3×Q8C2×D414yes—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—24no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—44no—
180C22×Dic3C2×D412yes—
181C2×C4×D3—24no—
182C2×C4×D3—24no—
183C2×C4×D3—24no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—44no—
191C2×C4×D3—44no—
192C2×C4×D3—44no—
193C2×C4×D3—44no—
194C3×M16—44no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C22⋊C4—42no—
203C3×C42—42no—
204#217—42no—
205(C2×D3)⋊C4—216no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—44no—
208C4⋊Dic3—24no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3C2×D414yes—
213Dic3⋊C4—216no—
214C4×Dic3—42no—
215#208—28no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8C2×D414yes—
219C25—31no—
220C22×Q8—22no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8#6714yes—
227C22×Q8C22×D312yes—
228C23×C4—21no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—32no—
232C23×C4—61no—
233C23×C4—62no—
234C23×C4—62no—
235C23×C4—62no—
236C23×C4—62no—
237C23×C4—62no—
238C23×C4—62no—
239C23×C4C22×D311yescharacteristic
240C2×M16—22no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—34no—
244C2×M16—62no—
245C2×M16—62no—
246C22×C8—32no—
247C22×C8C22×D312yes—
248C22⋊Q8—68no—
249C22⋊Q8—616no—
250C22⋊Q8—68no—
251#117—22no—
252#117—22no—
253#117—62no—
254#117—62no—
255C2×C4⋊C4—34no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—68no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4C22×D311yescharacteristic
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—68no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C42—21no—
271C2×C42—61no—
272#107—64no—
273#106—64no—
274#103—24no—
275#103—24no—
276#103—68no—
277C22⋊C8—68no—
278#95—62no—
279#95—62no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3#12012yes—
283C23×C3#12112yes—
284C23×C3#12312yes—
285C23×C3C22×D411yescharacteristic
286C22×D3—28no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—48no—
294C22×D3—48no—
295C22×D3—44no—
296C22×D3—44no—
297C22×D3#13714yes—
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—42no—
301C3×Q8—42no—
302C3×Q8—48no—
303C2×C3×C4—24no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—48no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12114yes—
313C2×C3×C4#12112yes—
314C2×C3×C4#12114yes—
315C2×C3×C4#12312yes—
316C2×C3×C4#12314yes—
317C2×C3×C4#12414yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—24no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—48no—
329C2×Dic3—44no—
330C2×Dic3—44no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—42no—
337C4×D3—42no—
338C4×D3—48no—
339C3×C8—42no—
340C3⋊C8—24no—
341C24—32no—
342C24—64no—
343C24—62no—
344C24—62no—
345C24—62no—
346C24—122no—
347C24D3×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8C2×#6714yes—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—32no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4—64no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—124no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4—124no—
387C22×C4—124no—
388C22×C4#23512yes—
389C22×C4#23512yes—
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D412yes—
392C22×C4D3×D412yes—
393C22×C4#23712yes—
394C22×C4C2×#6712yes—
395C22×C4C23×D311yescharacteristic
396M16—44no—
397M16—68no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—34no—
402C2×C8—62no—
403C2×C8—62no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—34no—
409C4⋊C4—616no—
410C4⋊C4—64no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C22⋊C4—42no—
415C22⋊C4—616no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—122no—
419C42—42no—
420C42—122no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—24no—
425C22×C3#39414yes—
426C22×C3C2×#12011yescharacteristic
427C22×C3C2×#12111yescharacteristic
428C22×C3C2×#12311yescharacteristic
429C22×C3C2×#13712yes—
430C22×C3#57512yes—
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—24no—
434C2×D3—416no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C3×C4—44no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4#39414yes—
443C3×C4C2×#12111yescharacteristic
444C3×C4C2×#12111yescharacteristic
445C3×C4C2×#12311yescharacteristic
446C3×C4C2×#12411yescharacteristic
447Dic3—42no—
448Dic3—44no—
449Dic3—82no—
450C23—22no—
451C23—22no—
452C23—34no—
453C23—68no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—128no—
461C23—128no—
462C23—124no—
463C23—124no—
464C23#88312yes—
465C23#88412yes—
466C23#93712yes—
467C23C2×D3×D411yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473C2×C4—24no—
474C2×C4—48no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—64no—
484C2×C4—62no—
485C2×C4—128no—
486C2×C4—124no—
487C2×C4—124no—
488C2×C4—128no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4#88312yes—
497C2×C4#88412yes—
498C2×C4#88414yes—
499C2×C4#88514yes—
500C2×C4#92914yes—
501C2×C4#93712yes—
502C2×C4#93914yes—
503C2×C4C2×#23511yescharacteristic
504C2×C4C2×#23511yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4C2×D3×D411yescharacteristic
507C2×C4C2×#23711yescharacteristic
508C2×C4C22×#6711yescharacteristic
509C8—42no—
510C8—64no—
511C2×C3—24no—
512C2×C3C2×#39411yescharacteristic
513C2×C3#300411yescharacteristic
514C2×C3#300511yescharacteristic
515C2×C3#305612yes—
516C2×C3#309512yes—
517D3—44no—
518C22—22no—
519C22—22no—
520C22—24no—
521C22—24no—
522C22—64no—
523C22—64no—
524C22—64no—
525C22—1216no—
526C22—122no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22#495914yes—
531C22C2×#88311yescharacteristic
532C22C2×#88411yescharacteristic
533C22D3×#13712yes—
534C22#577712yes—
535C22C2×#93711yescharacteristic
536C4—44no—
537C4—81no—
538C4—81no—
539C4—124no—
540C4—122no—
541C4—242no—
542C4#495914yes—
543C4C2×#88411yescharacteristic
544C4C2×#88511yescharacteristic
545C4C2×#92911yescharacteristic
546C4C2×#93911yescharacteristic
547C3#2348111yescharacteristic, complemented, Sylow
548C2—24no—
549C2—124no—
550C2#7896212yes—
551C2#7896512yes—
552C2C2×#495911yescharacteristic
553C2#8397911yescharacteristic
554C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenterC2×#419D3
hypocenterC3#23481
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
3. C2×#5225C2semidirect
6. C2×#4995C2semidirect
7. #71985C2semidirect
8. #70275C2semidirect
9. #70274C2semidirect
25. C2×C3×#103C22semidirect
26. C2×C3×#103C22semidirect
28. C2×#835C22semidirect
34. C2×#808C22semidirect
35. C2×#807C22semidirect
39. #4551C22semidirect
43. C2×#419D3semidirect
129. C2×#103C2×D3semidirect
130. C2×#103C2×D3semidirect
547. C3#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C2×C3×#4192
C2×#52252
C2×#51262
#822672
C2×#49952
#719852
#702752
#702742
#234813