Finite Group Databasebeta

← #11002192

Subgroups — #11002192

Order 768.

2450 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2C2×#5483C211yesmaximal, characteristic, complemented
3#84047C212yesmaximal
4#84031C212yesmaximal, complemented
5#82874C218yesmaximal, complemented
6#82871C218yesmaximal, complemented
7#82183C212yesmaximal, complemented
8#82178C212yesmaximal, complemented
9#82116C212yesmaximal, complemented
10#82114C212yesmaximal, complemented
11#82093C211yesmaximal, characteristic, complemented
12#36720—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C2×C3×#123—28no—
16C2×C3×#123C2218yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×#217C2212yes—
23C2×#935C2214yes—
24C2×#924C2212yes—
25C2×#924C2212yes—
26C2×#924C2214yescomplemented
27#5483—28no—
28C2×#878C2214yescomplemented
29C2×#878C2214yescomplemented
30#5247C2214yes—
31#5247C2212yes—
32#5222C2218yes—
33#5220C2214yes—
34#5212C2212yes—
35#5212C2214yescomplemented
36#5211C2214yes—
37#5210C2214yes—
38#5210C2214yes—
39#5210C2218yescomplemented
40C4×#217C2212yes—
41#4986C2214yescomplemented
42#4983C2218yescomplemented
43#4982C2218yescomplemented
44#4979C2214yescomplemented
45#4975C2214yescomplemented
46#4973C2214yescomplemented
47#4953C2214yescomplemented
48#4952C2218yescomplemented
49#4951C2218yescomplemented
50#4947C2214yescomplemented
51#4946C2214yescomplemented
52#4945C2218yescomplemented
53#4943C2214yescomplemented
54C2×#543—31no—
55C22×#123D311yescharacteristic
56#2848—32no—
57#2687—38no—
58#2680—38no—
59#2627—32no—
60#2561—32no—
61#2502—32no—
62#2483—32no—
63#2405—32no—
64#2312—31no—
65C25×C3C2311yescharacteristic
66C25×C3C2311yescharacteristic
67C22×C3×D4—24no—
68C22×C3×D4—44no—
69C22×C3×D4C2314yes—
70C23×C3×C4—24no—
71C23×C3×C4—44no—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2314yes—
74C23×C3×C4C2311yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3C2312yes—
79C23×Dic3C2312yes—
80C22×Dic6C2312yes—
81C22×Dic6C2312yes—
82C3×#123—232no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4D418yes—
90C2×C3×C22⋊C4C2314yes—
91C2×C3×C22⋊C4C2314yes—
92C2×C3×C22⋊C4C2314yes—
93C2×#217—28no—
94C2×#217C2318yes—
95C2×#217C2314yes—
96C2×#217C2314yes—
97C2×#217C2314yes—
98C2×#217C2314yes—
99C2×#217C2318yes—
100C2×#217C2314yes—
101#935—48no—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2314yes—
105#924—28no—
106#924—28no—
107#924—48no—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3C2314yes—
113#878—216no—
114#878—216no—
115C3⋊#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2314yes—
119C3⋊#95C2314yes—
120C3⋊#95C2318yes—
121C23×D4C2×D311yescharacteristic
122C24×C4C2×D311yescharacteristic
123#543—68no—
124C2×#123—28no—
125C2×#123—34no—
126C2×#123C2×D318yes—
127C2×C22⋊Q8—32no—
128C2×C22⋊Q8—32no—
129C2×C22⋊Q8—34no—
130C2×C22⋊Q8—34no—
131C2×C22⋊Q8—34no—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C22⋊C4—32no—
135C22×C22⋊C4C2×D311yescharacteristic
136C22×C22⋊C4C2×D311yescharacteristic
137C22×C22⋊C4C2×D311yescharacteristic
138#400—34no—
139#398—38no—
140#398—38no—
141#397—34no—
142#397—38no—
143#395—34no—
144#394—38no—
145#393—34no—
146#391—34no—
147#388—38no—
148#388—38no—
149#386—34no—
150#385—34no—
151#385—38no—
152#385—34no—
153#385—34no—
154#384—34no—
155#381—34no—
156#381—34no—
157#380—32no—
158#380—34no—
159#379—34no—
160#379—32no—
161C4×C22⋊C4—32no—
162C24×C3—22no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—42no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3C2×D414yes—
170C24×C3C2412yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C2×C3×D4—28no—
174C2×C3×D4—28no—
175C2×C3×D4—44no—
176C2×C3×D4—416no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—28no—
181C22×C3×C4—24no—
182C22×C3×C4—28no—
183C22×C3×C4—42no—
184C22×C3×C4—44no—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—416no—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2412yes—
192C22×C3×C4C2412yes—
193C22×C3×C4C2412yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2412yes—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—44no—
199C22×Dic3—44no—
200C22×Dic3C2418yes—
201C22×Dic3C2414yes—
202C22×Dic3C2414yes—
203C2×Dic6—24no—
204C2×Dic6—24no—
205C2×Dic6—44no—
206C2×Dic6—44no—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4C2×D4116yes—
212#217—28no—
213#217—28no—
214#217—216no—
215#217—28no—
216#217—44no—
217#217—44no—
218#217—48no—
219C4⋊Dic3—24no—
220C4⋊Dic3—24no—
221C4⋊Dic3—44no—
222Dic3⋊C4—28no—
223Dic3⋊C4—28no—
224Dic3⋊C4—48no—
225Dic3⋊C4—48no—
226C4×Dic3—44no—
227C25C22×D311yescharacteristic
228C25C22×D311yescharacteristic
229C22×Q8—32no—
230C22×Q8—32no—
231C22×D4—24no—
232C22×D4—44no—
233C22×D4C22×D314yes—
234C23×C4—24no—
235C23×C4—32no—
236C23×C4—32no—
237C23×C4—44no—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D314yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244#123—232no—
245#123—128no—
246C22⋊Q8—68no—
247C22⋊Q8—68no—
248C22⋊Q8—616no—
249C22⋊Q8—616no—
250C22⋊Q8—128no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—34no—
256C2×C4⋊C4—34no—
257C2×C4⋊C4—34no—
258C2×C4⋊C4—38no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4C22×D314yes—
261C2×C4⋊C4C22×D314yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—34no—
265C2×C22⋊C4—34no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—68no—
272C2×C22⋊C4#6718yes—
273C2×C22⋊C4C22×D314yes—
274C2×C22⋊C4C22×D314yes—
275C2×C22⋊C4C22×D314yes—
276C2×C42—34no—
277#95—34no—
278#95—38no—
279#95—34no—
280#95—38no—
281#95—38no—
282#95—38no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—28no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—28no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—416no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—88no—
300C23×C3C22×D414yes—
301C23×C3C22×D412yes—
302C23×C3H(5,𝔽2)14yes—
303C23×C32−1+414yes—
304C23×C3C2511yescharacteristic
305C3×D4—416no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—44no—
315C2×C3×C4—44no—
316C2×C3×C4—416no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4C22×D414yes—
321C2×C3×C4C22×D414yes—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—28no—
325C2×Dic3—28no—
326C2×Dic3—28no—
327C2×Dic3—28no—
328C2×Dic3—28no—
329C2×Dic3—84no—
330C2×Dic3—84no—
331Dic6—84no—
332Dic6—84no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24C2×#6714yes—
341C24C23×D312yes—
342C24C23×D312yes—
343C24C23×D311yescharacteristic
344C2×Q8—64no—
345C2×Q8—64no—
346C2×Q8—124no—
347C2×Q8—124no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—44no—
351C2×D4—416no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—38no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—416no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—124no—
370C22×C4—124no—
371C22×C4C2×#6714yes—
372C22×C4C2×#6714yes—
373C22×C4C23×D312yes—
374C22×C4C23×D312yes—
375C22×C4C23×D312yes—
376C22×C4C23×D312yes—
377C22×C4C23×D312yes—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—68no—
381C4⋊C4—68no—
382C4⋊C4—64no—
383C4⋊C4—64no—
384C4⋊C4—128no—
385C4⋊C4—124no—
386C4⋊C4—128no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—616no—
390C22⋊C4—68no—
391C22⋊C4—68no—
392C22⋊C4—68no—
393C22⋊C4—124no—
394C22⋊C4—124no—
395C22⋊C4—128no—
396C22⋊C4C2×#67116yes—
397C42—124no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—28no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—28no—
406C22×C3—416no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—44no—
410C22×C3—48no—
411C22×C3—88no—
412C22×C3#53614yes—
413C22×C3C23×D411yescharacteristic
414C22×C3C2×H(5,𝔽2)12yes—
415C22×C3C2×2−1+412yes—
416C22×C3#58512yes—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—42no—
422Dic3—44no—
423Dic3—44no—
424Dic3—48no—
425C23—22no—
426C23—22no—
427C23—28no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—28no—
436C23—44no—
437C23—44no—
438C23—416no—
439C23—44no—
440C23—44no—
441C23—88no—
442C23#100414yes—
443C23#101014yes—
444C23C22×#6714yes—
445C23C22×#6712yes—
446C23C24×D311yescharacteristic
447Q8—244no—
448Q8—244no—
449D4—416no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—68no—
465C2×C4—68no—
466C2×C4—68no—
467C2×C4—68no—
468C2×C4—68no—
469C2×C4—64no—
470C2×C4—64no—
471C2×C4—244no—
472C2×C4—244no—
473C2×C4C22×#6714yes—
474C2×C4C22×#6714yes—
475C2×C3—24no—
476C2×C3—44no—
477C2×C3—44no—
478C2×C3C2×#53611yescharacteristic
479C2×C3#348411yescharacteristic
480C2×C3#348511yescharacteristic
481C2×C3#353514yes—
482C22—22no—
483C22—22no—
484C22—28no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—28no—
490C22—416no—
491C22—42no—
492C22—42no—
493C22—44no—
494C22—48no—
495C22—88no—
496C22#582514yes—
497C22C2×#100412yes—
498C22C2×#101012yes—
499C22#596412yes—
500C22C23×#6711yescharacteristic
501C4—24no—
502C4—24no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—124no—
507C4—124no—
508C4—128no—
509C3#3672011yescharacteristic, complemented, Sylow
510C2—24no—
511C2—44no—
512C2—44no—
513C2#8600314yes—
514C2#8723911yescharacteristic
515C2C2×#582511yescharacteristic
516C2#8727311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC22×C3×#123C2
prime radicalC23×C3×D4C22
hypercenterC22×#123D3
hypocenterC3#36720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C2×#5483C2semidirect
4. #84031C2semidirect
5. #82874C2semidirect
6. #82871C2semidirect
7. #82183C2semidirect
8. #82178C2semidirect
9. #82116C2semidirect
10. #82114C2semidirect
11. #82093C2semidirect
26. C2×#924C22semidirect
28. C2×#878C22semidirect
29. C2×#878C22semidirect
35. #5212C22semidirect
39. #5210C22semidirect
41. #4986C22semidirect
42. #4983C22semidirect
43. #4982C22semidirect
44. #4979C22semidirect
45. #4975C22semidirect
46. #4973C22semidirect
47. #4953C22semidirect
48. #4952C22semidirect
49. #4951C22semidirect
50. #4947C22semidirect
51. #4946C22semidirect
52. #4945C22semidirect
53. #4943C22semidirect
509. C3#36720semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C2×#54832
#840472
#840312
#828742
#828712
#821832
#821782
#821162
#821142
#820932
#367203