Finite Group Databasebeta

← #10956205

Subgroups — #10956205

Order 768.

2682 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
1C2×#5521C211yesmaximal, characteristic
2C3×C4×#117C211yesmaximal, characteristic, complemented
3#82269C211yesmaximal, characteristic
4#82234C211yesmaximal, characteristic, complemented
5#82223C211yesmaximal, characteristic, complemented
6#82069C211yesmaximal, characteristic, complemented
7C2×#4939C211yesmaximal, characteristic
8#73646C214yesmaximal, complemented
9#73639C214yesmaximal, complemented
10#73574C218yesmaximal, complemented
11#73415C214yesmaximal, complemented
12#73299C214yesmaximal, complemented
13#34255—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×(C2×D3)⋊C4C2214yes—
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22#5521C4116yescomplemented
23C2×#893C2214yes—
24C2×D3×C22⋊C4C2214yes—
25C2×C4×D12C2214yes—
26C2×#872C2218yescomplemented
27C3×C4×C4⋊C4C2212yescomplemented
28C3×C4×C4⋊C4C2212yescomplemented
29C3×C4×C22⋊C4C2214yes—
30C3×#376C2212yescomplemented
31C3×C43C2212yescomplemented
32#5210C2214yes—
33#4988C2212yes—
34#4988C2212yes—
35#4977C2212yescomplemented
36#4975C2212yescomplemented
37Dic3×C4⋊C4C2214yescomplemented
38#4955C2214yes—
39#4946C2214yes—
40Dic3×C22⋊C4C2214yes—
41#4940C2218yescomplemented
42#4940C2218yescomplemented
43#4939C2212yes—
44#4939C2214yes—
45#4939C2214yes—
46#4939C2212yes—
47C4×(C2×D3)⋊C4C2218yescomplemented
48C4×C4⋊Dic3C2214yescomplemented
49C4×Dic3⋊C4C2214yescomplemented
50#4928C2214yescomplemented
51#4928C2212yescomplemented
52#4928C2212yescomplemented
53#4669C2218yescomplemented
54#4669C2218yescomplemented
55#4667C2218yescomplemented
56#4664C2218yescomplemented
57C2×#518—31no—
58#2357—34no—
59#2354—34no—
60#2342—38no—
61#2319—34no—
62#2317—31no—
63#2309—31no—
64#2296—31no—
65#2293—34no—
66C2×#390—31no—
67#2274—31no—
68C4×#117D311yescharacteristic, complemented
69C24×D3C2312yes—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×Dic3C2312yes—
73C22×D12—24no—
74C22×D12—24no—
75C22×D12C2×C418yes—
76C22×D12C2312yes—
77C22×D12C2312yes—
78C22×C4×D3C2318yes—
79C3×#117—24no—
80C3×#117C2×C418yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4C2312yes—
85C2×C3×C22⋊C4C2312yes—
86C2×C3×C42—24no—
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2312yes—
91C2×C3×C42C2314yes—
92C2×C3×C42C2314yes—
93C2×C3×C42C2312yes—
94C2×#217C2312yes—
95C2×#217C2312yes—
96C2×(C2×D3)⋊C4—48no—
97C2×(C2×D3)⋊C4C2318yes—
98C2×(C2×D3)⋊C4C2318yes—
99C2×(C2×D3)⋊C4C2318yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×(C2×D3)⋊C4C2318yes—
102#926—24no—
103#926C2×C418yes—
104C2×C4⋊Dic3—24no—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2314yes—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3C2312yes—
113C2×C4×Dic3C2318yes—
114C2×C4×Dic3C2312yes—
115#893C2×C4132yescomplemented
116D3×C22⋊C4C2×C4132yes—
117C4×D12C2×C4132yescomplemented
118#872—416no—
119C3×#95C2314yes—
120C3×#95C2314yes—
121C3⋊#95C2314yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C3⋊#95C2314yes—
125C23×D4—31no—
126#518—316no—
127C2×C4×D4—34no—
128C2×C4×D4—34no—
129C2×#117—31no—
130C2×#117—31no—
131C2×#117—38no—
132C2×#117C2×D311yescharacteristic
133C2×#117C2×D311yescharacteristic
134C22×C4⋊C4—31no—
135C22×C22⋊C4—34no—
136C22×C22⋊C4—34no—
137C22×C42C2×D311yescharacteristic
138#390—32no—
139#390—32no—
140#390—34no—
141#390—32no—
142#390—34no—
143#390—32no—
144#389—34no—
145#389—32no—
146#389—32no—
147#388—38no—
148#388—38no—
149#388—38no—
150#388—38no—
151#385—34no—
152#385—34no—
153#384—32no—
154#382—32no—
155#381—38no—
156#380—38no—
157#379—34no—
158C4×C4⋊C4—34no—
159C4×C4⋊C4—34no—
160C4×C4⋊C4—34no—
161C4×C4⋊C4C2×D312yescomplemented
162C4×C4⋊C4C2×D312yescomplemented
163C4×C22⋊C4—34no—
164C4×C22⋊C4—38no—
165C4×C22⋊C4C2×D314yes—
166#376C2×D312yescomplemented
167C43C2×D312yescomplemented
168C24×C3C2411yescharacteristic
169C23×D3—24no—
170C23×D3—24no—
171C23×D3—44no—
172C23×D3—44no—
173C23×D3C22×C418yes—
174C23×D3C2414yes—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—42no—
178C22×C3×C4C22×C414yes—
179C22×C3×C4#4114yes—
180C22×C3×C4#4114yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2412yes—
183C22×C3×C4C2412yes—
184C22×C3×C4C2414yes—
185C22×C3×C4C2412yes—
186C22×C3×C4C2411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—24no—
189C22×Dic3—44no—
190C22×Dic3C2414yes—
191C22×Dic3C2414yes—
192C22×Dic3C2414yes—
193C2×D12—232no—
194C2×D12—24no—
195C2×D12—24no—
196C2×D12—24no—
197C2×D12—24no—
198C2×D12C22×C418yes—
199C2×D12C22×C418yes—
200C2×C4×D3—48no—
201C2×C4×D3—48no—
202C2×C4×D3C22×C4132yes—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C22×C418yes—
206C3×C22⋊C4—24no—
207C3×C22⋊C4C22×C418yes—
208C3×C42—22no—
209C3×C42—24no—
210C3×C42—22no—
211C3×C42—24no—
212C3×C42—28no—
213C3×C42—44no—
214C3×C42C22×C418yes—
215#217—24no—
216#217C22×C418yes—
217(C2×D3)⋊C4—216no—
218(C2×D3)⋊C4—48no—
219(C2×D3)⋊C4—48no—
220(C2×D3)⋊C4—48no—
221(C2×D3)⋊C4C22×C4132yes—
222C4⋊Dic3—24no—
223C4⋊Dic3—24no—
224C4⋊Dic3—24no—
225C4⋊Dic3C22×C418yes—
226Dic3⋊C4—28no—
227Dic3⋊C4—48no—
228Dic3⋊C4—44no—
229C4×Dic3—24no—
230C4×Dic3—48no—
231C4×Dic3C22×C418yes—
232C25—32no—
233C22×D4—38no—
234C22×D4—32no—
235C22×D4—32no—
236C22×D4—64no—
237C22×D4—64no—
238C23×C4—38no—
239C23×C4—32no—
240C23×C4C22×D312yes—
241C23×C4C22×D311yescharacteristic
242C4×D4—332no—
243C4×D4—332no—
244#117—24no—
245#117—38no—
246#117—64no—
247#117—1216no—
248#117C4×D318yes—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—34no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—38no—
255C2×C4⋊C4—34no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C4⋊C4C22×D312yes—
260C2×C22⋊C4—32no—
261C2×C22⋊C4—38no—
262C2×C22⋊C4—332no—
263C2×C22⋊C4—32no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4—38no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—38no—
268C2×C22⋊C4—128no—
269C2×C22⋊C4C22×D312yes—
270C2×C22⋊C4C22×D312yes—
271C2×C42—24no—
272C2×C42—32no—
273C2×C42—32no—
274C2×C42—38no—
275C2×C42C22×D314yes—
276C2×C42C22×D312yes—
277C2×C42C22×D311yescharacteristic
278C2×C42C22×D311yescharacteristic
279C2×C42C22×D312yes—
280C2×C42C22×D314yes—
281C2×C42C22×D312yes—
282#95—34no—
283#95—38no—
284#95—38no—
285#95—34no—
286#95C22×D314yes—
287#95C22×D314yes—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3C23×C412yes—
291C23×C3C2×#4112yes—
292C23×C3C2×#4112yes—
293C23×C3C2511yescharacteristic
294C22×D3—24no—
295C22×D3—24no—
296C22×D3—28no—
297C22×D3—28no—
298C22×D3—216no—
299C22×D3—416no—
300C22×D3—88no—
301C22×D3—84no—
302C22×D3—84no—
303C22×D3C23×C418yes—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—48no—
316C2×C3×C4#117116yescomplemented
317C2×C3×C4C23×C418yes—
318C2×C3×C4C23×C412yes—
319C2×C3×C4C23×C412yes—
320C2×C3×C4C2×#4112yes—
321C2×C3×C4C2×#4112yes—
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2×#4112yes—
324C2×C3×C4H(5,𝔽2)14yes—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—28no—
331C2×Dic3—24no—
332C2×Dic3—84no—
333C2×Dic3C23×C418yes—
334D12—216no—
335D12—216no—
336C4×D3—432no—
337C24—34no—
338C24—38no—
339C24—64no—
340C24—64no—
341C24—124no—
342C24—124no—
343C24C23×D311yescharacteristic
344C2×D4—38no—
345C2×D4—38no—
346C2×D4—632no—
347C2×D4—64no—
348C2×D4—64no—
349C2×D4—64no—
350C2×D4—64no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—34no—
354C22×C4—332no—
355C22×C4—34no—
356C22×C4—34no—
357C22×C4—42no—
358C22×C4—64no—
359C22×C4—128no—
360C22×C4—128no—
361C22×C4—124no—
362C22×C4C2×C4×D314yes—
363C22×C4#23514yes—
364C22×C4#23514yes—
365C22×C4C23×D312yes—
366C22×C4C23×D312yes—
367C22×C4C23×D312yes—
368C22×C4C23×D314yes—
369C22×C4C23×D312yes—
370C22×C4C23×D311yescharacteristic
371C22×C4C23×D311yescharacteristic
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—38no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—68no—
379C4⋊C4—128no—
380C4⋊C4—124no—
381C4⋊C4C2×C4×D318yes—
382C22⋊C4—24no—
383C22⋊C4—38no—
384C22⋊C4—332no—
385C22⋊C4—64no—
386C22⋊C4—616no—
387C22⋊C4—128no—
388C22⋊C4—128no—
389C22⋊C4—128no—
390C22⋊C4C2×C4×D318yes—
391C42—22no—
392C42—24no—
393C42—22no—
394C42—24no—
395C42—28no—
396C42—38no—
397C42—44no—
398C42—64no—
399C42—128no—
400C42C2×C4×D318yes—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3C2×#11714yes—
406C22×C3C24×C411yescharacteristic
407C22×C3C22×#4111yescharacteristic
408C22×C3C22×#4111yescharacteristic
409C22×C3C2×H(5,𝔽2)12yes—
410C22×C3#58512yes—
411C2×D3—28no—
412C2×D3—216no—
413C2×D3—48no—
414C2×D3—44no—
415C2×D3—44no—
416C2×D3—816no—
417C3×C4—28no—
418C3×C4—24no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4C2×#11714yes—
423C3×C4C2×#11714yes—
424Dic3—28no—
425Dic3—44no—
426Dic3—44no—
427C23—22no—
428C23—22no—
429C23—38no—
430C23—64no—
431C23—64no—
432C23—68no—
433C23—68no—
434C23—616no—
435C23—1216no—
436C23—244no—
437C23—244no—
438C23—248no—
439C23C22×C4×D312yes—
440C23C2×#23512yes—
441C23C2×#23512yes—
442C23C24×D311yescharacteristic
443D4—616no—
444D4—616no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—38no—
457C2×C4—48no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—68no—
464C2×C4—64no—
465C2×C4—1232no—
466C2×C4—244no—
467C2×C4#872116yescomplemented
468C2×C4C22×C4×D318yes—
469C2×C4C22×C4×D312yes—
470C2×C4C22×C4×D312yes—
471C2×C4C2×#23512yes—
472C2×C4C2×#23512yes—
473C2×C4C2×#23512yes—
474C2×C4C2×#23512yes—
475C2×C4#100914yes—
476C2×C3—24no—
477C2×C3C22×#11711yescharacteristic
478C2×C3C4×H(5,𝔽2)12yes—
479C2×C3#349412yes—
480C2×C3#350811yescharacteristic
481C2×C3#351011yescharacteristic
482D3—48no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—616no—
488C22—68no—
489C22—124no—
490C22—124no—
491C22—128no—
492C22—2416no—
493C22C2×#87214yes—
494C22C23×C4×D311yescharacteristic
495C22C22×#23511yescharacteristic
496C22C22×#23511yescharacteristic
497C22C2×#100912yes—
498C22#596412yes—
499C4—28no—
500C4—24no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—68no—
505C4—124no—
506C4—124no—
507C4C2×#87214yes—
508C4C2×#87214yes—
509C3#3425511yescharacteristic, complemented, Sylow
510C2—24no—
511C2—128no—
512C2C22×#87211yescharacteristic
513C2#8574712yes—
514C2C4×#100912yes—
515C2#8584811yescharacteristic
516C2#8584911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×C4×#117C2
socle, cosocleC23×C3C25
prime radicalC23×D12C22
hypercenterC4×#117D3
hypocenterC3#34255
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. C3×C4×#117C2semidirect
4. #82234C2semidirect
5. #82223C2semidirect
6. #82069C2semidirect
8. #73646C2semidirect
9. #73639C2semidirect
10. #73574C2semidirect
11. #73415C2semidirect
12. #73299C2semidirect
22. #5521C4semidirect
26. C2×#872C22semidirect
27. C3×C4×C4⋊C4C22semidirect
28. C3×C4×C4⋊C4C22semidirect
30. C3×#376C22semidirect
31. C3×C43C22semidirect
35. #4977C22semidirect
36. #4975C22semidirect
37. Dic3×C4⋊C4C22semidirect
41. #4940C22semidirect
42. #4940C22semidirect
47. C4×(C2×D3)⋊C4C22semidirect
48. C4×C4⋊Dic3C22semidirect
49. C4×Dic3⋊C4C22semidirect
50. #4928C22semidirect
51. #4928C22semidirect
52. #4928C22semidirect
53. #4669C22semidirect
54. #4669C22semidirect
55. #4667C22semidirect
56. #4664C22semidirect
68. C4×#117D3semidirect
115. #893C2×C4semidirect
117. C4×D12C2×C4semidirect
161. C4×C4⋊C4C2×D3semidirect
162. C4×C4⋊C4C2×D3semidirect
166. #376C2×D3semidirect
167. C43C2×D3semidirect
316. C2×C3×C4#117semidirect
467. C2×C4#872semidirect
509. C3#34255semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
C3×C4×#1172
#822692
#822342
#822232
#820692
C2×#49392
#736462
#736392
#735742
#734152
#732992
#342553