Finite Group Databasebeta

← #13180975

Subgroups — #13180975

Order 1728.

996 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D3×#235C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
4#67×#157C214yesmaximal, complemented
5#11724259C212yesmaximal, complemented
6D3×C4⋊#157C212yesmaximal, complemented
7#11724247C212yesmaximal, complemented
8#11724245C212yesmaximal, complemented
9#10596650—31nomaximal
10C22×C3×#157C2212yes—
11C2×D3×#157C2214yes—
12#89076C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C32×#235C411yescharacteristic, complemented
15C3⋊D3×#67C2212yes—
16#89028C412yescomplemented
17C3⋊D3×D12C2211yescharacteristic
18C4×D3×C3⋊D3C2212yes—
19#89020C412yescomplemented
20#89018C411yescharacteristic, complemented
21C3⋊D3×Dic6C2211yescharacteristic
22#89014C411yescharacteristic, complemented
23#88994C2212yes—
24C4⋊(C3⋊#157)C2211yescharacteristic
25C4⋊(C3⋊#157)C2211yescharacteristic
26C4×C3⋊#157C2212yes—
27C3×C4⋊#157C2212yes—
28C3×C4⋊#157C2212yes—
29Dic3⋊#157C2214yes—
30(C2×D3)⋊#157C2214yes—
31Dic3×#157C2214yes—
32#41×C3⋊D3—31no—
33D3×#235—62no—
34Q8×#157—32no—
35D4×#157—32no—
36D4×#157—34no—
37#64286—31no—
38#64286—32no—
39C2×C4⋊#157—32no—
40C2×C4⋊#157D311yescharacteristic, complemented
41C22×C3×C3⋊D3C2311yescharacteristic
42C2×D3×C3⋊D3C2312yes—
43C2×C3⋊#157C2312yes—
44C2×C3⋊#157C2312yes—
45C2×C3×#157—42no—
46C2×C3×#157C2314yes—
47D3×#157—44no—
48C2×C33×C4C2×C411yescharacteristic, complemented
49#6312C2×C412yescomplemented
50C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic, complemented
51C4×C3⋊(C3⋊D3)C2×C412yescomplemented
52#6308C2×C411yescharacteristic, complemented
53C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
54C3×C4×C3⋊D3C2×C411yescharacteristic, complemented
55C3×C4×C3⋊D3C2×C411yescharacteristic, complemented
56C3×C4×C3⋊D3C2311yescharacteristic
57C3×C4×C3⋊D3C2311yescharacteristic
58C32×#67C2×C412yescomplemented
59C32×D12C2×C411yescharacteristic, complemented
60C32×C4×D3C2×C412yescomplemented
61C32×Dic6C2×C411yescharacteristic, complemented
62#6293C2×C412yescomplemented
63#6292C2×C412yescomplemented
64#6291C2×C412yescomplemented
65#6290C2×C412yescomplemented
66#6289C2×C412yescomplemented
67Dic3×C3⋊D3C2312yes—
68D3×C3⋊Dic3C2×C412yescomplemented
69#5504—91nomaximal
70C22×#157—34no—
71C22×#157C2×D312yes—
72C32×#41—31no—
73#3832—31no—
74Q8×C3⋊D3—31no—
75#3830—32no—
76#3830—31no—
77D4×C3⋊D3—32no—
78D4×C3⋊D3—31no—
79#3828—31no—
80#3828—32no—
81#3828—62no—
82#3828—122no—
83C2×C4×C3⋊D3—32no—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C3×#235—22no—
86C2×C3×C4×D3—62no—
87D3×#67—64no—
88#3804—64no—
89D3×D12—62no—
90C4×D32—64no—
91#3799—62no—
92#3798—64no—
93#3795—62no—
94D3×Dic6—62no—
95C22⋊#157—32no—
96C22⋊#157—34no—
97C4⋊#157—34no—
98C4⋊#157—31no—
99C4⋊#157—31no—
100C4⋊#157C2×D312yes—
101C4⋊#157C2×D312yes—
102C4×#157—34no—
103C4×#157—32no—
104C22×C33C22×C411yescharacteristic
105C2×C3⋊(C3⋊D3)C22×C412yes—
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3C22×C411yescharacteristic
109C2×C3×C3⋊D3C2411yescharacteristic
110C2×C32×D3C22×C412yes—
111D3×C3⋊D3—22no—
112D3×C3⋊D3—22no—
113C3⋊#157—22no—
114C3⋊#157—22no—
115C3×#157—24no—
116C33×C4C22×C411yescharacteristic
117C33×C4C22×C411yescharacteristic
118C3⋊(C3⋊Dic3)C22×C412yes—
119C3×C3⋊Dic3C22×C411yescharacteristic
120C3×C3⋊Dic3C22×C411yescharacteristic
121C32×Dic3C22×C412yes—
122D3×#41—182no—
123C2×#235—91no—
124C2×C3×C4⋊C4—91no—
125C4×#67—94no—
126#926—91no—
127#893—92no—
128#892—92no—
129D3×C4⋊C4—92no—
130#887—92no—
131C22×C3⋊D3—32no—
132C22×C3⋊D3C22×D311yescharacteristic
133C22×C3×D3—62no—
134C2×D32—64no—
135C2×#157—32no—
136C2×#157—32no—
137C2×#157—42no—
138C2×#157—124no—
139C2×#157C22×D314yes—
140C32×Q8—31no—
141C32×D4—32no—
142C32×D4—31no—
143C2×C32×C4—22no—
144C2×C32×C4—32no—
145C2×C32×C4—42no—
146C2×C32×C4C4×D311yescharacteristic, complemented
147#638—32no—
148#638—32no—
149#638—32no—
150#638—64no—
151#638—124no—
152C2×C3⋊Dic3—32no—
153C2×C3⋊Dic3C4×D311yescharacteristic, complemented
154C4⋊(C3⋊D3)—32no—
155C4⋊(C3⋊D3)—31no—
156C4⋊(C3⋊D3)—62no—
157C4⋊(C3⋊D3)—122no—
158C4×C3⋊D3—32no—
159C4×C3⋊D3—32no—
160C4×C3⋊D3—32no—
161C4×C3⋊D3—64no—
162C4×C3⋊D3—124no—
163C4×C3⋊D3C4×D311yescharacteristic, complemented
164C4×C3⋊D3C4×D311yescharacteristic, complemented
165C4×C3⋊D3C22×D311yescharacteristic
166C4×C3⋊D3C22×D311yescharacteristic
167#634—31no—
168#634—32no—
169#634—62no—
170#634—122no—
171C3×#67—24no—
172C2×C3×Dic3—62no—
173C3×D12—22no—
174C3×C4×D3—24no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×C4×D3—62no—
179C3×Dic6—22no—
180#627—64no—
181#626—64no—
182#626—64no—
183#625—64no—
184#624—64no—
185D3×Dic3—64no—
186D3×Dic3—64no—
187#520—271noSylow
188C2×C33—21no—
189C2×C33C23×C411yescharacteristic
190C3⋊(C3⋊D3)—22no—
191C3×C3⋊D3—21no—
192C3×C3⋊D3H(5,𝔽2)11yescharacteristic
193C3×C3⋊D32−1+411yescharacteristic
194C32×D3—22no—
195C3×#41—62no—
196C22×C3×C4—91no—
197C22×C3×C4—92no—
198C2×#67—92no—
199#239—182no—
200D3×Q8—182no—
201#237—182no—
202#237—184no—
203D3×D4—182no—
204D3×D4—184no—
205#235—91no—
206#235—92no—
207#235—91no—
208#235—92no—
209#235—91no—
210#235—184no—
211#235—182no—
212#235—362no—
213#235#15711yescharacteristic, complemented
214C2×D12—91no—
215C2×C4×D3—62no—
216C2×C4×D3—92no—
217C2×C4×D3—94no—
218C2×C4×D3—184no—
219C2×Dic6—91no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—92no—
222#217—92no—
223(C2×D3)⋊C4—94no—
224C4⋊Dic3—91no—
225C4⋊Dic3—91no—
226Dic3⋊C4—94no—
227C4×Dic3—94no—
228C4×Dic3—92no—
229C22×C32—22no—
230C22×C32—32no—
231C22×C32—42no—
232C22×C32C2×C4×D311yescharacteristic
233C2×C3⋊D3—21no—
234C2×C3⋊D3—21no—
235C2×C3⋊D3—32no—
236C2×C3⋊D3—64no—
237C2×C3⋊D3—62no—
238C2×C3⋊D3—62no—
239C2×C3⋊D3—124no—
240C2×C3⋊D3C2×C4×D311yescharacteristic
241C2×C3⋊D3C23×D311yescharacteristic
242C2×C3×D3—24no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—122no—
246C2×C3×D3—122no—
247D32—124no—
248D32—124no—
249#157—24no—
250#157—62no—
251#157—62no—
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—32no—
255C32×C4—42no—
256C32×C4—42no—
257C32×C4C2×C4×D311yescharacteristic
258C32×C4C2×C4×D311yescharacteristic
259C3⋊Dic3—32no—
260C3⋊Dic3—64no—
261C3⋊Dic3—124no—
262C3⋊Dic3C2×C4×D311yescharacteristic
263C3⋊Dic3C2×C4×D311yescharacteristic
264C3×Dic3—24no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C2×#41—271no—
268C4×Q8—272no—
269C4×D4—272no—
270C4×D4—274no—
271#117—272no—
272#117—271no—
273C2×C4⋊C4—91no—
274C2×C4⋊C4—272no—
275C33#52011yescharacteristic, complemented, Sylow
276C23×C3—91no—
277C22×D3—62no—
278C22×D3—92no—
279C22×D3—184no—
280C3×Q8—62no—
281C3×D4—62no—
282C3×D4—64no—
283C2×C3×C4—22no—
284C2×C3×C4—42no—
285C2×C3×C4—64no—
286C2×C3×C4—91no—
287C2×C3×C4—91no—
288C2×C3×C4—91no—
289C2×C3×C4—91no—
290C2×C3×C4—91no—
291C2×C3×C4—94no—
292C2×C3×C4—362no—
293C2×C3×C4C2×#15711yescharacteristic, complemented
294#67—92no—
295#67—92no—
296#67—92no—
297#67—184no—
298#67—184no—
299#67—184no—
300#67—364no—
301#67C2×#15712yescomplemented
302C2×Dic3—62no—
303C2×Dic3—92no—
304C2×Dic3—92no—
305C2×Dic3—92no—
306C2×Dic3—184no—
307D12—92no—
308D12—91no—
309D12—182no—
310D12—184no—
311D12—362no—
312D12C2×#15711yescharacteristic, complemented
313C4×D3—62no—
314C4×D3—62no—
315C4×D3—62no—
316C4×D3—62no—
317C4×D3—92no—
318C4×D3—92no—
319C4×D3—92no—
320C4×D3—184no—
321C4×D3—184no—
322C4×D3—184no—
323C4×D3—364no—
324C4×D3—364no—
325C4×D3C2×#15712yescomplemented
326Dic6—91no—
327Dic6—92no—
328Dic6—184no—
329Dic6—182no—
330Dic6—362no—
331Dic6C2×#15711yescharacteristic, complemented
332C2×C32—22no—
333C2×C32—21no—
334C2×C32—42no—
335C2×C32—42no—
336C2×C32—62no—
337C2×C32—82no—
338C2×C32C22×C4×D311yescharacteristic
339C3⋊D3—21no—
340C3⋊D3—62no—
341C3⋊D3—124no—
342C3⋊D3—244no—
343C3⋊D3#100411yescharacteristic
344C3⋊D3#100711yescharacteristic
345C3×D3—44no—
346C3×D3—62no—
347C3×D3—62no—
348C3×D3—122no—
349#41—31no—
350#41—272no—
351#41—272no—
352#41—271no—
353#41—271no—
354#41—271no—
355C2×Q8—271no—
356C2×D4—271no—
357C2×D4—272no—
358C22×C4—91no—
359C22×C4—92no—
360C22×C4—272no—
361C22×C4—274no—
362C4⋊C4—92no—
363C4⋊C4—92no—
364C4⋊C4—271no—
365C4⋊C4—274no—
366C4⋊C4—271no—
367C22⋊C4—274no—
368C22⋊C4—272no—
369C42—272no—
370C42—274no—
371C22×C3—22no—
372C22×C3—42no—
373C22×C3—64no—
374C22×C3—91no—
375C22×C3—91no—
376C22×C3—181no—
377C22×C3—181no—
378C22×C3C22×#15711yescharacteristic
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—92no—
382C2×D3—122no—
383C2×D3—122no—
384C2×D3—184no—
385C2×D3—182no—
386C2×D3—182no—
387C2×D3—364no—
388C2×D3—364no—
389C2×D3—364no—
390C2×D3C22×#15712yes—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—64no—
396C3×C4—91no—
397C3×C4—91no—
398C3×C4—184no—
399C3×C4C22×#15711yescharacteristic
400C3×C4C22×#15711yescharacteristic
401Dic3—62no—
402Dic3—62no—
403Dic3—92no—
404Dic3—184no—
405Dic3—182no—
406Dic3—182no—
407Dic3—364no—
408Dic3C22×#15712yes—
409C32—22no—
410C32—42no—
411C32#550411yescharacteristic, complemented
412C23—91no—
413C23—272no—
414Q8—31no—
415Q8—272no—
416Q8—271no—
417D4—32no—
418D4—31no—
419D4—272no—
420D4—272no—
421D4—272no—
422D4—272no—
423D4—271no—
424C2×C4—32no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—91no—
430C2×C4—94no—
431C2×C4—272no—
432C2×C4—272no—
433C2×C4—272no—
434C2×C4—272no—
435C2×C4—272no—
436C2×C4—272no—
437C2×C4—362no—
438C2×C4—1084no—
439C2×C4D3×#15711yescharacteristic, complemented
440C2×C3—21no—
441C2×C3—22no—
442C2×C3—42no—
443C2×C3—42no—
444C2×C3—82no—
445C2×C3—91no—
446C2×C3—91no—
447C2×C3—124no—
448C2×C3—181no—
449C2×C3C23×#15711yescharacteristic
450D3—22no—
451D3—62no—
452D3—62no—
453D3—122no—
454D3—182no—
455D3—364no—
456D3—724no—
457C22—32no—
458C22—91no—
459C22—91no—
460C22—181no—
461C22—181no—
462C22—272no—
463C22—542no—
464C22—542no—
465C22C2×D3×#15711yescharacteristic
466C4—32no—
467C4—91no—
468C4—91no—
469C4—184no—
470C4—272no—
471C4—542no—
472C4—542no—
473C4C2×D3×#15711yescharacteristic
474C4C2×D3×#15711yescharacteristic
475C3—22no—
476C3—42no—
477C3#1059665011yescharacteristic, complemented
478C2—21no—
479C2—62no—
480C2—91no—
481C2—91no—
482C2—181no—
483C2—542no—
484C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C2×C4
prime radicalC3⋊D3×#235C2
cosocleC2×C3×C3⋊D3C24
hypercenterC2×C4D3×#157
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C2×C3×C4⋊#157C2semidirect
4. #67×#157C2semidirect
5. #11724259C2semidirect
6. D3×C4⋊#157C2semidirect
7. #11724247C2semidirect
8. #11724245C2semidirect
12. #89076C4semidirect
14. C32×#235C4semidirect
16. #89028C4semidirect
19. #89020C4semidirect
20. #89018C4semidirect
22. #89014C4semidirect
40. C2×C4⋊#157D3semidirect
48. C2×C33×C4C2×C4semidirect
49. #6312C2×C4semidirect
50. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
51. C4×C3⋊(C3⋊D3)C2×C4semidirect
52. #6308C2×C4semidirect
53. C2×C3×C3⋊Dic3C2×C4semidirect
54. C3×C4×C3⋊D3C2×C4semidirect
55. C3×C4×C3⋊D3C2×C4semidirect
58. C32×#67C2×C4semidirect
59. C32×D12C2×C4semidirect
60. C32×C4×D3C2×C4semidirect
61. C32×Dic6C2×C4semidirect
62. #6293C2×C4semidirect
63. #6292C2×C4semidirect
64. #6291C2×C4semidirect
65. #6290C2×C4semidirect
66. #6289C2×C4semidirect
68. D3×C3⋊Dic3C2×C4semidirect
146. C2×C32×C4C4×D3semidirect
153. C2×C3⋊Dic3C4×D3semidirect
163. C4×C3⋊D3C4×D3semidirect
164. C4×C3⋊D3C4×D3semidirect
213. #235#157semidirect
275. C33#520semidirect
293. C2×C3×C4C2×#157semidirect
301. #67C2×#157semidirect
312. D12C2×#157semidirect
325. C4×D3C2×#157semidirect
331. Dic6C2×#157semidirect
411. C32#5504semidirect
439. C2×C4D3×#157semidirect
477. C3#10596650semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
#117245142
C2×C3×C4⋊#1572
#67×#1572
#117242592
D3×C4⋊#1572
#117242472
#117242452
#105966503
#55049