Finite Group Databasebeta

← #13168687

Subgroups — #13168687

Order 1728.

1430 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3798C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic, complemented
3#11723424C212yesmaximal, complemented
4#11723404C211yesmaximal, characteristic, complemented
5C3×D3×C4⋊Dic3C212yesmaximal, complemented
6#11722941C212yesmaximal, complemented
7#11722927C214yesmaximal, complemented
8Dic3×#625C212yesmaximal, complemented
9#10594986—32nomaximal
10#10594509—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2212yes—
13C2×#6294C2212yes—
14C2×C3×#627C2211yescharacteristic
15C2×C3×#626C2212yes—
16C2×C3×#625C2211yescharacteristic
17C2×C3×D3×Dic3C2214yes—
18C3×#3798C418yescomplemented
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C32×C4⋊Dic3C2211yescharacteristic
22#88815C2212yes—
23#88803C2212yes—
24(C2×D3)⋊(C3⋊Dic3)C2214yes—
25Dic3×C3⋊Dic3C2212yes—
26Dic3×C3⋊Dic3C2214yes—
27C3×Dic3⋊Dic3C2214yes—
28C2×C3×#235—32no—
29C2×#3798D311yescharacteristic
30#64135—32no—
31#64101—31no—
32C2×C3×C4⋊Dic3—32no—
33C3×D3×C4⋊C4—32no—
34Dic3×#67—34no—
35Dic3×#67—34no—
36#63904—34no—
37C4×#625—32no—
38#63900—31no—
39#63898—34no—
40D3×C4⋊Dic3—32no—
41D3×C4⋊Dic3D312yescomplemented
42#63843—34no—
43#63841—32no—
44#63841—32no—
45#63841—32no—
46#63841—62no—
47#63841—61no—
48#63838—32no—
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C32×D3C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)C2312yes—
53C2×C3×C3⋊Dic3C2314yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C3×C4×C3⋊D3C2×C414yescomplemented
56C2×C32×Dic3C2312yes—
57C2×C32×Dic3C2312yes—
58C32×C4×D3C2×C418yescomplemented
59#6294—42no—
60C3×#627C2×C414yescomplemented
61C3×#626C2×C418yescomplemented
62C3×#625C2×C414yescomplemented
63C3×D3×Dic3—44no—
64#5827—91no—
65#5504—92no—
66C22×C32×C4—32no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×C3×#67—32no—
69C2×C3×#67—32no—
70C22×C3×Dic3—34no—
71C3×#235—316no—
72C2×C3×D12—32no—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3C2×D312yes—
78C2×C3×C4×D3C2×D312yes—
79C2×C3×Dic6—32no—
80C2×#627C2×D311yescharacteristic
81C2×#626C2×D312yes—
82C2×#625C2×D311yescharacteristic
83C2×#624—32no—
84C2×D3×Dic3—34no—
85C2×D3×Dic3C2×D314yes—
86#3798Dic318yescomplemented
87C32×C4⋊C4—31no—
88#3757—34no—
89#3752—32no—
90C4⋊(C3⋊Dic3)—31no—
91C4⋊(C3⋊Dic3)—32no—
92C4⋊(C3⋊Dic3)—61no—
93C4⋊(C3⋊Dic3)—62no—
94C4⋊(C3⋊Dic3)—121no—
95C4⋊(C3⋊Dic3)C2×D312yes—
96C4×C3⋊Dic3—32no—
97C4×C3⋊Dic3—34no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—32no—
100C3×C4⋊Dic3—32no—
101C3×C4⋊Dic3—34no—
102C3×C4⋊Dic3—62no—
103C3×C4⋊Dic3C2×D312yes—
104C3×Dic3⋊C4—34no—
105#3724—32no—
106Dic3⋊Dic3—34no—
107Dic3⋊Dic3C2×D314yes—
108(C2×D3)⋊Dic3—34no—
109(C2×D3)⋊Dic3—34no—
110(C2×D3)⋊Dic3—34no—
111(C2×D3)⋊Dic3—62no—
112(C2×D3)⋊Dic3—64no—
113Dic32—34no—
114Dic32—34no—
115Dic32—34no—
116Dic32—64no—
117Dic32—62no—
118C22×C33C2411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3C22×C412yes—
122C2×C32×D3—22no—
123C2×C32×D3—22no—
124C2×C32×D3C22×C414yes—
125C33×C4C22×C412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3—24no—
128C3×C3⋊Dic3C22×C412yes—
129C32×Dic3—22no—
130C32×Dic3C22×C414yes—
131C2×C3×#41—91no—
132C2×#235—32no—
133C2×C3×C4⋊C4—92no—
134Q8×Dic3—92no—
135D4×Dic3—94no—
136D4×Dic3—92no—
137C4×#67—94no—
138C4×#67—94no—
139#926—92no—
140#926—92no—
141#926—91no—
142#926—182no—
143C2×C4⋊Dic3—32no—
144C2×C4⋊Dic3—91no—
145#893—94no—
146#892—92no—
147#892—92no—
148#892—181no—
149D3×C4⋊C4—32no—
150D3×C4⋊C4—92no—
151#887—94no—
152C23×C32—32no—
153C22×C3⋊D3C22×D311yescharacteristic
154C22×C3×D3—22no—
155C22×C3×D3—32no—
156C22×C3×D3—61no—
157C22×C3×D3C22×D312yes—
158C22×C3×D3C22×D312yes—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—32no—
162C2×C32×C4—38no—
163C2×C32×C4—32no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—34no—
169C2×C3⋊Dic3—64no—
170C2×C3⋊Dic3—62no—
171C2×C3⋊Dic3—122no—
172C2×C3⋊Dic3C22×D314yes—
173C2×C3⋊Dic3C22×D311yescharacteristic
174C4×C3⋊D3C2×Dic314yescomplemented
175C3×#67—38no—
176C3×#67—38no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—34no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—124no—
185C2×C3×Dic3C22×D314yes—
186C2×C3×Dic3C22×D312yes—
187C2×C3×Dic3C22×D312yes—
188C3×D12—38no—
189C3×C4×D3—28no—
190C3×C4×D3—38no—
191C3×C4×D3—64no—
192C3×C4×D3—124no—
193C3×C4×D3C4×D318yescomplemented
194C3×C4×D3C2×Dic318yescomplemented
195C3×Dic6—38no—
196#627C2×Dic314yescomplemented
197#626C2×Dic318yescomplemented
198#625C2×Dic314yescomplemented
199#624—122no—
200D3×Dic3—44no—
201D3×Dic3—124no—
202#520—271noSylow
203C2×C33C23×C411yescharacteristic
204C2×C33H(5,𝔽2)11yescharacteristic
205C2×C332−1+411yescharacteristic
206C3×C3⋊D3—22no—
207C32×D3—24no—
208C3×#41—98no—
209C2×C3×Q8—91no—
210C2×C3×D4—92no—
211C2×C3×D4—91no—
212C22×C3×C4—32no—
213C22×C3×C4—32no—
214C22×C3×C4—62no—
215C22×C3×C4—94no—
216C22×C3×C4—91no—
217C2×#67—32no—
218C2×#67—32no—
219C22×Dic3—34no—
220C22×Dic3—92no—
221#235—316no—
222C2×D12—32no—
223C2×C4×D3—32no—
224C2×C4×D3—34no—
225C2×C4×D3—61no—
226C2×C4×D3—94no—
227C2×C4×D3D3212yes—
228C2×Dic6—32no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—94no—
232C3×C4⋊C4—92no—
233#217—94no—
234#217—94no—
235#217—92no—
236#217—184no—
237(C2×D3)⋊C4—94no—
238(C2×D3)⋊C4—94no—
239(C2×D3)⋊C4—182no—
240C4⋊Dic3—32no—
241C4⋊Dic3—32no—
242C4⋊Dic3—34no—
243C4⋊Dic3—62no—
244C4⋊Dic3—92no—
245C4⋊Dic3—92no—
246C4⋊Dic3—91no—
247C4⋊Dic3—181no—
248C4⋊Dic3—182no—
249C4⋊Dic3—361no—
250C4⋊Dic3D3211yescharacteristic
251Dic3⋊C4—34no—
252Dic3⋊C4—94no—
253C4×Dic3—94no—
254C4×Dic3—94no—
255C4×Dic3—94no—
256C4×Dic3—94no—
257C4×Dic3—92no—
258C4×Dic3—182no—
259C4×Dic3—184no—
260C22×C32—21no—
261C22×C32—22no—
262C22×C32—34no—
263C22×C32—41no—
264C22×C32—62no—
265C22×C32—62no—
266C22×C32C23×D312yes—
267C22×C32C23×D311yescharacteristic
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3⋊D3C22×Dic312yes—
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—22no—
274C2×C3×D3—22no—
275C2×C3×D3—24no—
276C2×C3×D3—34no—
277C2×C3×D3—42no—
278C2×C3×D3—42no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3C2×C4×D314yes—
285C2×C3×D3C22×Dic314yes—
286C32×C4—22no—
287C32×C4—24no—
288C32×C4—34no—
289C32×C4—42no—
290C32×C4—62no—
291C32×C4C2×C4×D314yes—
292C32×C4C22×Dic312yes—
293C3⋊Dic3—24no—
294C3⋊Dic3—64no—
295C3⋊Dic3—62no—
296C3⋊Dic3—124no—
297C3⋊Dic3—122no—
298C3⋊Dic3—242no—
299C3⋊Dic3C22×Dic312yes—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—34no—
303C3×Dic3—42no—
304C3×Dic3—62no—
305C3×Dic3—64no—
306C3×Dic3—64no—
307C3×Dic3—124no—
308C3×Dic3C2×C4×D314yes—
309C3×Dic3C22×Dic314yes—
310C2×#41—91no—
311C4×Q8—272no—
312C4×D4—274no—
313C4×D4—272no—
314#117—271no—
315#117—272no—
316C2×C4⋊C4—92no—
317C2×C4⋊C4—271no—
318C33#52011yescharacteristic, complemented, Sylow
319C23×C3—32no—
320C23×C3—32no—
321C23×C3—62no—
322C23×C3—91no—
323C22×D3—32no—
324C22×D3—61no—
325C22×D3C2×D3212yes—
326C3×Q8—94no—
327C3×D4—98no—
328C3×D4—94no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—38no—
334C2×C3×C4—38no—
335C2×C3×C4—34no—
336C2×C3×C4—41no—
337C2×C3×C4—68no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—91no—
341C2×C3×C4—94no—
342C2×C3×C4—94no—
343C2×C3×C4—364no—
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3212yes—
346#67—38no—
347#67—38no—
348C2×Dic3—32no—
349C2×Dic3—34no—
350C2×Dic3—34no—
351C2×Dic3—61no—
352C2×Dic3—64no—
353C2×Dic3—94no—
354C2×Dic3—92no—
355C2×Dic3—124no—
356C2×Dic3—182no—
357C2×Dic3—184no—
358C2×Dic3—362no—
359C2×Dic3—362no—
360C2×Dic3C2×D3212yes—
361C2×Dic3C2×D3212yes—
362D12—38no—
363C4×D3—38no—
364C4×D3—64no—
365C4×D3—124no—
366C4×D3—364no—
367C4×D3D3×Dic318yescomplemented
368Dic6—38no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—22no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—22no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—41no—
378C2×C32—64no—
379C2×C32C22×C4×D312yes—
380C2×C32#100412yes—
381C2×C32#100712yes—
382C2×C32#100911yescharacteristic
383C2×C32#101011yescharacteristic
384C2×C32C23×Dic311yescharacteristic
385C3⋊D3—22no—
386C3×D3—24no—
387C3×D3—24no—
388C3×D3—44no—
389C3×D3—64no—
390C3×D3—122no—
391#41—98no—
392C2×Q8—91no—
393C2×D4—92no—
394C2×D4—91no—
395C22×C4—32no—
396C22×C4—94no—
397C22×C4—91no—
398C22×C4—272no—
399C4⋊C4—31no—
400C4⋊C4—94no—
401C4⋊C4—92no—
402C4⋊C4—272no—
403C4⋊C4—271no—
404C22⋊C4—272no—
405C22⋊C4—274no—
406C42—274no—
407C42—272no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—34no—
411C22×C3—34no—
412C22×C3—41no—
413C22×C3—64no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—92no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—181no—
422C22×C3—181no—
423C22×C3C22×D3212yes—
424C22×C3C22×D3211yescharacteristic
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—34no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3C2×D3×Dic314yes—
434C3×C4—22no—
435C3×C4—24no—
436C3×C4—34no—
437C3×C4—34no—
438C3×C4—42no—
439C3×C4—64no—
440C3×C4—64no—
441C3×C4—92no—
442C3×C4—122no—
443C3×C4—184no—
444C3×C4C4×D3212yes—
445C3×C4C2×D3×Dic314yes—
446Dic3—22no—
447Dic3—34no—
448Dic3—62no—
449Dic3—64no—
450Dic3—64no—
451Dic3—124no—
452Dic3—184no—
453Dic3—182no—
454Dic3—362no—
455Dic3—364no—
456Dic3—722no—
457Dic3C2×D3×Dic314yes—
458C32—22no—
459C32—21no—
460C32—41no—
461C32#550412yescomplemented
462C32#582711yescharacteristic, complemented
463C23—32no—
464C23—91no—
465Q8—94no—
466D4—98no—
467D4—94no—
468C2×C4—32no—
469C2×C4—38no—
470C2×C4—32no—
471C2×C4—91no—
472C2×C4—94no—
473C2×C4—94no—
474C2×C4—272no—
475C2×C4—364no—
476C2×C4—1082no—
477C2×C4D3311yescharacteristic
478C2×C3—22no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—22no—
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—64no—
488C2×C3—64no—
489C2×C3—124no—
490C2×C3—182no—
491C2×C3#6429712yes—
492C2×C3C2×C4×D3211yescharacteristic
493C2×C3#6430712yes—
494C2×C3#6431311yescharacteristic
495C2×C3#6431611yescharacteristic
496C2×C3C22×D3×Dic312yes—
497D3—24no—
498D3—64no—
499D3—122no—
500C22—34no—
501C22—62no—
502C22—62no—
503C22—92no—
504C22—181no—
505C22—181no—
506C22C2×D3311yescharacteristic
507C4—34no—
508C4—62no—
509C4—92no—
510C4—184no—
511C4—542no—
512C4D32×Dic312yes—
513C3—21no—
514C3—22no—
515C3—41no—
516C3#1059450911yescharacteristic, complemented
517C3#1059498612yescomplemented
518C2—64no—
519C2—182no—
520C2#1172438011yescharacteristic
521C2#1172439511yescharacteristic
522C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C3×#3798C2
hypercenterC2×C4D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723689C2semidirect
3. #11723424C2semidirect
4. #11723404C2semidirect
5. C3×D3×C4⋊Dic3C2semidirect
6. #11722941C2semidirect
7. #11722927C2semidirect
8. Dic3×#625C2semidirect
18. C3×#3798C4semidirect
41. D3×C4⋊Dic3D3semidirect
55. C3×C4×C3⋊D3C2×C4semidirect
58. C32×C4×D3C2×C4semidirect
60. C3×#627C2×C4semidirect
61. C3×#626C2×C4semidirect
62. C3×#625C2×C4semidirect
86. #3798Dic3semidirect
174. C4×C3⋊D3C2×Dic3semidirect
193. C3×C4×D3C4×D3semidirect
194. C3×C4×D3C2×Dic3semidirect
196. #627C2×Dic3semidirect
197. #626C2×Dic3semidirect
198. #625C2×Dic3semidirect
318. C33#520semidirect
367. C4×D3D3×Dic3semidirect
461. C32#5504semidirect
462. C32#5827semidirect
516. C3#10594509semidirect
517. C3#10594986semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37982
#117236892
#117234242
#117234042
C3×D3×C4⋊Dic32
#117229412
#117229272
Dic3×#6252
#105949863
#105945093