Finite Group Databasebeta

← #13169264

Subgroups — #13169264

Order 1728.

1646 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1113C211yesmaximal, characteristic
2#11723501C212yesmaximal, complemented
3#11723437C211yesmaximal, characteristic, complemented
4C3×#63894C212yesmaximal, complemented
5C3×#63849C211yesmaximal, characteristic, complemented
6#11722934C214yesmaximal, complemented
7#11722905C214yesmaximal, complemented
8#10594604—32nomaximal
9#10593996—31nomaximal
10C22×#1113C2212yes—
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6295C2212yes—
14C4⋊#1113C418yescomplemented
15C2×Dic3×C3⋊D3C2214yescomplemented
16C2×C3×#624C2214yescomplemented
17C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C32×C4⋊Dic3C2212yescomplemented
19C32×C4×Dic3C2211yescharacteristic, complemented
20#88807C2212yescomplemented
21#88803C2212yes—
22#88803C2214yescomplemented
23C3×#3722C2212yes—
24C3×#3722C2214yescomplemented
25C3×Dic32C2212yescomplemented
26C2×#3802—31no—
27C2×D3×D12—32no—
28#64102—32no—
29#64081—31no—
30C3×#893—32no—
31C3×#892—32no—
32C3×C4×D12—32no—
33#63921—34no—
34D3×(C2×D3)⋊C4—34no—
35D3×(C2×D3)⋊C4—34no—
36C4×#626—34no—
37#63898—32no—
38#63898—32no—
39#63898—62no—
40#63894D312yescomplemented
41#63880—34no—
42#63879—34no—
43#63855—32no—
44#63855—61no—
45#63849D311yescharacteristic, complemented
46C22×C3×C3⋊D3C2311yescharacteristic
47C22×C3×C3⋊D3C2314yes—
48C2×#1113—44no—
49C2×#1113C2×C418yescomplemented
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)C2312yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×C4⋊(C3⋊D3)C2×C418yescomplemented
54C3×C4×C3⋊D3C2×C414yescomplemented
55C2×C32×Dic3C2312yes—
56C2×C32×Dic3C2314yes—
57#6295C2×C418yescomplemented
58Dic3×C3⋊D3—44no—
59C3×#624—44no—
60#5545—92no—
61#5521—91no—
62C22×D32—34no—
63C22×D32—32no—
64C2×C4⋊(C3⋊D3)C2×D312yes—
65C2×C4×C3⋊D3—34no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×D12—32no—
68C2×C3×D12—32no—
69C2×C3×D12—62no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—61no—
74C2×#626—34no—
75C2×#625—32no—
76C2×#624C2×D314yescomplemented
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79C2×D3×Dic3—64no—
80#3802—38no—
81D3×D12—316no—
82C32×C4⋊C4—32no—
83C32×C42—31no—
84#3752—32no—
85#3752—34no—
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—32no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—62no—
91C4×C3⋊Dic3—121no—
92C3×(C2×D3)⋊C4—34no—
93C3×(C2×D3)⋊C4—34no—
94C3×(C2×D3)⋊C4—34no—
95C3×C4⋊Dic3—22no—
96C3×C4⋊Dic3C2×D312yescomplemented
97C3×C4⋊Dic3C2×D312yescomplemented
98C3×C4×Dic3—21no—
99C3×C4×Dic3—34no—
100C3×C4×Dic3C2×D312yescomplemented
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—62no—
104#3722C2×D312yes—
105#3722C2×D314yescomplemented
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—34no—
109(C2×D3)⋊Dic3—34no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—64no—
112Dic32C2×D312yescomplemented
113C22×C33C2411yescharacteristic
114C2×C3×C3⋊D3—24no—
115C2×C3×C3⋊D3—24no—
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3C22×C418yes—
118C2×C3×C3⋊D3C22×C412yes—
119#1113—48no—
120C33×C4C22×C412yes—
121C3⋊(C3⋊Dic3)—22no—
122C3×C3⋊Dic3C22×C412yes—
123C32×Dic3—22no—
124C32×Dic3—24no—
125C2×D3×D4—92no—
126C22×D12—91no—
127C3×C4×D4—92no—
128C3×#117—91no—
129D4×Dic3—92no—
130C4×#67—94no—
131C2×(C2×D3)⋊C4—94no—
132#926—91no—
133#893—32no—
134#893—92no—
135#893—92no—
136#893—182no—
137#892—32no—
138#67⋊C4—94no—
139D3×C22⋊C4—94no—
140D3×C22⋊C4—94no—
141C4×D12—32no—
142C4×D12—94no—
143#872—92no—
144#872—181no—
145C22×C3⋊D3C22×D311yescharacteristic
146C22×C3⋊D3C22×D314yes—
147C22×C3×D3—32no—
148C22×C3×D3—34no—
149C22×C3×D3—34no—
150C22×C3×D3—61no—
151C22×C3×D3—64no—
152C2×D32—316no—
153C2×D32—38no—
154C2×D32—124no—
155C2×D32—124no—
156C2×D32—124no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4—34no—
160C2×C32×C4—32no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—62no—
170C2×C3⋊Dic3—122no—
171C2×C3⋊Dic3C22×D311yescharacteristic
172C4⋊(C3⋊D3)C4×D318yescomplemented
173C4×C3⋊D3—124no—
174C4×C3⋊D3C4×D314yescomplemented
175C2×C3×Dic3—22no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3C22×D314yes—
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C22×D314yes—
182C2×C3×Dic3C22×D312yes—
183C3×D12—38no—
184C3×D12—38no—
185C3×D12—68no—
186C3×C4×D3—38no—
187C3×C4×D3—64no—
188C3×C4×D3—124no—
189C3×C4×D3—124no—
190#626—316no—
191#625—38no—
192#624—44no—
193D3×Dic3—124no—
194D3×Dic3—124no—
195D3×Dic3—128no—
196#518—271noSylow
197C2×C33C23×C411yescharacteristic
198C2×C33H(5,𝔽2)12yes—
199C3×C3⋊D3—22no—
200C3×C3⋊D3—28no—
201C23×D3—92no—
202C23×D3—94no—
203C2×C3×D4—92no—
204C22×C3×C4—94no—
205C22×C3×C4—91no—
206C2×#67—94no—
207C22×Dic3—94no—
208D3×D4—916no—
209C2×D12—32no—
210C2×D12—32no—
211C2×D12—62no—
212C2×D12—98no—
213C2×D12—92no—
214C2×C4×D3—32no—
215C2×C4×D3—34no—
216C2×C4×D3—34no—
217C2×C4×D3—61no—
218C2×C4×D3—94no—
219C2×C4×D3—94no—
220C2×C4×D3—184no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—32no—
223C3×C4⋊C4—62no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—94no—
226C3×C42—32no—
227C3×C42—61no—
228C3×C42—92no—
229#217—92no—
230#217—94no—
231(C2×D3)⋊C4—34no—
232(C2×D3)⋊C4—34no—
233(C2×D3)⋊C4—34no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—94no—
237(C2×D3)⋊C4—94no—
238(C2×D3)⋊C4—182no—
239(C2×D3)⋊C4—184no—
240C4⋊Dic3—92no—
241C4⋊Dic3D3212yescomplemented
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—182no—
245C4×Dic3—34no—
246C4×Dic3—92no—
247C4×Dic3—91no—
248C4×Dic3—181no—
249C4×Dic3—182no—
250C4×Dic3—361no—
251C4×Dic3D3211yescharacteristic, complemented
252C22×C32—21no—
253C22×C32—22no—
254C22×C32—41no—
255C22×C32C23×D312yes—
256C22×C32C23×D311yescharacteristic
257C2×C3⋊D3—24no—
258C2×C3⋊D3—24no—
259C2×C3⋊D3—22no—
260C2×C3⋊D3C2×C4×D318yes—
261C2×C3⋊D3C2×C4×D312yes—
262C2×C3×D3—38no—
263C2×C3×D3—38no—
264C2×C3×D3—34no—
265C2×C3×D3—62no—
266C2×C3×D3—68no—
267C2×C3×D3—64no—
268C2×C3×D3—62no—
269C2×C3×D3—64no—
270C2×C3×D3—64no—
271C2×C3×D3—64no—
272C2×C3×D3—62no—
273C2×C3×D3—124no—
274C2×C3×D3—122no—
275C2×C3×D3—124no—
276D32—128no—
277D32—1216no—
278C32×C4—22no—
279C32×C4—24no—
280C32×C4—42no—
281C32×C4—62no—
282C32×C4—64no—
283C32×C4C2×C4×D314yes—
284C32×C4C2×C4×D312yes—
285C3⋊Dic3—62no—
286C3⋊Dic3—64no—
287C3⋊Dic3—62no—
288C3⋊Dic3—62no—
289C3⋊Dic3—122no—
290C3⋊Dic3—122no—
291C3⋊Dic3—124no—
292C3⋊Dic3C2×C4×D312yes—
293C3×Dic3—24no—
294C3×Dic3—22no—
295C3×Dic3—24no—
296C3×Dic3—22no—
297C3×Dic3—34no—
298C3×Dic3—44no—
299C3×Dic3—42no—
300C3×Dic3—62no—
301C22×D4—271no—
302C4×D4—92no—
303C4×D4—274no—
304C4×D4—272no—
305#117—91no—
306#117—271no—
307C2×C22⋊C4—274no—
308C33#51811yescharacteristic, complemented, Sylow
309C23×C3—91no—
310C23×C3—94no—
311C22×D3—32no—
312C22×D3—34no—
313C22×D3—34no—
314C22×D3—61no—
315C22×D3—64no—
316C22×D3—98no—
317C22×D3—916no—
318C22×D3—364no—
319C22×D3—364no—
320C22×D3—364no—
321C3×D4—98no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—34no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—34no—
328C2×C3×C4—41no—
329C2×C3×C4—62no—
330C2×C3×C4—64no—
331C2×C3×C4—91no—
332C2×C3×C4—94no—
333C2×C3×C4—364no—
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3212yes—
336#67—916no—
337C2×Dic3—32no—
338C2×Dic3—61no—
339C2×Dic3—92no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—182no—
343C2×Dic3—182no—
344C2×Dic3—182no—
345C2×Dic3—362no—
346C2×Dic3—364no—
347C2×Dic3C2×D3214yes—
348C2×Dic3C2×D3212yes—
349D12—38no—
350D12—38no—
351D12—68no—
352D12—98no—
353C4×D3—38no—
354C4×D3—64no—
355C4×D3—124no—
356C4×D3—124no—
357C4×D3—364no—
358C4×D3—364no—
359C4×D3—368no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—22no—
365C2×C32—41no—
366C2×C32—42no—
367C2×C32C22×C4×D311yescharacteristic
368C2×C32C22×C4×D312yes—
369C2×C32#100412yes—
370C2×C32#100912yes—
371C2×C32#100912yes—
372C3⋊D3—22no—
373C3⋊D3—28no—
374C3×D3—68no—
375C3×D3—64no—
376C3×D3—68no—
377C3×D3—128no—
378C3×D3—122no—
379C24—272no—
380C2×D4—92no—
381C2×D4—278no—
382C2×D4—272no—
383C22×C4—94no—
384C22×C4—91no—
385C22×C4—274no—
386C4⋊C4—32no—
387C4⋊C4—272no—
388C22⋊C4—92no—
389C22⋊C4—94no—
390C22⋊C4—272no—
391C22⋊C4—274no—
392C42—31no—
393C42—92no—
394C42—271no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—41no—
398C22×C3—98no—
399C22×C3—92no—
400C22×C3—182no—
401C22×C3—184no—
402C22×C3—184no—
403C22×C3C22×D3211yescharacteristic
404C22×C3C22×D3212yes—
405C2×D3—34no—
406C2×D3—38no—
407C2×D3—38no—
408C2×D3—62no—
409C2×D3—68no—
410C2×D3—64no—
411C2×D3—64no—
412C2×D3—64no—
413C2×D3—62no—
414C2×D3—64no—
415C2×D3—62no—
416C2×D3—124no—
417C2×D3—124no—
418C2×D3—122no—
419C2×D3—3616no—
420C2×D3—368no—
421C3×C4—22no—
422C3×C4—24no—
423C3×C4—42no—
424C3×C4—64no—
425C3×C4—62no—
426C3×C4—64no—
427C3×C4—62no—
428C3×C4—92no—
429C3×C4—124no—
430C3×C4—122no—
431C3×C4C4×D3214yes—
432C3×C4C4×D3212yes—
433Dic3—22no—
434Dic3—24no—
435Dic3—34no—
436Dic3—62no—
437Dic3—184no—
438Dic3—182no—
439Dic3—182no—
440Dic3—182no—
441Dic3—362no—
442Dic3—362no—
443Dic3—364no—
444C32—22no—
445C32—21no—
446C32—41no—
447C32#552111yescharacteristic, complemented
448C32#554512yescomplemented
449C23—91no—
450C23—94no—
451C23—278no—
452C23—1084no—
453D4—98no—
454D4—278no—
455C2×C4—32no—
456C2×C4—34no—
457C2×C4—91no—
458C2×C4—94no—
459C2×C4—272no—
460C2×C4—364no—
461C2×C4—1084no—
462C2×C4D3311yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—42no—
469C2×C3—41no—
470C2×C3—188no—
471C2×C3—182no—
472C2×C3C2×C4×D3212yes—
473C2×C3C2×C4×D3211yescharacteristic
474C2×C3#6430712yes—
475C2×C3#6431412yes—
476C2×C3#6432012yes—
477D3—64no—
478D3—68no—
479D3—68no—
480D3—122no—
481D3—128no—
482C22—92no—
483C22—98no—
484C22—184no—
485C22—184no—
486C22—182no—
487C22—1088no—
488C22C2×D3311yescharacteristic
489C4—62no—
490C4—64no—
491C4—92no—
492C4—542no—
493C4#8894912yes—
494C3—22no—
495C3—21no—
496C3—41no—
497C3#1059399611yescharacteristic, complemented
498C3#1059460412yescomplemented
499C2—182no—
500C2—188no—
501C2#1172438512yes—
502C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723501C2semidirect
3. #11723437C2semidirect
4. C3×#63894C2semidirect
5. C3×#63849C2semidirect
6. #11722934C2semidirect
7. #11722905C2semidirect
14. C4⋊#1113C4semidirect
15. C2×Dic3×C3⋊D3C22semidirect
16. C2×C3×#624C22semidirect
17. C4×C3⋊(C3⋊Dic3)C22semidirect
18. C32×C4⋊Dic3C22semidirect
19. C32×C4×Dic3C22semidirect
20. #88807C22semidirect
22. #88803C22semidirect
24. C3×#3722C22semidirect
25. C3×Dic32C22semidirect
40. #63894D3semidirect
45. #63849D3semidirect
49. C2×#1113C2×C4semidirect
53. C3×C4⋊(C3⋊D3)C2×C4semidirect
54. C3×C4×C3⋊D3C2×C4semidirect
57. #6295C2×C4semidirect
76. C2×#624C2×D3semidirect
96. C3×C4⋊Dic3C2×D3semidirect
97. C3×C4⋊Dic3C2×D3semidirect
100. C3×C4×Dic3C2×D3semidirect
105. #3722C2×D3semidirect
112. Dic32C2×D3semidirect
172. C4⋊(C3⋊D3)C4×D3semidirect
174. C4×C3⋊D3C4×D3semidirect
241. C4⋊Dic3D32semidirect
251. C4×Dic3D32semidirect
308. C33#518semidirect
447. C32#5521semidirect
448. C32#5545semidirect
497. C3#10593996semidirect
498. C3#10594604semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
#117235012
#117234372
C3×#638942
C3×#638492
#117229342
#117229052
#105946043
#105939963