Finite Group Databasebeta

← #13175401

Subgroups — #13175401

Order 1728.

2436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723915C211yesmaximal, characteristic, complemented
4C3×#64081C211yesmaximal, characteristic, complemented
5C32×#875C211yesmaximal, characteristic, complemented
6#11723542C214yesmaximal, complemented
7#11723525C214yesmaximal, complemented
8#11723524C211yesmaximal, characteristic, complemented
9#11723440C211yesmaximal, characteristic, complemented
10#10596145—31nomaximal
11#10594001—34nomaximal
12C22×D3×C3⋊D3C2212yes—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2214yescomplemented
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic, complemented
19C2×#6292C2212yes—
20C2×#6291C2212yes—
21C2×#6291C2214yescomplemented
22C2×#6289C2212yescomplemented
23C3⋊D3×D12—24no—
24#89018—24no—
25C33×C42C2211yescharacteristic, complemented
26C3×#3752C2212yes—
27C3×#3750C2212yescomplemented
28C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
29C32×(C2×D3)⋊C4C2214yescomplemented
30#88804C2214yescomplemented
31C2×#3832—31no—
32C2×D4×C3⋊D3—31no—
33C2×D3×D12—34no—
34C2×#3799—34no—
35C32×#124—31no—
36#64156—31no—
37#64149—31no—
38#64093—34no—
39#64091—34no—
40#64083—31no—
41#64083—34no—
42#64083—64no—
43#64081D311yescharacteristic, complemented
44C3×#875D314yescomplemented
45C3×#872—34no—
46#63923—316no—
47#63911—34no—
48#63910—316no—
49#63864—34no—
50C22×C3⋊(C3⋊D3)C2312yes—
51C22×C3⋊(C3⋊D3)C2312yes—
52C22×C3×C3⋊D3C2311yescharacteristic
53C22×C32×D3C2312yes—
54C2×D3×C3⋊D3—24no—
55C2×D3×C3⋊D3—44no—
56C2×C33×C4C2312yes—
57C2×C33×C4C2311yescharacteristic
58C4⋊(C3⋊(C3⋊D3))—22no—
59C4⋊(C3⋊(C3⋊D3))—22no—
60C4⋊(C3⋊(C3⋊D3))—44no—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C3×C4×C3⋊D3D414yescomplemented
64C2×C32×Dic3C2312yes—
65C32×D12—22no—
66C32×Dic6—22no—
67#6292—24no—
68#6291—24no—
69#6291—44no—
70#6289—24no—
71#5674—94no—
72#5525—91no—
73C23×C3⋊D3—32no—
74C22×D32—38no—
75C2×C32×Q8—31no—
76C2×C32×D4—31no—
77C2×#638—32no—
78C2×#638—34no—
79#3832—64no—
80D4×C3⋊D3—64no—
81C2×C4⋊(C3⋊D3)—32no—
82C2×C4⋊(C3⋊D3)—31no—
83C2×C4⋊(C3⋊D3)—31no—
84C2×C4⋊(C3⋊D3)—34no—
85C2×C4⋊(C3⋊D3)—34no—
86C2×C4⋊(C3⋊D3)—34no—
87C2×C4⋊(C3⋊D3)—316no—
88C2×C4⋊(C3⋊D3)—64no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C4⋊(C3⋊D3)—616no—
91C2×C4×C3⋊D3—32no—
92C2×C4×C3⋊D3C2×D311yescharacteristic
93C2×C3×D12C2×D314yes—
94C2×C3×C4×D3—34no—
95C2×C3×Dic6C2×D314yescomplemented
96C2×#626—38no—
97C2×#626—38no—
98C2×#626—316no—
99C2×#624—38no—
100D3×D12—616no—
101#3799—616no—
102C32×C22⋊C4—34no—
103C32×C42—24no—
104C32×C42C2×D314yescomplemented
105C32×C42C2×D311yescharacteristic, complemented
106#3752—34no—
107#3752C2×D312yes—
108#3750C2×D312yescomplemented
109C4×C3⋊Dic3C2×D311yescharacteristic, complemented
110C3×(C2×D3)⋊C4—38no—
111C3×(C2×D3)⋊C4C2×D3116yescomplemented
112C3×Dic3⋊C4—38no—
113C3×C4×Dic3—34no—
114#3722—316no—
115C22×C33C2411yescharacteristic
116C2×C3⋊(C3⋊D3)—22no—
117C2×C3⋊(C3⋊D3)—22no—
118C2×C3⋊(C3⋊D3)—24no—
119C2×C3⋊(C3⋊D3)—24no—
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3C2×D412yes—
122C2×C32×D3—22no—
123C2×C32×D3—24no—
124D3×C3⋊D3—48no—
125C33×C4—22no—
126C33×C4C2×D412yes—
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3C2×D412yes—
129C32×Dic3—22no—
130C2×#239—94no—
131C2×D3×D4—94no—
132C2×#235—91no—
133C22×D12—91no—
134C3×#124—34no—
135C3×#117—91no—
136#947—94no—
137#940—94no—
138#895—94no—
139#884—916no—
140#882—916no—
141#882—94no—
142#875—91no—
143#875C3⋊D311yescharacteristic, complemented
144#874—91no—
145#874—94no—
146#874—91no—
147#874—184no—
148#872—34no—
149C23×C32—32no—
150C22×C3⋊D3—32no—
151C22×C3⋊D3—32no—
152C22×C3⋊D3—38no—
153C22×C3⋊D3—38no—
154C22×C3⋊D3—68no—
155C22×C3⋊D3—68no—
156C22×C3⋊D3—64no—
157C22×C3⋊D3—124no—
158C22×C3⋊D3C22×D311yescharacteristic
159C22×C3×D3—34no—
160C22×C3×D3C22×D318yes—
161C2×D32—616no—
162C2×D32—1216no—
163C32×Q8—62no—
164C32×D4—62no—
165C2×C32×C4—28no—
166C2×C32×C4—24no—
167C2×C32×C4—32no—
168C2×C32×C4C22×D318yes—
169C2×C32×C4C22×D314yes—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172#638—64no—
173#638—124no—
174C2×C3⋊Dic3C22×D312yes—
175C2×C3⋊Dic3C22×D311yescharacteristic
176C4⋊(C3⋊D3)—64no—
177C4⋊(C3⋊D3)—62no—
178C4⋊(C3⋊D3)—62no—
179C4⋊(C3⋊D3)—68no—
180C4⋊(C3⋊D3)—68no—
181C4⋊(C3⋊D3)—128no—
182C4⋊(C3⋊D3)—128no—
183C4⋊(C3⋊D3)—124no—
184C4⋊(C3⋊D3)—1216no—
185C4⋊(C3⋊D3)—1232no—
186C4×C3⋊D3—64no—
187C4×C3⋊D3D1214yescomplemented
188C2×C3×Dic3—34no—
189C2×C3×Dic3—38no—
190C2×C3×Dic3C22×D318yes—
191C3×D12—28no—
192C3×C4×D3—316no—
193C3×Dic6—28no—
194#626—616no—
195#626—616no—
196#626—1216no—
197#624—616no—
198#535—271noSylow
199C2×C33C22×D411yescharacteristic
200C2×C33H(5,𝔽2)12yes—
201C3⋊(C3⋊D3)—42no—
202C3⋊(C3⋊D3)—42no—
203C3×C3⋊D3—22no—
204C32×D3—42no—
205C23×D3—98no—
206C23×D3—92no—
207C2×C3×Q8—34no—
208C2×C3×D4—34no—
209C22×C3×C4—91no—
210C2×#67—98no—
211C2×#67—916no—
212C2×#67—92no—
213#239—1816no—
214D3×D4—1816no—
215#235—184no—
216C2×D12—98no—
217C2×D12—92no—
218C2×D12—94no—
219C2×D12—94no—
220C2×D12—91no—
221C2×D12—91no—
222C2×D12—94no—
223C2×D12—916no—
224C2×D12—94no—
225C2×D12—184no—
226C2×D12—184no—
227C2×D12—1816no—
228C2×D12—184no—
229C2×D12C2×C3⋊D311yescharacteristic
230C2×C4×D3—34no—
231C2×C4×D3—98no—
232C2×C4×D3—92no—
233C2×Dic6C2×C3⋊D311yescharacteristic, complemented
234C3×C4⋊C4—92no—
235C3×C22⋊C4—316no—
236C3×C22⋊C4—92no—
237C3×C42—24no—
238C3×C42—91no—
239C3×C42D3214yescomplemented
240C3×C42C2×C3⋊D311yescharacteristic, complemented
241(C2×D3)⋊C4—38no—
242(C2×D3)⋊C4—916no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4C2×C3⋊D314yescomplemented
245Dic3⋊C4—38no—
246C4×Dic3—34no—
247C22×C32—24no—
248C22×C32—62no—
249C22×C32—64no—
250C22×C32C23×D314yes—
251C22×C32C23×D311yescharacteristic
252C2×C3⋊D3—22no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—64no—
256C2×C3⋊D3—64no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—616no—
260C2×C3⋊D3—616no—
261C2×C3⋊D3—128no—
262C2×C3⋊D3—128no—
263C2×C3⋊D3—1216no—
264C2×C3⋊D3—1216no—
265C2×C3⋊D3—128no—
266C2×C3⋊D3C2×D1212yes—
267C2×C3×D3—28no—
268C2×C3×D3—216no—
269C2×C3×D3—38no—
270C2×C3×D3—68no—
271D32—1232no—
272C32×C4—216no—
273C32×C4—28no—
274C32×C4—22no—
275C32×C4—28no—
276C32×C4—62no—
277C32×C4C2×D1212yes—
278C32×C4D3×D418yes—
279C3⋊Dic3—22no—
280C3⋊Dic3C2×D1212yes—
281C3×Dic3—28no—
282C3×Dic3—38no—
283C3×Dic3—68no—
284C2×#41—271no—
285C22×D4—271no—
286#127—271no—
287#127—271no—
288#124—31no—
289#124—271no—
290#121—274no—
291#120—274no—
292#117—91no—
293C33#53511yescharacteristic, complemented, Sylow
294C23×C3—38no—
295C23×C3—91no—
296C22×D3—34no—
297C22×D3—98no—
298C22×D3—98no—
299C22×D3—92no—
300C22×D3—92no—
301C22×D3—188no—
302C22×D3—188no—
303C22×D3—184no—
304C22×D3—1816no—
305C22×D3—364no—
306C22×D3—3616no—
307C22×D3C22×C3⋊D312yes—
308C3×Q8—68no—
309C3×D4—68no—
310C2×C3×C4—28no—
311C2×C3×C4—24no—
312C2×C3×C4—38no—
313C2×C3×C4—91no—
314C2×C3×C4—94no—
315C2×C3×C4—92no—
316C2×C3×C4C2×D3218yes—
317C2×C3×C4C2×D3214yes—
318C2×C3×C4C22×C3⋊D312yes—
319C2×C3×C4C22×C3⋊D311yescharacteristic
320#67—184no—
321#67—1816no—
322#67—3616no—
323C2×Dic3—34no—
324C2×Dic3—38no—
325C2×Dic3C22×C3⋊D312yes—
326D12—22no—
327D12—1816no—
328D12—184no—
329D12—188no—
330D12—188no—
331D12—182no—
332D12—182no—
333D12—368no—
334D12—368no—
335D12—3616no—
336D12—364no—
337D12—3632no—
338D12—364no—
339C4×D3—316no—
340C4×D3—184no—
341C4×D3—1816no—
342Dic6—22no—
343C2×C32—24no—
344C2×C32—28no—
345C2×C32—122no—
346C2×C32C22×D1211yescharacteristic
347C2×C32C2×D3×D414yes—
348C2×C32#100918yes—
349C2×C32#100912yes—
350C3⋊D3—22no—
351C3⋊D3—122no—
352C3⋊D3—122no—
353C3⋊D3—128no—
354C3⋊D3—128no—
355C3⋊D3—248no—
356C3⋊D3—248no—
357C3×D3—48no—
358C3×D3—68no—
359C24—272no—
360#41—544no—
361C2×Q8—31no—
362C2×D4—31no—
363C2×D4—272no—
364C2×D4—272no—
365C2×D4—274no—
366C2×D4—271no—
367C2×D4—271no—
368C2×D4—274no—
369C2×D4—544no—
370C22×C4—91no—
371C22×C4—272no—
372C4⋊C4—92no—
373C22⋊C4—34no—
374C22⋊C4—92no—
375C22⋊C4—274no—
376C42—91no—
377C42D3×C3⋊D311yescharacteristic, complemented
378C22×C3—24no—
379C22×C3—68no—
380C22×C3—616no—
381C22×C3—92no—
382C22×C3—182no—
383C22×C3C22×D3214yes—
384C22×C3C23×C3⋊D311yescharacteristic
385C2×D3—22no—
386C2×D3—24no—
387C2×D3—38no—
388C2×D3—68no—
389C2×D3—188no—
390C2×D3—188no—
391C2×D3—1816no—
392C2×D3—1816no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—184no—
396C2×D3—184no—
397C2×D3—368no—
398C2×D3—368no—
399C2×D3—3616no—
400C2×D3—3616no—
401C2×D3—368no—
402C2×D3—3632no—
403C3×C4—216no—
404C3×C4—28no—
405C3×C4—22no—
406C3×C4—28no—
407C3×C4—68no—
408C3×C4—92no—
409C3×C4—182no—
410C3×C4D3×D1218yes—
411C3×C4D4×C3⋊D312yes—
412Dic3—22no—
413Dic3—38no—
414Dic3—68no—
415C32—24no—
416C32#552511yescharacteristic, complemented
417C32#567414yescomplemented
418C23—32no—
419C23—91no—
420C23—272no—
421C23—272no—
422C23—544no—
423C23—1084no—
424Q8—62no—
425D4—62no—
426D4—544no—
427D4—544no—
428D4—542no—
429D4—542no—
430D4—1084no—
431D4—1084no—
432C2×C4—32no—
433C2×C4—91no—
434C2×C4—94no—
435C2×C4—92no—
436C2×C4—544no—
437C2×C4C2×D3×C3⋊D312yes—
438C2×C4C2×D3×C3⋊D311yescharacteristic
439C2×C3—24no—
440C2×C3—28no—
441C2×C3—128no—
442C2×C3—182no—
443C2×C3C2×D3×D1214yes—
444C2×C3#6430818yes—
445C2×C3C2×D4×C3⋊D311yescharacteristic
446C2×C3#6436612yes—
447D3—42no—
448D3—68no—
449D3—362no—
450D3—362no—
451D3—368no—
452D3—368no—
453D3—728no—
454D3—728no—
455C22—62no—
456C22—64no—
457C22—92no—
458C22—182no—
459C22—542no—
460C22—542no—
461C22—544no—
462C22—544no—
463C22—1088no—
464C22C22×D3×C3⋊D311yescharacteristic
465C4—22no—
466C4—62no—
467C4—92no—
468C4—182no—
469C4C3⋊D3×D1212yes—
470C3—24no—
471C3#1059400114yescomplemented
472C3#1059614511yescharacteristic, complemented
473C2—122no—
474C2—182no—
475C2—1082no—
476C2—1082no—
477C2C2×C3⋊D3×D1211yescharacteristic
478C2#1172458112yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×D12C2semidirect
2. C2×#89018C2semidirect
3. #11723915C2semidirect
4. C3×#64081C2semidirect
5. C32×#875C2semidirect
6. #11723542C2semidirect
7. #11723525C2semidirect
8. #11723524C2semidirect
9. #11723440C2semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C2×C32×Dic6C22semidirect
21. C2×#6291C22semidirect
22. C2×#6289C22semidirect
25. C33×C42C22semidirect
27. C3×#3750C22semidirect
28. C3×C4×C3⋊Dic3C22semidirect
29. C32×(C2×D3)⋊C4C22semidirect
30. #88804C22semidirect
43. #64081D3semidirect
44. C3×#875D3semidirect
63. C3×C4×C3⋊D3D4semidirect
95. C2×C3×Dic6C2×D3semidirect
104. C32×C42C2×D3semidirect
105. C32×C42C2×D3semidirect
108. #3750C2×D3semidirect
109. C4×C3⋊Dic3C2×D3semidirect
111. C3×(C2×D3)⋊C4C2×D3semidirect
143. #875C3⋊D3semidirect
187. C4×C3⋊D3D12semidirect
233. C2×Dic6C2×C3⋊D3semidirect
239. C3×C42D32semidirect
240. C3×C42C2×C3⋊D3semidirect
244. (C2×D3)⋊C4C2×C3⋊D3semidirect
293. C33#535semidirect
377. C42D3×C3⋊D3semidirect
416. C32#5525semidirect
417. C32#5674semidirect
471. C3#10594001semidirect
472. C3#10596145semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×#890182
#117239152
C3×#640812
C32×#8752
#117235422
#117235252
#117235242
#117234402
#105961453
#105940013