Finite Group Databasebeta

← #13161444

Subgroups — #13161444

Order 1728.

1932 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88864C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×Dic3⋊(C3⋊Dic3)C212yesmaximal, complemented
5#11722061C212yesmaximal, complemented
6#10592649—31nomaximal
7#10591718—34nomaximal
8C24×C33C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3—22no—
12C22×C3×C3⋊Dic3—24no—
13C22×C3×C3⋊Dic3C2212yes—
14C22×C32×Dic3C2212yes—
15#88864—22no—
16C32×#217C414yescomplemented
17Dic3⋊(C3⋊Dic3)—24no—
18C23×C3⋊Dic3D311yescharacteristic
19C23×C3×Dic3—34no—
20C2×C32×C22⋊C4—31no—
21C2×#3757—31no—
22C2×#3757—34no—
23C2×#3757—64no—
24C2×C4⋊(C3⋊Dic3)—32no—
25C2×C3×#217D314yes—
26C2×Dic3⋊Dic3—38no—
27#63658—32no—
28#63579—38no—
29C23×C33—22no—
30C23×C33—22no—
31C23×C33C2×C412yes—
32C23×C33D412yes—
33C23×C33Q812yes—
34C23×C33C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3⋊(C3⋊Dic3)—21no—
37C2×C3⋊(C3⋊Dic3)—22no—
38C2×C3⋊(C3⋊Dic3)—22no—
39C2×C3×C3⋊Dic3—28no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3D414yes—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C414yescomplemented
46#5210—91no—
47#4950—94no—
48C24×C32—24no—
49C24×C32C2×D311yescharacteristic
50C24×C32C2×D314yes—
51C22×C32×C4—32no—
52C22×C3⋊Dic3—22no—
53C22×C3⋊Dic3—24no—
54C22×C3⋊Dic3—31no—
55C22×C3⋊Dic3—31no—
56C22×C3⋊Dic3—34no—
57C22×C3⋊Dic3—34no—
58C22×C3⋊Dic3—64no—
59C22×C3⋊Dic3—64no—
60C22×C3⋊Dic3C2×D312yes—
61C22×C3×Dic3—38no—
62C22×C3×Dic3—616no—
63C22×C3×Dic3—68no—
64C22×C3×Dic3C2×D318yes—
65C32×C22⋊C4—34no—
66#3757—616no—
67#3757—68no—
68#3757—62no—
69C4⋊(C3⋊Dic3)—64no—
70C3×#217Dic3116yescomplemented
71Dic3⋊Dic3—616no—
72C22×C33—24no—
73C22×C33—22no—
74C22×C33—24no—
75C22×C33—22no—
76C22×C33C4⋊C414yescomplemented
77C22×C33C22×C411yescharacteristic
78C22×C33C2×D411yescharacteristic
79C22×C33C2×D412yes—
80C22×C33C2×Q811yescharacteristic
81C22×C33#4112yes—
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—24no—
85C32×Dic3—24no—
86C23×C3×C4—91no—
87C23×Dic3—34no—
88C2×C3×C22⋊C4—34no—
89C2×#217—94no—
90C2×#217—91no—
91C2×#217—184no—
92C2×#217C3⋊D311yescharacteristic
93C2×C4⋊Dic3—98no—
94C2×Dic3⋊C4—92no—
95C3⋊#95—98no—
96C3⋊#95—92no—
97C23×C32—22no—
98C23×C32—22no—
99C23×C32—28no—
100C23×C32—28no—
101C23×C32—28no—
102C23×C32—24no—
103C23×C32—28no—
104C23×C32—28no—
105C23×C32—48no—
106C23×C32—48no—
107C23×C32Dic618yes—
108C23×C32Dic612yes—
109C23×C32C4×D312yes—
110C23×C32D1218yes—
111C23×C32C2×Dic318yes—
112C23×C32#6712yes—
113C23×C32C22×D314yes—
114C23×C32C22×D311yescharacteristic
115C2×C32×C4—34no—
116C2×C32×C4—62no—
117C2×C32×C4—62no—
118C2×C3⋊Dic3—28no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—68no—
122C2×C3⋊Dic3—68no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—62no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—68no—
130C2×C3⋊Dic3—68no—
131C2×C3⋊Dic3—128no—
132C2×C3⋊Dic3—128no—
133C2×C3⋊Dic3#6714yes—
134C2×C3×Dic3—28no—
135C2×C3×Dic3—28no—
136C2×C3×Dic3—316no—
137C2×C3×Dic3—632no—
138C2×C3×Dic3—68no—
139C2×C3×Dic3—68no—
140C2×C3×Dic3C2×Dic3116yescomplemented
141#385—271noSylow
142C2×C33—24no—
143C2×C33C2×C4⋊C411yescharacteristic
144C2×C33C4×D412yes—
145C2×C33#12011yescharacteristic
146C2×C33C22⋊Q812yes—
147C2×C33#12311yescharacteristic
148C24×C3—24no—
149C24×C3D3214yes—
150C24×C3C2×C3⋊D311yescharacteristic
151C22×C3×C4—38no—
152C22×C3×C4—92no—
153C22×C3×C4—184no—
154C22×C3×C4—182no—
155C22×Dic3—38no—
156C22×Dic3—616no—
157C22×Dic3—68no—
158C22×Dic3—94no—
159C22×Dic3—94no—
160C22×Dic3—91no—
161C22×Dic3—91no—
162C22×Dic3—184no—
163C22×Dic3—184no—
164C22×Dic3C2×C3⋊D312yes—
165C3×C22⋊C4—316no—
166#217—1816no—
167#217—188no—
168#217—182no—
169#217C3⋊Dic314yescomplemented
170C4⋊Dic3—1816no—
171Dic3⋊C4—184no—
172C22×C32—24no—
173C22×C32—24no—
174C22×C32—22no—
175C22×C32—22no—
176C22×C32—216no—
177C22×C32—24no—
178C22×C32—28no—
179C22×C32—24no—
180C22×C32—24no—
181C22×C32—28no—
182C22×C32—216no—
183C22×C32—28no—
184C22×C32—28no—
185C22×C32—216no—
186C22×C32—416no—
187C22×C32—48no—
188C22×C32—416no—
189C22×C32—48no—
190C22×C32Dic3⋊C414yescomplemented
191C22×C32C4⋊Dic3116yescomplemented
192C22×C32C2×Dic614yes—
193C22×C32C2×Dic611yescharacteristic
194C22×C32C2×C4×D311yescharacteristic
195C22×C32C2×D1214yes—
196C22×C32#23512yes—
197C22×C32D3×D418yes—
198C22×C32#23718yes—
199C22×C32C22×Dic314yes—
200C22×C32C2×#6711yescharacteristic
201C22×C32C2×#6712yes—
202C32×C4—64no—
203C3⋊Dic3—24no—
204C3⋊Dic3—128no—
205C3⋊Dic3—128no—
206C3⋊Dic3—124no—
207C3⋊Dic3—124no—
208C3⋊Dic3—121no—
209C3⋊Dic3—121no—
210C3×Dic3—216no—
211C3×Dic3—616no—
212C23×C4—91no—
213C2×C4⋊C4—272no—
214C2×C22⋊C4—31no—
215C2×C22⋊C4—271no—
216#95—272no—
217C33#38511yescharacteristic, complemented, Sylow
218C23×C3—28no—
219C23×C3—28no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—28no—
223C23×C3—28no—
224C23×C3—28no—
225C23×C3—24no—
226C23×C3—48no—
227C23×C3—48no—
228C23×C3D3×Dic318yes—
229C23×C3#62618yes—
230C23×C3#62718yes—
231C23×C3#63412yes—
232C23×C3C4⋊(C3⋊D3)12yes—
233C23×C3C2×C3⋊Dic312yes—
234C23×C3C2×D3214yes—
235C23×C3C22×C3⋊D311yescharacteristic
236C2×C3×C4—316no—
237C2×C3×C4—68no—
238C2×C3×C4—68no—
239C2×C3×C4—94no—
240C2×C3×C4—188no—
241C2×C3×C4—182no—
242C2×C3×C4—182no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—316no—
246C2×Dic3—632no—
247C2×Dic3—68no—
248C2×Dic3—68no—
249C2×Dic3—188no—
250C2×Dic3—188no—
251C2×Dic3—188no—
252C2×Dic3—184no—
253C2×Dic3—184no—
254C2×Dic3—188no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—182no—
258C2×Dic3—182no—
259C2×Dic3—368no—
260C2×Dic3—368no—
261C2×Dic3C2×C3⋊Dic314yescomplemented
262C2×C32—24no—
263C2×C32—28no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—28no—
267C2×C32—24no—
268C2×C32—216no—
269C2×C32—416no—
270C2×C32#87818yes—
271C2×C32#88214yes—
272C2×C32#88614yes—
273C2×C32C2×Dic3⋊C411yescharacteristic
274C2×C32#92412yes—
275C2×C32C2×C4⋊Dic314yes—
276C2×C32C4×#6712yes—
277C2×C32#92911yescharacteristic
278C2×C32D4×Dic318yes—
279C2×C32#95311yescharacteristic
280C24D3×C3⋊D311yescharacteristic
281C22×C4—32no—
282C22×C4—92no—
283C22×C4—184no—
284C22×C4—182no—
285C22×C4—271no—
286C22×C4—271no—
287C4⋊C4—544no—
288C22⋊C4—34no—
289C22⋊C4—542no—
290C22×C3—24no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—24no—
294C22×C3—216no—
295C22×C3—28no—
296C22×C3—216no—
297C22×C3—28no—
298C22×C3—24no—
299C22×C3—28no—
300C22×C3—24no—
301C22×C3—24no—
302C22×C3—28no—
303C22×C3—216no—
304C22×C3—48no—
305C22×C3—416no—
306C22×C3—48no—
307C22×C3—416no—
308C22×C3Dic3⋊Dic3116yescomplemented
309C22×C3C4⋊(C3⋊Dic3)14yescomplemented
310C22×C3C2×D3×Dic314yes—
311C22×C3#380418yes—
312C22×C3C2×#62614yes—
313C22×C3C2×#62714yes—
314C22×C3D3×#6718yes—
315C22×C3C2×#63411yescharacteristic
316C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
317C22×C3D4×C3⋊D312yes—
318C22×C3#383012yes—
319C22×C3C22×C3⋊Dic311yescharacteristic
320C3×C4—616no—
321C3×C4—184no—
322Dic3—24no—
323Dic3—616no—
324Dic3—368no—
325Dic3—368no—
326Dic3—361no—
327Dic3—361no—
328Dic3—364no—
329Dic3—364no—
330C32—24no—
331C32#495014yescomplemented
332C32#521011yescharacteristic, complemented
333C23—22no—
334C23—22no—
335C23D3×C3⋊Dic312yes—
336C23#629212yes—
337C23#629312yes—
338C23C2×D3×C3⋊D311yescharacteristic
339C2×C4—34no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—94no—
343C2×C4—188no—
344C2×C4—182no—
345C2×C4—182no—
346C2×C4—542no—
347C2×C4—541no—
348C2×C4—541no—
349C2×C4—542no—
350C2×C3—24no—
351C2×C3—216no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—28no—
355C2×C3—24no—
356C2×C3—28no—
357C2×C3—416no—
358C2×C3#6396418yes—
359C2×C3C2×Dic3⋊Dic314yes—
360C2×C3#6396614yes—
361C2×C3Dic3×#6718yes—
362C2×C3#6397714yes—
363C2×C3#6408712yes—
364C2×C3#6409111yescharacteristic
365C2×C3#6409511yescharacteristic
366C2×C3C2×C4⋊(C3⋊Dic3)11yescharacteristic
367C2×C3D4×C3⋊Dic312yes—
368C22—22no—
369C22—24no—
370C22—22no—
371C22—24no—
372C22Dic3⋊(C3⋊Dic3)14yescomplemented
373C22C2×D3×C3⋊Dic311yescharacteristic
374C22#8902812yes—
375C22C2×#629211yescharacteristic
376C22C2×#629311yescharacteristic
377C22C3⋊D3×#6712yes—
378C4—64no—
379C4—184no—
380C4—1081no—
381C4—1081no—
382C3—24no—
383C3#1059171814yescomplemented
384C3#1059264911yescharacteristic, complemented
385C2—24no—
386C2C2×Dic3⋊(C3⋊Dic3)11yescharacteristic
387C2#1172360811yescharacteristic
388C2#67×C3⋊Dic312yes—
389C2#1172363611yescharacteristic
390C2#1172364112yes—
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
Fitting, prime radicalC24×C33C22
socle, cosocleC23×C33C23
hypercenterC24D3×C3⋊D3
hypocenterC33#385
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊(C3⋊Dic3)C2semidirect
5. #11722061C2semidirect
16. C32×#217C4semidirect
45. C2×C32×Dic3C2×C4semidirect
70. C3×#217Dic3semidirect
76. C22×C33C4⋊C4semidirect
140. C2×C3×Dic3C2×Dic3semidirect
169. #217C3⋊Dic3semidirect
190. C22×C32Dic3⋊C4semidirect
191. C22×C32C4⋊Dic3semidirect
217. C33#385semidirect
261. C2×Dic3C2×C3⋊Dic3semidirect
308. C22×C3Dic3⋊Dic3semidirect
309. C22×C3C4⋊(C3⋊Dic3)semidirect
331. C32#4950semidirect
332. C32#5210semidirect
372. C22Dic3⋊(C3⋊Dic3)semidirect
383. C3#10591718semidirect
384. C3#10592649semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×#888642
C2×C32×#2172
C2×Dic3⋊(C3⋊Dic3)2
#117220612
#105926493
#105917183