Finite Group Databasebeta

← #13160339

Subgroups — #13160339

Order 1728.

1252 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2#11723967C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722038C212yesmaximal, complemented
5#11722032C212yesmaximal, complemented
6#10592932—31nomaximal
7#10591084—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C2×C32×#67—21no—
10C32×#235—22no—
11C32×#235C414yescomplemented
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C2×C32×Dic6C2211yescharacteristic
15#88864—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2212yes—
17C4×C3⋊(C3⋊Dic3)—21no—
18C2×C3×C3⋊(C3⋊C8)—22no—
19C2×C3×C3⋊(C3⋊C8)C2212yes—
20C2×C32×#41—31no—
21C2×C3×#235D314yes—
22#64135—31no—
23#64135—34no—
24#64135—64no—
25C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
26C22×C3×C3⋊C8—34no—
27#63662—32no—
28#63659—32no—
29#63566—38no—
30#63563—38no—
31C23×C33D411yescharacteristic
32C22×C32×D3—21no—
33C2×C33×C4C2×C412yes—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4D412yes—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)—22no—
38C32×#67—24no—
39C2×C32×Dic3—21no—
40C32×D12—21no—
41C32×D12C2×C412yes—
42C32×C4×D3—24no—
43C32×Dic6—21no—
44C32×Dic6C2×C412yes—
45C3×C3⋊(C3⋊C8)—22no—
46#5233—94no—
47#5113—91no—
48C32×#41—34no—
49C32×#41—62no—
50C2×C32×Q8—31no—
51C2×C32×D4—31no—
52C2×C32×D4—61no—
53C22×C32×C4—24no—
54C22×C32×C4—61no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D314yes—
57C2×C3×#67—24no—
58C3×#235—28no—
59C3×#235Dic3116yescomplemented
60C2×C3×D12C2×D314yes—
61C2×C3×C4×D3—24no—
62C2×C3×Dic6C2×D314yes—
63#3757—68no—
64#3757—61no—
65#3757—64no—
66C4⋊(C3⋊Dic3)—32no—
67C4⋊(C3⋊Dic3)—38no—
68C4⋊(C3⋊Dic3)—68no—
69C4×C3⋊Dic3—68no—
70C4×C3⋊Dic3—61no—
71C4×C3⋊Dic3—64no—
72C2×C3⋊(C3⋊C8)—22no—
73C2×C3⋊(C3⋊C8)C2×D312yes—
74C2×C3×C3⋊C8—38no—
75C2×C3×C3⋊C8—68no—
76C22×C33—22no—
77C22×C33C22⋊C412yes—
78C22×C33C2×D411yescharacteristic
79C2×C32×D3—22no—
80C2×C32×D3—42no—
81C33×C4C22⋊C412yes—
82C33×C4C22×C411yescharacteristic
83C33×C4C2×D411yescharacteristic
84C3⋊(C3⋊Dic3)—42no—
85C32×Dic3—22no—
86C2×C3×#41—34no—
87C2×#235C3⋊D311yescharacteristic
88C22×C3×C8—91no—
89#926—94no—
90#926—91no—
91#926—184no—
92C22×C3⋊C8—34no—
93#835—98no—
94#832—98no—
95#821—92no—
96#816—92no—
97C23×C32—24no—
98C23×C32—61no—
99C23×C32D1211yescharacteristic
100C23×C32#6714yes—
101C22×C3×D3—24no—
102C32×Q8—32no—
103C32×Q8—61no—
104C32×D4—32no—
105C32×D4—64no—
106C32×D4—61no—
107C2×C32×C4—28no—
108C2×C32×C4—28no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—61no—
112C2×C32×C4—64no—
113C2×C32×C4C4×D312yes—
114C2×C32×C4D1211yescharacteristic
115C2×C32×C4C2×Dic318yes—
116C2×C32×C4#6718yes—
117C2×C32×C4#6714yes—
118C2×C32×C4#6712yes—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—616no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—68no—
124C3×#67—216no—
125C2×C3×Dic3—24no—
126C3×D12—24no—
127C3×D12C2×Dic318yes—
128C3×C4×D3—216no—
129C3×Dic6—24no—
130C3×Dic6C2×Dic318yes—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—68no—
133#416—271noSylow
134C2×C33—22no—
135C2×C33C2×C22⋊C411yescharacteristic
136C2×C33#13512yes—
137C32×D3—42no—
138C3×#41—316no—
139C3×#41—68no—
140C2×C3×Q8—34no—
141C2×C3×D4—34no—
142C2×C3×D4—64no—
143C22×C3×C4—24no—
144C22×C3×C4—64no—
145C22×C3×C4D3214yes—
146C22×C3×C4C2×C3⋊D311yescharacteristic
147C2×#67—21no—
148#235—22no—
149#235C3⋊Dic314yescomplemented
150C2×D12C2×C3⋊D311yescharacteristic
151C2×C4×D3—21no—
152C2×Dic6C2×C3⋊D311yescharacteristic
153C2×C3×C8—92no—
154C2×C3×C8—182no—
155#217—188no—
156#217—184no—
157#217—181no—
158C4⋊Dic3—98no—
159C4⋊Dic3—92no—
160C4⋊Dic3—188no—
161C4×Dic3—188no—
162C4×Dic3—184no—
163C4×Dic3—181no—
164C2×C3⋊C8—38no—
165C2×C3⋊C8—68no—
166C22×C32—22no—
167C22×C32—28no—
168C22×C32—24no—
169C22×C32—28no—
170C22×C32—48no—
171C22×C32—62no—
172C22×C32—122no—
173C22×C32(C2×D3)⋊C412yes—
174C22×C32#21718yes—
175C22×C32C2×D1211yescharacteristic
176C22×C32C2×#6714yes—
177C2×C3×D3—28no—
178C2×C3×D3—48no—
179C32×C4—28no—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—62no—
183C32×C4(C2×D3)⋊C412yes—
184C32×C4#21718yes—
185C32×C4C2×C4×D311yescharacteristic
186C32×C4C22×Dic314yes—
187C32×C4C2×#6714yes—
188C32×C4C2×#6711yescharacteristic
189C3⋊Dic3—1216no—
190C3⋊Dic3—122no—
191C3⋊Dic3—128no—
192C3×Dic3—28no—
193C2×#41—31no—
194C22×C8—91no—
195#117—271no—
196#103—272no—
197#102—272no—
198C33#41611yescharacteristic, complemented, Sylow
199C23×C3—24no—
200C23×C3—64no—
201C23×C3#62614yes—
202C23×C3#63811yescharacteristic
203C22×D3—21no—
204C3×Q8—38no—
205C3×Q8—64no—
206C3×D4—38no—
207C3×D4—616no—
208C3×D4—64no—
209C2×C3×C4—28no—
210C2×C3×C4—28no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—64no—
214C2×C3×C4—616no—
215C2×C3×C4D3×Dic318yes—
216C2×C3×C4#62518yes—
217C2×C3×C4#62614yes—
218C2×C3×C4C2×C3⋊Dic312yes—
219C2×C3×C4#63812yes—
220C2×C3×C4#63811yescharacteristic
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223#67—24no—
224C2×Dic3—21no—
225C2×Dic3—1816no—
226C2×Dic3—188no—
227C2×Dic3—182no—
228D12—21no—
229D12C2×C3⋊Dic312yes—
230C4×D3—24no—
231Dic6—21no—
232Dic6C2×C3⋊Dic312yes—
233C3×C8—182no—
234C3⋊C8—68no—
235C2×C32—22no—
236C2×C32—28no—
237C2×C32—24no—
238C2×C32—28no—
239C2×C32—48no—
240C2×C32—122no—
241C2×C32#90412yes—
242C2×C32C2×(C2×D3)⋊C411yescharacteristic
243C2×C32#95018yes—
244C2×C32C2×#21714yes—
245C3×D3—48no—
246#41—34no—
247#41—62no—
248C2×Q8—31no—
249C2×D4—31no—
250C2×D4—61no—
251C22×C4—61no—
252C22×C4D3×C3⋊D311yescharacteristic
253C2×C8—92no—
254C2×C8—182no—
255C4⋊C4—272no—
256C22⋊C4—541no—
257C42—541no—
258C22×C3—28no—
259C22×C3—22no—
260C22×C3—24no—
261C22×C3—28no—
262C22×C3—48no—
263C22×C3—68no—
264C22×C3—128no—
265C22×C3(C2×D3)⋊Dic318yes—
266C22×C3#375712yes—
267C22×C3C2×#62614yes—
268C22×C3C2×#63811yescharacteristic
269C2×D3—22no—
270C2×D3—42no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—28no—
274C3×C4—68no—
275C3×C4(C2×D3)⋊Dic318yes—
276C3×C4#375712yes—
277C3×C4C2×D3×Dic314yes—
278C3×C4C2×#62514yes—
279C3×C4C22×C3⋊Dic311yescharacteristic
280C3×C4C2×#63811yescharacteristic
281Dic3—22no—
282Dic3—3616no—
283Dic3—368no—
284Dic3—362no—
285C32—24no—
286C32#511311yescharacteristic, complemented
287C32#523314yescomplemented
288C23—61no—
289C23#629111yescharacteristic
290Q8—32no—
291Q8—61no—
292D4—32no—
293D4—64no—
294D4—61no—
295C2×C4—61no—
296C2×C4—64no—
297C2×C4—542no—
298C2×C4D3×C3⋊Dic312yes—
299C2×C4#629012yes—
300C2×C4#629111yescharacteristic
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—182no—
303C2×C3—28no—
304C2×C3—22no—
305C2×C3—24no—
306C2×C3—28no—
307C2×C3—48no—
308C2×C3—128no—
309C2×C3#6382918yes—
310C2×C3C2×(C2×D3)⋊Dic314yes—
311C2×C3#6415912yes—
312C2×C3C2×#375711yescharacteristic
313D3—42no—
314C22—22no—
315C22—62no—
316C22—122no—
317C22(C2×D3)⋊(C3⋊Dic3)12yes—
318C22C2×#629111yescharacteristic
319C4—62no—
320C4—1082no—
321C4(C2×D3)⋊(C3⋊Dic3)12yes—
322C4C2×D3×C3⋊Dic311yescharacteristic
323C4C2×#629011yescharacteristic
324C3—24no—
325C3#1059108414yescomplemented
326C3#1059293211yescharacteristic, complemented
327C2—22no—
328C2—122no—
329C2#1172337712yes—
330C2C2×(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC2×C4#6291
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C32×#235C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11723967C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. #11722038C2semidirect
5. #11722032C2semidirect
11. C32×#235C4semidirect
25. C22×C3⋊(C3⋊C8)D3semidirect
59. C3×#235Dic3semidirect
149. #235C3⋊Dic3semidirect
198. C33#416semidirect
286. C32#5113semidirect
287. C32#5233semidirect
325. C3#10591084semidirect
326. C3#10592932semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
#117239672
C22×C3×C3⋊(C3⋊C8)2
#117220382
#117220322
#105929323
#105910843