Finite Group Databasebeta

← C3×#10590787 #13157777

Subgroups — C3×#10590787 #13157777

Order 1728.

522 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#627C211yesmaximal, characteristic
2C3×C8×C3⋊Dic3C211yesmaximal, characteristic
3C32×#815C212yesmaximal
4C3×#63572C211yesmaximal, characteristic
5C3×Dic3⋊(C3⋊C8)C212yesmaximal
6C3×#4701—32nomaximal
7#10590787C311yesmaximal, characteristic, direct product factor
8C2×C3×#627C411yescharacteristic
9C2×C33×C8C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4×Dic3C2212yes—
13C2×C32×C3⋊C8C2212yes—
14C3×#3724C411yescharacteristic
15C3×Dic3⋊Dic3C412yes—
16C3×C4×Dic6—32no—
17C4×#627C2×C311yescharacteristic
18C32×#97—32no—
19C8×C3⋊Dic3C2×C311yescharacteristic
20C3×#815—22no—
21C3×#815D312yes—
22C3×#815C2×C312yes—
23C3×Dic3⋊C8—32no—
24C3×Dic3⋊C8—32no—
25C3×C8×Dic3—32no—
26C3×C8×Dic3—61no—
27C3×C4⋊(C3⋊C8)—32no—
28#63572C2×C311yescharacteristic
29Dic3⋊(C3⋊C8)C2×C312yes—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C32×Dic3C2×C412yes—
34C2×C32×Dic3C2×C412yes—
35C3×#627—22no—
36C33×C8Q812yescomplemented
37C3×C3⋊(C3⋊C8)—21no—
38C32×C3⋊C8—22no—
39C3×#446—91no—
40#4701—32no—
41C2×C3×Dic6—32no—
42C2×#627C3×C411yescharacteristic
43C2×C32×C8—21no—
44C2×C32×C8—22no—
45C2×C32×C8—32no—
46C2×C32×C8—41no—
47C2×C32×C8C2×D312yes—
48C2×C32×C8C22×C311yescharacteristic
49C32×C42—32no—
50C4×C3⋊Dic3C22×C311yescharacteristic
51C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
52C3×C4⋊Dic3—32no—
53C3×Dic3⋊C4—32no—
54C3×Dic3⋊C4—32no—
55C3×C4×Dic3—22no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C3×C4×Dic3C2×D312yes—
59C3×C4×Dic3C22×C312yes—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—61no—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C22×C312yes—
65#3724C3×C411yescharacteristic
66Dic3⋊Dic3C3×C412yes—
67C22×C33C22×C411yescharacteristic
68C33×C4C2×Q811yescharacteristic
69C33×C4#4111yescharacteristic
70C3×C3⋊Dic3—21no—
71C3×C3⋊Dic3M1612yes—
72C32×Dic3—22no—
73C32×Dic3—22no—
74C3×C4×Q8—91no—
75C4×Dic6—32no—
76C3×C4⋊C8—92no—
77C3×C4⋊C8—91no—
78C3×#97—32no—
79C3×#97—32no—
80C3×#97—62no—
81C3×C4×C8—91no—
82#815C3×D312yes—
83Dic3⋊C8—32no—
84Dic3⋊C8—32no—
85C8×Dic3—32no—
86C8×Dic3—61no—
87C4⋊(C3⋊C8)—32no—
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—32no—
91C2×C32×C4—32no—
92C2×C32×C4—41no—
93C2×C32×C4C22×D312yes—
94C2×C32×C4C23×C311yescharacteristic
95C2×C3⋊Dic3C2×C3×C411yescharacteristic
96C2×C3⋊Dic3C2×C3×C411yescharacteristic
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—32no—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3C4×D312yes—
104C2×C3×Dic3C4×D312yes—
105C2×C3×Dic3C2×C3×C412yes—
106C2×C3×Dic3C2×C3×C412yes—
107C3×Dic6—64no—
108#627—22no—
109C32×C8—22no—
110C32×C8—24no—
111C32×C8—42no—
112C32×C8—62no—
113C32×C8Dic614yescomplemented
114C32×C8C3×Q812yescomplemented
115C3⋊(C3⋊C8)—21no—
116C3×C3⋊C8—24no—
117C3×C3⋊C8—22no—
118C3×C3⋊C8—22no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—62no—
121#446—91noSylow
122C2×C33C4×Q811yescharacteristic
123C2×C33C2×M1611yescharacteristic
124C2×C33#13111yescharacteristic
125C2×C3×Q8—91no—
126C2×Dic6—32no—
127C2×C3×C8—21no—
128C2×C3×C8—22no—
129C2×C3×C8—32no—
130C2×C3×C8—32no—
131C2×C3×C8—41no—
132C2×C3×C8—62no—
133C2×C3×C8—91no—
134C2×C3×C8D3211yescharacteristic
135C2×C3×C8C2×C3×D312yes—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—92no—
138C3×C42—32no—
139C3×C42—32no—
140C3×C42—62no—
141C3×C42—91no—
142C4⋊Dic3—32no—
143Dic3⋊C4—32no—
144Dic3⋊C4—32no—
145C4×Dic3—32no—
146C4×Dic3—61no—
147C4×Dic3C2×C3×D312yes—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—61no—
150C2×C3⋊C8C2×C3×D312yes—
151C22×C32—21no—
152C22×C32—22no—
153C22×C32—41no—
154C22×C32C2×C4×D312yes—
155C22×C32C22×C3×C411yescharacteristic
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—22no—
159C32×C4—22no—
160C32×C4—41no—
161C32×C4—41no—
162C32×C4—62no—
163C32×C4—62no—
164C32×C4C2×Dic612yes—
165C32×C4#23512yes—
166C32×C4C2×C3×Q811yescharacteristic
167C32×C4C3×#4111yescharacteristic
168C3⋊Dic3—21no—
169C3⋊Dic3C3×M1612yes—
170C3×Dic3—22no—
171C3×Dic3—22no—
172C3×Dic3—22no—
173C3×Dic3—22no—
174C3×Dic3—34no—
175C3×Dic3—42no—
176C3×Dic3—42no—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C4×Q8—91no—
181C4⋊C8—92no—
182C4⋊C8—91no—
183#97—32no—
184C4×C8—91no—
185C33#44611yescharacteristic, complemented, Sylow
186C3×Q8—182no—
187C2×C3×C4—22no—
188C2×C3×C4—21no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4C2×D3211yescharacteristic
199C2×C3×C4C22×C3×D312yes—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3C3×C4×D312yes—
205C2×Dic3C3×C4×D312yes—
206Dic6—64no—
207C3×C8—24no—
208C3×C8—22no—
209C3×C8—42no—
210C3×C8—64no—
211C3×C8—62no—
212C3×C8—62no—
213C3×C8—181no—
214C3×C8#62712yescomplemented
215C3×C8C3×Dic614yescomplemented
216C3⋊C8—22no—
217C3⋊C8—62no—
218C3⋊C8—62no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—22no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32C4×Dic612yes—
229C2×C32D3×M1612yes—
230C2×C32#90712yes—
231C2×C32C3×C4×Q811yescharacteristic
232C2×C32C2×C3×M1611yescharacteristic
233C2×C32C3×#13111yescharacteristic
234C2×Q8—91no—
235C2×C8—32no—
236C2×C8—91no—
237C2×C8C3×D3211yescharacteristic
238C4⋊C4—91no—
239C4⋊C4—92no—
240C42—32no—
241C42—91no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3C4×D3211yescharacteristic
246C22×C3C2×C3×C4×D312yes—
247C3×C4—21no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—62no—
254C3×C4—62no—
255C3×C4—62no—
256C3×C4—62no—
257C3×C4—92no—
258C3×C4—122no—
259C3×C4—122no—
260C3×C4—181no—
261C3×C4#379811yescharacteristic
262C3×C4C2×#62711yescharacteristic
263C3×C4C2×C3×Dic612yes—
264C3×C4C3×#23512yes—
265Dic3—22no—
266Dic3—22no—
267Dic3—34no—
268Dic3—62no—
269Dic3—62no—
270Dic3—62no—
271C32—22no—
272C32—21no—
273C32—41no—
274C32#470112yescomplemented
275C32C3×#44611yescharacteristic, complemented
276Q8—182no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4C2×C3×D3211yescharacteristic
282C8—62no—
283C8—181no—
284C8C3×#62712yescomplemented
285C2×C3—21no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3#6379111yescharacteristic
295C2×C3#6380511yescharacteristic
296C2×C3C4×#62711yescharacteristic
297C2×C3C3×C4×Dic612yes—
298C2×C3C3×D3×M1612yes—
299C2×C3C3×#90712yes—
300C22C3×C4×D3211yescharacteristic
301C4—62no—
302C4—62no—
303C4—92no—
304C4—181no—
305C4C3×#379811yescharacteristic
306C4C2×C3×#62711yescharacteristic
307C3—22no—
308C3—21no—
309C3—41no—
310C3#1059078711yescharacteristic, direct product factor
311C3C3×#470112yescomplemented
312C2C3×#6379111yescharacteristic
313C2C3×#6380511yescharacteristic
314C2C3×C4×#62711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C22×C4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#446
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
7. #10590787C3direct (N × H)
36. C33×C8Q8semidirect
113. C32×C8Dic6semidirect
114. C32×C8C3×Q8semidirect
185. C33#446semidirect
214. C3×C8#627semidirect
215. C3×C8C3×Dic6semidirect
274. C32#4701semidirect
275. C32C3×#446semidirect
284. C8C3×#627semidirect
310. C3#10590787direct (N × H)
311. C3C3×#4701semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6272
C3×C8×C3⋊Dic32
C32×#8152
C3×#635722
C3×Dic3⋊(C3⋊C8)2
C3×#47013
#105907873