Subgroups — C₃²×#1814 #11768863

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#421	C₂	1	2	yes	maximal
5	C₃²×#344	C₂	1	2	yes	maximal, complemented
6	C₃×#1814	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
8	C₃²×#131	—	2	1	no	—
9	C₃²×#131	—	2	1	no	—
10	C₃²×#131	C₄	1	2	yes	complemented
11	C₃²×#131	C₄	1	2	yes	complemented
12	C₂×C₃²×M₁₆	—	2	1	no	—
13	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
14	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₈	—	2	1	no	—
16	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
17	C₃²×#117	C₂²	1	2	yes	—
18	C₃²×#108	—	2	1	no	—
19	C₃²×#108	—	2	1	no	—
20	C₃²×#107	—	2	1	no	—
21	C₃²×#106	—	2	1	no	—
22	C₃²×#104	—	2	4	no	—
23	C₂×C₃×#131	C₂×C₃	1	4	yes	—
24	C₂×C₃×#108	C₂×C₃	1	4	yes	complemented
25	C₃×#427	C₂×C₃	1	4	yes	complemented
26	C₃×#421	C₂×C₃	1	8	yes	—
27	C₃×#344	C₂×C₃	1	8	yes	complemented
28	C₃²×#41	—	2	1	no	—
29	C₃²×#41	—	2	1	no	—
30	C₃²×#41	C₂×C₄	1	2	yes	—
31	C₃²×#41	D₄	1	2	yes	—
32	C₂×C₃²×Q₈	Q₈	1	1	yes	characteristic
33	C₂×C₃²×D₄	—	2	1	no	—
34	C₂×C₃²×D₄	Q₈	1	1	yes	characteristic
35	C₂²×C₃²×C₄	—	2	1	no	—
36	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
37	C₃²×M₁₆	—	2	1	no	—
38	C₃²×M₁₆	—	2	2	no	—
39	C₃²×M₁₆	—	2	2	no	—
40	C₃²×M₁₆	—	2	2	no	—
41	C₃²×M₁₆	—	2	2	no	—
42	C₃²×M₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₂×C₃²×C₈	—	2	2	no	—
45	C₂×C₃²×C₈	—	2	2	no	—
46	C₂×C₃²×C₈	—	2	2	no	—
47	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
50	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
51	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
52	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
53	C₃²×C₄⋊C₄	—	2	2	no	—
54	C₃²×C₂²⋊C₄	—	2	2	no	—
55	C₃²×C₄²	—	2	2	no	—
56	#1814	C₃²	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
58	C₃×#131	—	2	4	no	—
59	C₃×#131	—	2	4	no	—
60	C₃×#131	C₃×C₄	1	8	yes	complemented
61	C₃×#131	C₃×C₄	1	8	yes	complemented
62	C₂×C₃×M₁₆	—	2	4	no	—
63	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
64	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
65	C₂²×C₃×C₈	—	2	4	no	—
66	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
67	C₃×#117	C₂²×C₃	1	8	yes	—
68	C₃×#108	—	2	4	no	—
69	C₃×#108	—	2	4	no	—
70	C₃×#107	—	2	4	no	—
71	C₃×#106	—	2	4	no	—
72	C₃×#104	—	2	16	no	—
73	C₂³×C₃²	—	2	1	no	—
74	C₂³×C₃²	#41	1	1	yes	characteristic
75	C₃²×Q₈	—	2	1	no	—
76	C₃²×Q₈	C₄⋊C₄	1	2	yes	—
77	C₃²×D₄	—	2	2	no	—
78	C₃²×D₄	—	2	2	no	—
79	C₃²×D₄	—	2	1	no	—
80	C₃²×D₄	C₄⋊C₄	1	2	yes	—
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	4	2	no	—
85	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₃²×C₄	#41	1	1	yes	characteristic
91	C₃²×C₈	—	2	2	no	—
92	C₃²×C₈	—	4	2	no	—
93	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×#131	C₂×C₃²	1	1	yes	characteristic
98	C₂×#108	C₂×C₃²	1	1	yes	characteristic, complemented
99	#427	C₂×C₃²	1	1	yes	characteristic, complemented
100	#421	C₂×C₃²	1	2	yes	—
101	#344	C₂×C₃²	1	2	yes	complemented
102	C₃×#41	—	2	4	no	—
103	C₃×#41	—	2	4	no	—
104	C₃×#41	C₂×C₃×C₄	1	8	yes	—
105	C₃×#41	C₃×D₄	1	8	yes	—
106	C₂×C₃×Q₈	C₃×Q₈	1	4	yes	—
107	C₂×C₃×D₄	—	2	4	no	—
108	C₂×C₃×D₄	C₃×Q₈	1	4	yes	—
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
111	C₃×M₁₆	—	2	4	no	—
112	C₃×M₁₆	—	2	8	no	—
113	C₃×M₁₆	—	2	8	no	—
114	C₃×M₁₆	—	2	8	no	—
115	C₃×M₁₆	—	2	8	no	—
116	C₃×M₁₆	C₂×C₃×C₄	1	8	yes	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	—	2	8	no	—
119	C₂×C₃×C₈	—	2	8	no	—
120	C₂×C₃×C₈	—	2	8	no	—
121	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
122	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
123	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
124	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
125	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
126	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
127	C₃×C₄⋊C₄	—	2	8	no	—
128	C₃×C₂²⋊C₄	—	2	8	no	—
129	C₃×C₄²	—	2	8	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	—	4	1	no	—
132	C₂²×C₃²	—	4	2	no	—
133	C₂²×C₃²	#117	1	1	yes	characteristic
134	C₂²×C₃²	#121	1	1	yes	characteristic
135	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
136	C₃²×C₄	—	2	2	no	—
137	C₃²×C₄	—	8	2	no	—
138	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₃²×C₄	#121	1	1	yes	characteristic
141	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
143	#131	—	2	1	no	—
144	#131	—	2	1	no	—
145	#131	C₃²×C₄	1	2	yes	complemented
146	#131	C₃²×C₄	1	2	yes	complemented
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
149	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
152	#117	C₂²×C₃²	1	2	yes	—
153	#108	—	2	1	no	—
154	#108	—	2	1	no	—
155	#107	—	2	1	no	—
156	#106	—	2	1	no	—
157	#104	—	2	4	no	—
158	C₂³×C₃	—	2	4	no	—
159	C₂³×C₃	C₃×#41	1	4	yes	—
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	C₃×C₄⋊C₄	1	8	yes	—
162	C₃×D₄	—	2	8	no	—
163	C₃×D₄	—	2	8	no	—
164	C₃×D₄	—	2	4	no	—
165	C₃×D₄	C₃×C₄⋊C₄	1	8	yes	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	8	no	—
169	C₂×C₃×C₄	—	4	8	no	—
170	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
171	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
174	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
175	C₂×C₃×C₄	C₃×#41	1	4	yes	—
176	C₃×C₈	—	2	8	no	—
177	C₃×C₈	—	4	8	no	—
178	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
179	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
180	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
181	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
182	C₂×C₃²	—	2	1	no	—
183	C₂×C₃²	—	2	1	no	—
184	C₂×C₃²	—	2	1	no	—
185	C₂×C₃²	—	4	2	no	—
186	C₂×C₃²	#380	1	1	yes	characteristic
187	#41	—	2	1	no	—
188	#41	—	2	1	no	—
189	#41	C₂×C₃²×C₄	1	2	yes	—
190	#41	C₃²×D₄	1	2	yes	—
191	C₂×Q₈	C₃²×Q₈	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₃²×Q₈	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	C₂×C₃²×C₄	1	2	yes	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
208	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
209	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
210	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
211	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₂²×C₃	—	2	8	no	—
216	C₂²×C₃	—	4	4	no	—
217	C₂²×C₃	—	4	8	no	—
218	C₂²×C₃	C₃×#117	1	4	yes	—
219	C₂²×C₃	C₃×#121	1	4	yes	—
220	C₂²×C₃	C₃×C₂²⋊Q₈	1	4	yes	—
221	C₃×C₄	—	2	8	no	—
222	C₃×C₄	—	8	8	no	—
223	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
224	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
225	C₃×C₄	C₃×#121	1	4	yes	—
226	C₃×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
227	C₃²	#1814	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	2	1	no	—
229	C₂³	C₃²×#41	1	1	yes	characteristic
230	Q₈	—	2	1	no	—
231	Q₈	C₃²×C₄⋊C₄	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	C₃²×C₄⋊C₄	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
245	C₂×C₄	C₃²×#41	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	4	2	no	—
248	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
249	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
250	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
251	C₈	C₃²×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	—	4	8	no	—
256	C₂×C₃	C₃×#380	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	C₃²×#117	1	1	yes	characteristic
261	C₂²	C₃²×#121	1	1	yes	characteristic
262	C₂²	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	8	2	no	—
265	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
266	C₄	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
267	C₄	C₃²×#121	1	1	yes	characteristic
268	C₄	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
269	C₃	C₃×#1814	1	4	yes	direct product factor
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	C₃²×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	#121
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂×C₃²	#380
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#108	C₂	semidirect
3. C₃²×#427	C₂	semidirect
5. C₃²×#344	C₂	semidirect
6. C₃×#1814	C₃	direct (N × H)
10. C₃²×#131	C₄	semidirect
11. C₃²×#131	C₄	semidirect
24. C₂×C₃×#108	C₂×C₃	semidirect
25. C₃×#427	C₂×C₃	semidirect
27. C₃×#344	C₂×C₃	semidirect
56. #1814	C₃²	direct (N × H)
60. C₃×#131	C₃×C₄	semidirect
61. C₃×#131	C₃×C₄	semidirect
98. C₂×#108	C₂×C₃²	semidirect
99. #427	C₂×C₃²	semidirect
101. #344	C₂×C₃²	semidirect
145. #131	C₃²×C₄	semidirect
146. #131	C₃²×C₄	semidirect
227. C₃²	#1814	direct (N × H)
269. C₃	C₃×#1814	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₂×C₃²×#108	2
C₃²×#427	2
C₃²×#421	2
C₃²×#344	2
C₃×#1814	3

Subgroups — C32×#1814 #11768863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1814 #11768863