Subgroups — C₃²×D₃×#108 #13162328

Order 1728.

756 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₃×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃³×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#822	C₂	1	2	yes	maximal, complemented
5	C₃²×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#108	—	3	1	no	maximal
7	C₃×D₃×#108	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₃³×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃²×#208	C₂²	1	2	yes	—
12	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×ΓL(1,𝔽₂₅)	—	2	2	no	—
14	C₃²×D₃×C₈	—	2	2	no	—
15	C₃²×D₃×C₈	C₄	1	2	yes	—
16	C₃²×D₃×C₈	C₄	1	2	yes	—
17	C₂×C₃²×M₁₆	—	3	2	no	—
18	C₂²×C₃²×C₈	—	3	1	no	—
19	C₃×D₃×M₁₆	C₂×C₃	1	8	yes	—
20	C₂×C₃×D₃×C₈	C₂×C₃	1	4	yes	—
21	C₃²×#108	—	2	4	no	—
22	C₃²×#108	—	3	2	no	—
23	C₃²×#108	—	3	1	no	—
24	C₃²×#108	D₃	1	1	yes	characteristic, direct product factor
25	C₃²×#108	C₂×C₃	1	4	yes	complemented
26	C₃×#822	C₂×C₃	1	8	yes	complemented
27	C₃×#819	C₂×C₃	1	4	yes	complemented
28	C₂²×C₃²×D₃	Q₈	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	Q₈	1	1	yes	characteristic
31	C₃²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
32	C₃²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄×D₃	D₄	1	1	yes	characteristic
34	C₃²×C₄×D₃	D₄	1	1	yes	characteristic
35	C₃³×C₈	—	2	2	no	—
36	C₃³×C₈	C₂×C₄	1	2	yes	—
37	C₃²×C₃⋊C₈	—	2	2	no	—
38	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×#108	—	3	4	no	—
40	D₃×#108	C₃²	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	—	3	1	no	—
42	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
43	C₃²×M₁₆	—	2	8	no	—
44	C₃²×M₁₆	—	3	2	no	—
45	C₃²×M₁₆	—	6	2	no	—
46	C₃²×M₁₆	C₂×D₃	1	2	yes	—
47	C₃²×M₁₆	C₂²×C₃	1	8	yes	—
48	C₂×C₃²×C₈	—	2	4	no	—
49	C₂×C₃²×C₈	—	3	1	no	—
50	C₂×C₃²×C₈	—	3	2	no	—
51	C₂×C₃²×C₈	—	3	2	no	—
52	C₂×C₃²×C₈	—	6	2	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
55	C₃×#208	C₂²×C₃	1	8	yes	—
56	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
57	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
58	C₃×D₃×C₈	—	2	8	no	—
59	C₃×D₃×C₈	C₃×C₄	1	8	yes	—
60	C₃×D₃×C₈	C₃×C₄	1	8	yes	—
61	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃²×D₃	—	2	2	no	—
63	C₂×C₃²×D₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×D₃	C₄⋊C₄	1	1	yes	characteristic
65	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
67	C₃²×Dic₃	C₄⋊C₄	1	1	yes	characteristic
68	C₃²×Dic₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃×M₁₆	—	3	8	no	—
70	C₂²×C₃×C₈	—	3	4	no	—
71	D₃×M₁₆	C₂×C₃²	1	2	yes	—
72	C₂×D₃×C₈	C₂×C₃²	1	1	yes	characteristic
73	C₃×#108	—	2	4	no	—
74	C₃×#108	—	3	8	no	—
75	C₃×#108	—	3	4	no	—
76	C₃×#108	C₃×D₃	1	4	yes	direct product factor
77	C₃×#108	C₂×C₃²	1	1	yes	characteristic, complemented
78	#822	C₂×C₃²	1	2	yes	complemented
79	#819	C₂×C₃²	1	1	yes	characteristic, complemented
80	C₂³×C₃²	—	3	1	no	—
81	C₂²×C₃×D₃	C₃×Q₈	1	4	yes	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	3	1	no	—
84	C₂×C₃²×C₄	—	3	1	no	—
85	C₂×C₃²×C₄	—	3	1	no	—
86	C₂×C₃²×C₄	—	3	1	no	—
87	C₂×C₃²×C₄	—	3	1	no	—
88	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
90	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
91	C₃×C₄×D₃	C₂×C₃×C₄	1	4	yes	—
92	C₃×C₄×D₃	C₂×C₃×C₄	1	4	yes	—
93	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
94	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
95	C₃²×C₈	—	2	2	no	—
96	C₃²×C₈	—	2	8	no	—
97	C₃²×C₈	—	2	8	no	—
98	C₃²×C₈	—	3	2	no	—
99	C₃²×C₈	—	4	8	no	—
100	C₃²×C₈	—	6	2	no	—
101	C₃²×C₈	C₄×D₃	1	2	yes	—
102	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
103	C₃×C₃⋊C₈	—	2	8	no	—
104	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
105	C₂×#108	—	3	1	no	Sylow
106	C₂×C₃³	—	2	1	no	—
107	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃²×D₃	—	2	1	no	—
109	C₃²×D₃	#108	1	2	yes	direct product factor
110	C₂²×C₃×C₄	—	3	4	no	—
111	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
112	C₃×M₁₆	—	2	8	no	—
113	C₃×M₁₆	—	3	8	no	—
114	C₃×M₁₆	—	6	8	no	—
115	C₃×M₁₆	C₂×C₃×D₃	1	8	yes	—
116	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	—	3	4	no	—
119	C₂×C₃×C₈	—	3	8	no	—
120	C₂×C₃×C₈	—	3	8	no	—
121	C₂×C₃×C₈	—	6	8	no	—
122	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
123	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
124	#208	C₂²×C₃²	1	2	yes	—
125	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
126	ΓL(1,𝔽₂₅)	—	2	2	no	—
127	D₃×C₈	—	2	2	no	—
128	D₃×C₈	C₃²×C₄	1	2	yes	—
129	D₃×C₈	C₃²×C₄	1	2	yes	—
130	C₂²×C₃²	—	2	4	no	—
131	C₂²×C₃²	—	3	1	no	—
132	C₂²×C₃²	—	3	1	no	—
133	C₂²×C₃²	—	6	2	no	—
134	C₂²×C₃²	D₃×Q₈	1	1	yes	characteristic
135	C₂²×C₃²	C₂×C₃×Q₈	1	4	yes	—
136	C₂×C₃×D₃	—	2	8	no	—
137	C₂×C₃×D₃	C₃×C₄⋊C₄	1	4	yes	—
138	C₂×C₃×D₃	C₃×C₄⋊C₄	1	4	yes	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	2	4	no	—
141	C₃²×C₄	—	3	1	no	—
142	C₃²×C₄	—	3	1	no	—
143	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
144	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
145	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
146	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
147	C₃×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
148	C₃×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
149	C₂×M₁₆	—	3	2	no	—
150	C₂²×C₈	—	3	1	no	—
151	#108	—	3	2	no	—
152	#108	—	3	1	no	—
153	#108	C₃²×D₃	1	1	yes	characteristic, direct product factor
154	C₃³	C₂×#108	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₃	—	3	4	no	—
156	C₂²×D₃	C₃²×Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	4	no	—
160	C₂×C₃×C₄	—	3	4	no	—
161	C₂×C₃×C₄	—	3	4	no	—
162	C₂×C₃×C₄	—	3	4	no	—
163	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
164	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
165	C₂×Dic₃	C₃²×Q₈	1	1	yes	characteristic
166	C₄×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
167	C₄×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
168	C₄×D₃	C₃²×D₄	1	1	yes	characteristic
169	C₄×D₃	C₃²×D₄	1	1	yes	characteristic
170	C₃×C₈	—	2	8	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	2	8	no	—
173	C₃×C₈	—	3	8	no	—
174	C₃×C₈	—	4	8	no	—
175	C₃×C₈	—	6	8	no	—
176	C₃×C₈	C₃×C₄×D₃	1	8	yes	—
177	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
178	C₃⋊C₈	—	2	2	no	—
179	C₃⋊C₈	C₂×C₃²×C₄	1	2	yes	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	3	2	no	—
184	C₂×C₃²	—	4	4	no	—
185	C₂×C₃²	—	6	1	no	—
186	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
188	C₃×D₃	—	2	4	no	—
189	C₃×D₃	C₃×#108	1	8	yes	direct product factor
190	C₂²×C₄	—	3	1	no	—
191	M₁₆	—	3	2	no	—
192	M₁₆	—	6	2	no	—
193	M₁₆	C₂×C₃²×D₃	1	2	yes	—
194	C₂×C₈	—	3	1	no	—
195	C₂×C₈	—	3	2	no	—
196	C₂×C₈	—	3	2	no	—
197	C₂×C₈	—	6	2	no	—
198	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	3	4	no	—
201	C₂²×C₃	—	3	4	no	—
202	C₂²×C₃	—	6	8	no	—
203	C₂²×C₃	C₃×D₃×Q₈	1	4	yes	—
204	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
207	C₂×D₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	3	4	no	—
211	C₃×C₄	—	3	4	no	—
212	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
213	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
214	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
215	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
216	Dic₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
217	Dic₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
218	C₃²	—	2	4	no	—
219	C₃²	D₃×#108	1	1	yes	characteristic, direct product factor
220	C₃²	C₂×C₃×#108	1	4	yes	complemented
221	C₂³	—	3	1	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	3	1	no	—
225	C₂×C₄	—	3	1	no	—
226	C₂×C₄	—	3	1	no	—
227	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	3	2	no	—
230	C₈	—	6	2	no	—
231	C₈	C₃²×C₄×D₃	1	2	yes	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	4	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	3	8	no	—
236	C₂×C₃	—	4	4	no	—
237	C₂×C₃	—	6	4	no	—
238	C₂×C₃	C₃×D₃×C₄⋊C₄	1	4	yes	—
239	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
240	D₃	—	2	1	no	—
241	D₃	C₃²×#108	1	2	yes	direct product factor
242	C₂²	—	3	1	no	—
243	C₂²	—	3	1	no	—
244	C₂²	—	6	2	no	—
245	C₂²	C₃²×D₃×Q₈	1	1	yes	characteristic
246	C₄	—	3	1	no	—
247	C₄	—	3	1	no	—
248	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
249	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
250	C₃	—	2	4	no	—
251	C₃	C₃×D₃×#108	1	4	yes	direct product factor
252	C₃	C₂×C₃²×#108	1	1	yes	characteristic, complemented
253	C₂	—	2	1	no	—
254	C₂	—	3	2	no	—
255	C₂	—	6	1	no	—
256	C₂	C₃²×D₃×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#108	C₂
socle	C₂×C₃³	C₂×C₄⋊C₄
prime radical	C₂²×C₃²×D₃	Q₈
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#108	D₃
hypocenter	C₃	C₂×C₃²×#108
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#108

Complements

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

N	H ≅ G/N	type
3. C₃³×#108	C₂	semidirect
4. C₃²×#822	C₂	semidirect
5. C₃²×#819	C₂	semidirect
7. C₃×D₃×#108	C₃	direct (N × H)
24. C₃²×#108	D₃	direct (N × H)
25. C₃²×#108	C₂×C₃	semidirect
26. C₃×#822	C₂×C₃	semidirect
27. C₃×#819	C₂×C₃	semidirect
40. D₃×#108	C₃²	direct (N × H)
76. C₃×#108	C₃×D₃	direct (N × H)
77. C₃×#108	C₂×C₃²	semidirect
78. #822	C₂×C₃²	semidirect
79. #819	C₂×C₃²	semidirect
109. C₃²×D₃	#108	direct (N × H)
153. #108	C₃²×D₃	direct (N × H)
154. C₃³	C₂×#108	semidirect
189. C₃×D₃	C₃×#108	direct (N × H)
219. C₃²	D₃×#108	direct (N × H)
220. C₃²	C₂×C₃×#108	semidirect
241. D₃	C₃²×#108	direct (N × H)
251. C₃	C₃×D₃×#108	direct (N × H)
252. C₃	C₂×C₃²×#108	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₃×M₁₆	2
C₂×C₃²×D₃×C₈	2
C₃³×#108	2
C₃²×#822	2
C₃²×#819	2
C₂×C₃²×#108	3
C₃×D₃×#108	3

Subgroups — C32×D3×#108 #13162328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×#108 #13162328