Subgroups — C₃×#10596193 #13179474

Order 1728.

1544 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#64153	C₂	1	4	yes	maximal
3	C₃×#64097	C₂	1	4	yes	maximal
4	C₃×#64096	C₂	1	4	yes	maximal
5	C₃×C₄×#634	C₂	1	2	yes	maximal
6	#10596193	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5722	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	2	yes	—
9	C₂×C₃×#634	C₂²	1	4	yes	—
10	C₂×C₃×#634	C₂²	1	2	yes	—
11	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
14	C₃×#3750	C₂²	1	4	yes	—
15	C₃×#3750	C₂²	1	8	yes	—
16	C₃×C₄×C₃⋊Dic₃	C₂²	1	4	yes	—
17	C₃×C₄×C₃⋊Dic₃	C₂²	1	4	yes	—
18	C₃²×C₄⋊Q₈	—	2	4	no	—
19	C₃²×C₄⋊Q₈	D₃	1	4	yes	—
20	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
21	#64153	C₂×C₃	1	4	yes	—
22	#64097	C₂×C₃	1	4	yes	—
23	#64096	C₂×C₃	1	4	yes	—
24	C₄×#634	C₂×C₃	1	2	yes	—
25	C₃×#944	—	3	16	no	—
26	C₃×#888	—	3	16	no	—
27	C₃×#887	—	3	16	no	—
28	C₃×C₄×Dic₆	—	3	8	no	—
29	C₃³×Q₈	—	4	2	no	—
30	C₂×C₃³×C₄	C₂³	1	4	yes	—
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	2	yes	—
33	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
34	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
35	C₃×#634	—	4	4	no	—
36	C₃×#634	Q₈	1	8	yes	complemented
37	C₃×Q₈²	—	9	1	no	—
38	#5722	—	3	4	no	—
39	C₂×C₃²×Q₈	—	2	8	no	—
40	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
41	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
42	C₂×#634	C₂²×C₃	1	4	yes	—
43	C₂×#634	C₂²×C₃	1	2	yes	—
44	C₂×C₃×Dic₆	—	3	8	no	—
45	C₂×C₃×Dic₆	—	3	16	no	—
46	C₃²×C₄⋊C₄	—	2	16	no	—
47	C₃²×C₄⋊C₄	C₂×D₃	1	16	yes	—
48	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₃²×C₄²	—	2	4	no	—
50	C₃²×C₄²	C₂×D₃	1	4	yes	—
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
53	#3750	C₂²×C₃	1	4	yes	—
54	#3750	C₂²×C₃	1	8	yes	—
55	C₄×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
56	C₄×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
57	C₃×C₄⋊Dic₃	—	3	8	no	—
58	C₃×Dic₃⋊C₄	—	3	32	no	—
59	C₃×Dic₃⋊C₄	—	3	16	no	—
60	C₃×C₄×Dic₃	—	3	16	no	—
61	C₃×C₄×Dic₃	—	3	16	no	—
62	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
63	C₃³×C₄	—	2	1	no	—
64	C₃³×C₄	—	2	4	no	—
65	C₃³×C₄	C₂×Q₈	1	4	yes	—
66	C₃×C₃⋊Dic₃	—	2	4	no	—
67	C₃×C₃⋊Dic₃	C₂×Q₈	1	8	yes	—
68	C₃×C₄⋊Q₈	—	2	4	no	—
69	C₃×C₄⋊Q₈	—	9	4	no	—
70	C₃×C₄⋊Q₈	—	9	4	no	—
71	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	—
72	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
73	C₃×C₄×Q₈	—	9	4	no	—
74	C₃×C₄×Q₈	—	9	2	no	—
75	#944	—	3	16	no	—
76	#888	—	3	16	no	—
77	#887	—	3	16	no	—
78	C₄×Dic₆	—	3	8	no	—
79	C₃²×Q₈	—	4	8	no	—
80	C₃²×Q₈	—	4	2	no	—
81	C₃²×Q₈	—	4	16	no	—
82	C₂×C₃²×C₄	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	16	no	—
85	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
87	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
88	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
89	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
91	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
92	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
93	#634	—	4	4	no	—
94	#634	C₃×Q₈	1	8	yes	complemented
95	C₂×C₃×Dic₃	—	3	16	no	—
96	C₂×C₃×Dic₃	—	3	16	no	—
97	C₃×Dic₆	—	3	32	no	—
98	C₃×Dic₆	—	12	16	no	—
99	Q₈²	—	9	1	no	Sylow
100	C₂×C₃³	C₂²×Q₈	1	2	yes	—
101	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	2	8	no	—
103	C₂×C₃×Q₈	—	9	2	no	—
104	C₂×C₃×Q₈	—	9	4	no	—
105	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
106	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
107	C₂×Dic₆	—	3	8	no	—
108	C₂×Dic₆	—	3	16	no	—
109	C₃×C₄⋊C₄	—	2	16	no	—
110	C₃×C₄⋊C₄	—	9	8	no	—
111	C₃×C₄⋊C₄	—	9	4	no	—
112	C₃×C₄⋊C₄	—	9	2	no	—
113	C₃×C₄⋊C₄	C₂×C₃×D₃	1	16	yes	—
114	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
115	C₃×C₄²	—	2	4	no	—
116	C₃×C₄²	—	9	4	no	—
117	C₃×C₄²	—	9	4	no	—
118	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
119	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
120	C₄⋊Dic₃	—	3	8	no	—
121	Dic₃⋊C₄	—	3	16	no	—
122	Dic₃⋊C₄	—	3	32	no	—
123	C₄×Dic₃	—	3	16	no	—
124	C₄×Dic₃	—	3	16	no	—
125	C₂²×C₃²	—	2	4	no	—
126	C₂²×C₃²	C₂³×D₃	1	4	yes	—
127	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
128	C₃²×C₄	—	2	8	no	—
129	C₃²×C₄	—	2	4	no	—
130	C₃²×C₄	—	2	4	no	—
131	C₃²×C₄	—	2	1	no	—
132	C₃²×C₄	—	2	16	no	—
133	C₃²×C₄	—	2	16	no	—
134	C₃²×C₄	—	4	16	no	—
135	C₃²×C₄	D₃×Q₈	1	16	yes	—
136	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
137	C₃⋊Dic₃	—	2	4	no	—
138	C₃⋊Dic₃	C₂×C₃×Q₈	1	8	yes	—
139	C₃×Dic₃	—	3	32	no	—
140	C₃×Dic₃	—	6	16	no	—
141	C₄⋊Q₈	—	9	4	no	—
142	C₄⋊Q₈	—	9	4	no	—
143	C₄⋊Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
144	C₄×Q₈	—	9	4	no	—
145	C₄×Q₈	—	9	2	no	—
146	C₃³	Q₈²	1	1	yes	characteristic, complemented, Sylow
147	C₃×Q₈	—	4	8	no	—
148	C₃×Q₈	—	4	2	no	—
149	C₃×Q₈	—	4	16	no	—
150	C₃×Q₈	—	9	8	no	—
151	C₃×Q₈	—	36	4	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	16	no	—
155	C₂×C₃×C₄	—	9	4	no	—
156	C₂×C₃×C₄	—	9	4	no	—
157	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
159	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
160	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
161	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
163	C₂×Dic₃	—	3	16	no	—
164	C₂×Dic₃	—	3	16	no	—
165	Dic₆	—	3	32	no	—
166	Dic₆	—	12	16	no	—
167	C₂×C₃²	—	2	4	no	—
168	C₂×C₃²	—	2	8	no	—
169	C₂×C₃²	#1004	1	4	yes	—
170	C₂×C₃²	C₂×D₃×Q₈	1	8	yes	—
171	C₂×C₃²	C₂²×C₃×Q₈	1	2	yes	—
172	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×Q₈	—	9	2	no	—
174	C₂×Q₈	—	9	4	no	—
175	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
176	C₄⋊C₄	—	9	8	no	—
177	C₄⋊C₄	—	9	4	no	—
178	C₄⋊C₄	—	9	2	no	—
179	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	4	yes	—
180	C₄²	—	9	4	no	—
181	C₄²	—	9	4	no	—
182	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
185	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
186	C₃×C₄	—	2	8	no	—
187	C₃×C₄	—	2	16	no	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	16	no	—
192	C₃×C₄	—	4	16	no	—
193	C₃×C₄	—	9	8	no	—
194	C₃×C₄	—	18	4	no	—
195	C₃×C₄	C₃×D₃×Q₈	1	16	yes	—
196	C₃×C₄	Q₈×C₃⋊D₃	1	4	yes	—
197	Dic₃	—	3	32	no	—
198	Dic₃	—	6	16	no	—
199	C₃²	—	2	4	no	—
200	C₃²	#5722	1	4	yes	complemented
201	C₃²	C₃×Q₈²	1	1	yes	characteristic, complemented
202	Q₈	—	4	2	no	—
203	Q₈	—	9	8	no	—
204	Q₈	—	36	4	no	—
205	C₂×C₄	—	9	4	no	—
206	C₂×C₄	—	9	4	no	—
207	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
208	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
210	C₂×C₃	—	2	4	no	—
211	C₂×C₃	—	2	8	no	—
212	C₂×C₃	C₃×#1004	1	4	yes	—
213	C₂×C₃	C₂×C₃×D₃×Q₈	1	8	yes	—
214	C₂×C₃	#64361	1	1	yes	characteristic
215	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	2	yes	—
216	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	4	no	—
219	C₄	—	9	8	no	—
220	C₄	—	18	4	no	—
221	C₄	C₃×Q₈×C₃⋊D₃	1	4	yes	—
222	C₃	—	2	4	no	—
223	C₃	C₃×#5722	1	4	yes	complemented
224	C₃	#10596193	1	1	yes	characteristic, direct product factor
225	C₂	C₃×#64361	1	1	yes	characteristic
226	C₂	C₂×C₃×Q₈×C₃⋊D₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂⁴
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×Q₈²
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. #10596193	C₃	direct (N × H)
36. C₃×#634	Q₈	semidirect
94. #634	C₃×Q₈	semidirect
146. C₃³	Q₈²	semidirect
200. C₃²	#5722	semidirect
201. C₃²	C₃×Q₈²	semidirect
223. C₃	C₃×#5722	semidirect
224. C₃	#10596193	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₃×#64153	2
C₃×#64097	2
C₃×#64096	2
C₃×C₄×#634	2
#10596193	3
C₃×#5722	3

Subgroups — C3×#10596193 #13179474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596193 #13179474