Subgroups — C₃×#33393 #10652000

Order 768.

1102 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	C₂×C₃×#578	C₂	1	1	yes	maximal, characteristic
2	C₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2148	C₂	1	6	yes	maximal, complemented
5	C₂×C₃×#362	C₂	1	6	yes	maximal, complemented
6	#33393	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×2₋¹⁺⁴	—	2	1	no	—
8	C₃×#578	—	2	4	no	—
9	C₃×#578	C₄	1	8	yes	—
10	C₂×C₃×#137	—	2	3	no	—
11	C₂×C₃×#135	C₂²	1	3	yes	—
12	C₂²×C₃×Q₁₆	C₂²	1	3	yes	—
13	C₃×#568	—	2	1	no	—
14	C₂×C₃×#131	C₂²	1	1	yes	characteristic
15	C₃×#504	—	2	4	no	—
16	C₂×C₃×M₃₂	C₂²	1	6	yes	—
17	C₃×#430	—	2	3	no	—
18	C₂×C₃×#108	C₂²	1	6	yes	—
19	C₃×#362	C₂²	1	12	yes	—
20	C₃×#361	C₂²	1	4	yes	—
21	C₂×#578	C₂×C₃	1	1	yes	characteristic
22	#3407	C₂×C₃	1	1	yes	characteristic, complemented
23	#2916	C₂×C₃	1	1	yes	characteristic, complemented
24	#2148	C₂×C₃	1	6	yes	complemented
25	C₂×#362	C₂×C₃	1	6	yes	complemented
26	C₃×2₋¹⁺⁴	—	2	8	no	—
27	C₃×2₋¹⁺⁴	—	4	4	no	—
28	C₂×C₃×#41	—	2	3	no	—
29	C₂×C₃×#41	—	4	3	no	—
30	C₂×C₃×#41	D₄	1	1	yes	characteristic
31	C₂²×C₃×Q₈	—	2	3	no	—
32	C₂²×C₃×Q₈	—	4	1	no	—
33	C₃×#137	—	2	24	no	—
34	C₃×#135	—	2	6	no	—
35	C₃×#135	C₂×C₄	1	12	yes	—
36	C₂×C₃×Q₁₆	—	2	3	no	—
37	C₂×C₃×Q₁₆	—	2	6	no	—
38	C₂×C₃×Q₁₆	C₂×C₄	1	12	yes	—
39	C₂×C₃×Q₁₆	C₂³	1	3	yes	—
40	C₂×C₃×SD₁₆	—	2	3	no	—
41	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
42	C₃×#131	—	4	4	no	—
43	C₃×#131	C₂×C₄	1	4	yes	—
44	C₃×#131	D₄	1	4	yes	—
45	C₂×C₃×M₁₆	—	2	6	no	—
46	C₂×C₃×M₁₆	D₄	1	3	yes	—
47	C₂²×C₃×C₈	C₂³	1	3	yes	—
48	C₃×M₃₂	—	2	6	no	—
49	C₃×M₃₂	C₂³	1	4	yes	—
50	C₂×C₃×C₁₆	—	2	6	no	—
51	C₃×#108	—	2	6	no	—
52	C₃×#108	C₂³	1	4	yes	—
53	C₂×2₋¹⁺⁴	—	2	1	no	—
54	#578	—	2	4	no	—
55	#578	C₃×C₄	1	8	yes	—
56	C₂×#137	—	2	3	no	—
57	C₂×#135	C₂²×C₃	1	3	yes	—
58	C₂²×Q₁₆	C₂²×C₃	1	3	yes	—
59	#568	—	2	1	no	—
60	C₂×#131	C₂²×C₃	1	1	yes	characteristic
61	#504	—	2	4	no	—
62	C₂×M₃₂	C₂²×C₃	1	6	yes	—
63	#430	—	2	3	no	—
64	C₂×#108	C₂²×C₃	1	6	yes	—
65	#362	C₂²×C₃	1	12	yes	—
66	#361	C₂²×C₃	1	4	yes	—
67	C₃×#41	—	2	12	no	—
68	C₃×#41	—	4	24	no	—
69	C₃×#41	—	4	6	no	—
70	C₃×#41	C₂²⋊C₄	1	4	yes	—
71	C₃×#41	D₈	1	4	yes	—
72	C₂×C₃×Q₈	—	2	12	no	—
73	C₂×C₃×Q₈	—	2	3	no	—
74	C₂×C₃×Q₈	—	4	8	no	—
75	C₂×C₃×Q₈	—	4	6	no	—
76	C₂×C₃×Q₈	—	4	3	no	—
77	C₂×C₃×Q₈	SD₁₆	1	1	yes	characteristic
78	C₂×C₃×D₄	—	2	1	no	—
79	C₂×C₃×D₄	—	4	3	no	—
80	C₂×C₃×D₄	SD₁₆	1	3	yes	—
81	C₂²×C₃×C₄	—	4	3	no	—
82	C₂²×C₃×C₄	—	4	3	no	—
83	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
84	C₃×Q₁₆	—	2	12	no	—
85	C₃×Q₁₆	—	2	3	no	—
86	C₃×Q₁₆	C₂²×C₄	1	6	yes	—
87	C₃×SD₁₆	—	2	12	no	—
88	C₃×D₈	—	2	1	no	—
89	C₃×D₈	C₂²×C₄	1	2	yes	—
90	C₃×M₁₆	—	2	4	no	—
91	C₃×M₁₆	—	4	6	no	—
92	C₃×M₁₆	C₂²⋊C₄	1	12	yes	—
93	C₂×C₃×C₈	—	4	6	no	—
94	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
95	C₂×C₃×C₈	C₂×D₄	1	3	yes	—
96	C₂×C₃×C₈	C₂×D₄	1	6	yes	—
97	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
98	C₃×C₁₆	—	2	4	no	—
99	2₋¹⁺⁴	—	2	8	no	—
100	2₋¹⁺⁴	—	4	4	no	—
101	C₂×#41	—	2	3	no	—
102	C₂×#41	—	4	3	no	—
103	C₂×#41	C₃×D₄	1	1	yes	characteristic
104	C₂²×Q₈	—	2	3	no	—
105	C₂²×Q₈	—	4	1	no	—
106	#137	—	2	24	no	—
107	#135	—	2	6	no	—
108	#135	C₂×C₃×C₄	1	12	yes	—
109	C₂×Q₁₆	—	2	3	no	—
110	C₂×Q₁₆	—	2	6	no	—
111	C₂×Q₁₆	C₂×C₃×C₄	1	12	yes	—
112	C₂×Q₁₆	C₂³×C₃	1	3	yes	—
113	C₂×SD₁₆	—	2	3	no	—
114	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
115	#131	—	4	4	no	—
116	#131	C₂×C₃×C₄	1	4	yes	—
117	#131	C₃×D₄	1	4	yes	—
118	C₂×M₁₆	—	2	6	no	—
119	C₂×M₁₆	C₃×D₄	1	3	yes	—
120	C₂²×C₈	C₂³×C₃	1	3	yes	—
121	M₃₂	—	2	6	no	—
122	M₃₂	C₂³×C₃	1	4	yes	—
123	C₂×C₁₆	—	2	6	no	—
124	#108	—	2	6	no	—
125	#108	C₂³×C₃	1	4	yes	—
126	C₂³×C₃	—	4	1	no	—
127	C₂³×C₃	C₂×SD₁₆	1	3	yes	—
128	C₃×Q₈	—	2	6	no	—
129	C₃×Q₈	—	4	3	no	—
130	C₃×Q₈	—	4	12	no	—
131	C₃×Q₈	#102	1	4	yes	—
132	C₃×D₄	—	2	2	no	—
133	C₃×D₄	—	4	12	no	—
134	C₃×D₄	—	4	1	no	—
135	C₃×D₄	#102	1	12	yes	—
136	C₂×C₃×C₄	—	4	3	no	—
137	C₂×C₃×C₄	—	4	12	no	—
138	C₂×C₃×C₄	—	4	12	no	—
139	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	6	yes	—
140	C₂×C₃×C₄	C₂×D₈	1	6	yes	—
141	C₂×C₃×C₄	C₂×SD₁₆	1	3	yes	—
142	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₃×C₈	—	4	4	no	—
144	C₃×C₈	C₂×C₂²⋊C₄	1	6	yes	—
145	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
146	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
147	#41	—	2	12	no	—
148	#41	—	4	24	no	—
149	#41	—	4	6	no	—
150	#41	C₃×C₂²⋊C₄	1	4	yes	—
151	#41	C₃×D₈	1	4	yes	—
152	C₂×Q₈	—	2	12	no	—
153	C₂×Q₈	—	2	3	no	—
154	C₂×Q₈	—	4	8	no	—
155	C₂×Q₈	—	4	6	no	—
156	C₂×Q₈	—	4	3	no	—
157	C₂×Q₈	C₃×SD₁₆	1	1	yes	characteristic
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	4	3	no	—
160	C₂×D₄	C₃×SD₁₆	1	3	yes	—
161	C₂²×C₄	—	4	3	no	—
162	C₂²×C₄	—	4	3	no	—
163	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
164	Q₁₆	—	2	12	no	—
165	Q₁₆	—	2	3	no	—
166	Q₁₆	C₂²×C₃×C₄	1	6	yes	—
167	SD₁₆	—	2	12	no	—
168	D₈	—	2	1	no	—
169	D₈	C₂²×C₃×C₄	1	2	yes	—
170	M₁₆	—	2	4	no	—
171	M₁₆	—	4	6	no	—
172	M₁₆	C₃×C₂²⋊C₄	1	12	yes	—
173	C₂×C₈	—	4	6	no	—
174	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
175	C₂×C₈	C₂×C₃×D₄	1	3	yes	—
176	C₂×C₈	C₂×C₃×D₄	1	6	yes	—
177	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₂²×C₃	—	4	3	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	8	2	no	—
182	C₂²×C₃	C₂×#102	1	6	yes	—
183	C₂²×C₃	C₂²×SD₁₆	1	1	yes	characteristic
184	C₃×C₄	—	4	6	no	—
185	C₃×C₄	C₂×#102	1	6	yes	—
186	C₃×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
187	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
188	C₂³	—	4	1	no	—
189	C₂³	C₂×C₃×SD₁₆	1	3	yes	—
190	Q₈	—	2	6	no	—
191	Q₈	—	4	3	no	—
192	Q₈	—	4	12	no	—
193	Q₈	C₃×#102	1	4	yes	—
194	D₄	—	2	2	no	—
195	D₄	—	4	12	no	—
196	D₄	—	4	1	no	—
197	D₄	C₃×#102	1	12	yes	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	12	no	—
200	C₂×C₄	—	4	12	no	—
201	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	6	yes	—
202	C₂×C₄	C₂×C₃×D₈	1	6	yes	—
203	C₂×C₄	C₂×C₃×SD₁₆	1	3	yes	—
204	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
205	C₈	—	4	4	no	—
206	C₈	C₂×C₃×C₂²⋊C₄	1	6	yes	—
207	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
208	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	6	no	—
211	C₂×C₃	—	8	2	no	—
212	C₂×C₃	C₂²×#102	1	1	yes	characteristic
213	C₂²	—	4	3	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	C₂×C₃×#102	1	6	yes	—
217	C₂²	C₂²×C₃×SD₁₆	1	1	yes	characteristic
218	C₄	—	4	6	no	—
219	C₄	C₂×C₃×#102	1	6	yes	—
220	C₄	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
222	C₃	#33393	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	1	no	—
224	C₂	—	2	6	no	—
225	C₂	—	8	2	no	—
226	C₂	C₂²×C₃×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#102
commutator	C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#578	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×#41 ◁ C₂×C₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#3407	C₂	semidirect
3. C₃×#2916	C₂	semidirect
4. C₃×#2148	C₂	semidirect
5. C₂×C₃×#362	C₂	semidirect
6. #33393	C₃	direct (N × H)
22. #3407	C₂×C₃	semidirect
23. #2916	C₂×C₃	semidirect
24. #2148	C₂×C₃	semidirect
25. C₂×#362	C₂×C₃	semidirect
222. C₃	#33393	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×C₃×#578	2
C₃×#3407	2
C₃×#2916	2
C₃×#2148	2
C₂×C₃×#362	2
#33393	3

Subgroups — C3×#33393 #10652000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33393 #10652000