Subgroups — C₂²×C₃×#433 #10678723

Order 768.

2310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#433	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#433	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₈	C₂²	1	2	yes	—
10	C₂³×C₃×C₈	C₂²	1	2	yes	—
11	C₂×C₃×#117	C₄	1	12	yes	—
12	C₂×C₃×#117	C₂²	1	12	yes	—
13	C₂²×C₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₂²×C₃×C₂²⋊C₄	C₄	1	2	yes	—
15	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#433	C₂²	1	64	yes	direct product factor
18	C₂×C₃×C₄⋊C₈	C₂²	1	24	yes	complemented
19	C₂×C₃×C₂²⋊C₈	C₂²	1	24	yes	—
20	C₂×C₃×#97	C₂²	1	12	yes	complemented
21	C₂×C₃×C₄×C₈	C₂²	1	12	yes	complemented
22	C₂²×#117	C₂×C₃	1	1	yes	characteristic
23	C₂×#433	C₂×C₃	1	24	yes	direct product factor
24	C₂²×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
25	C₂²×C₂²⋊C₈	C₂×C₃	1	2	yes	—
26	C₂²×#97	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₃×C₄	—	2	8	no	—
30	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
32	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
33	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
34	C₂³×C₃×C₄	C₂³	1	6	yes	—
35	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₈	—	2	8	no	—
37	C₂²×C₃×C₈	—	2	8	no	—
38	C₂²×C₃×C₈	C₂³	1	12	yes	—
39	C₂²×C₃×C₈	C₂³	1	12	yes	—
40	C₃×#117	C₂×C₄	1	48	yes	—
41	C₃×#117	C₂³	1	16	yes	—
42	C₂×C₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
43	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	24	yes	—
44	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
45	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
46	C₂×C₃×C₄²	C₂³	1	6	yes	—
47	C₂×C₃×C₄²	C₂³	1	6	yes	—
48	C₃×C₄⋊C₈	C₂³	1	32	yes	complemented
49	C₃×C₂²⋊C₈	C₂³	1	32	yes	—
50	C₃×#97	C₂³	1	16	yes	complemented
51	C₃×C₄×C₈	C₂³	1	16	yes	complemented
52	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₈	C₂²×C₃	1	2	yes	—
54	C₂³×C₈	C₂²×C₃	1	2	yes	—
55	C₂×#117	C₃×C₄	1	12	yes	—
56	C₂×#117	C₂²×C₃	1	12	yes	—
57	C₂²×C₄⋊C₄	C₃×C₄	1	2	yes	—
58	C₂²×C₂²⋊C₄	C₃×C₄	1	2	yes	—
59	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	#433	C₂²×C₃	1	64	yes	direct product factor
62	C₂×C₄⋊C₈	C₂²×C₃	1	24	yes	complemented
63	C₂×C₂²⋊C₈	C₂²×C₃	1	24	yes	—
64	C₂×#97	C₂²×C₃	1	12	yes	complemented
65	C₂×C₄×C₈	C₂²×C₃	1	12	yes	complemented
66	C₂⁴×C₃	—	2	4	no	—
67	C₂⁴×C₃	—	2	8	no	—
68	C₂⁴×C₃	C₂²×C₄	1	6	yes	—
69	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	—	2	4	no	—
71	C₂²×C₃×C₄	—	2	8	no	—
72	C₂²×C₃×C₄	—	2	8	no	—
73	C₂²×C₃×C₄	—	2	24	no	—
74	C₂²×C₃×C₄	—	2	24	no	—
75	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
76	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
77	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
78	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
79	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
80	C₂²×C₃×C₄	#41	1	8	yes	—
81	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
82	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
83	C₂×C₃×C₈	—	2	24	no	—
84	C₂×C₃×C₈	—	2	24	no	—
85	C₂×C₃×C₈	C₂⁴	1	8	yes	—
86	C₂×C₃×C₈	C₂⁴	1	8	yes	—
87	C₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
88	C₃×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
89	C₃×C₄²	C₂²×C₄	1	12	yes	—
90	C₃×C₄²	C₂²×C₄	1	12	yes	—
91	C₃×C₄²	C₂⁴	1	4	yes	—
92	C₃×C₄²	C₂⁴	1	4	yes	—
93	C₂⁵	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂³×C₄	—	2	8	no	—
95	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
96	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
97	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂³×C₄	C₂×C₃×C₄	1	6	yes	—
99	C₂³×C₄	C₂³×C₃	1	6	yes	—
100	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	—	2	8	no	—
103	C₂²×C₈	C₂³×C₃	1	12	yes	—
104	C₂²×C₈	C₂³×C₃	1	12	yes	—
105	#117	C₂×C₃×C₄	1	48	yes	—
106	#117	C₂³×C₃	1	16	yes	—
107	C₂×C₄⋊C₄	C₂×C₃×C₄	1	24	yes	—
108	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	24	yes	—
109	C₂×C₄²	C₂×C₃×C₄	1	6	yes	—
110	C₂×C₄²	C₂×C₃×C₄	1	6	yes	—
111	C₂×C₄²	C₂³×C₃	1	6	yes	—
112	C₂×C₄²	C₂³×C₃	1	6	yes	—
113	C₄⋊C₈	C₂³×C₃	1	32	yes	complemented
114	C₂²⋊C₈	C₂³×C₃	1	32	yes	—
115	#97	C₂³×C₃	1	16	yes	complemented
116	C₄×C₈	C₂³×C₃	1	16	yes	complemented
117	C₂³×C₃	—	2	12	no	—
118	C₂³×C₃	—	2	24	no	—
119	C₂³×C₃	—	4	16	no	—
120	C₂³×C₃	#117	1	4	yes	—
121	C₂³×C₃	#131	1	8	yes	—
122	C₂³×C₃	C₂³×C₄	1	4	yes	—
123	C₂³×C₃	C₂³×C₄	1	3	yes	—
124	C₂×C₃×C₄	—	2	12	no	—
125	C₂×C₃×C₄	—	2	24	no	—
126	C₂×C₃×C₄	—	2	24	no	—
127	C₂×C₃×C₄	—	2	24	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	#117	1	12	yes	—
130	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
131	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
132	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
133	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
134	C₂×C₃×C₄	C₂×#41	1	12	yes	—
135	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
136	C₃×C₈	—	2	8	no	—
137	C₃×C₈	—	2	8	no	—
138	C₂⁴	—	2	4	no	—
139	C₂⁴	—	2	8	no	—
140	C₂⁴	C₂²×C₃×C₄	1	6	yes	—
141	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	24	no	—
146	C₂²×C₄	—	2	24	no	—
147	C₂²×C₄	C₂²×C₃×C₄	1	12	yes	—
148	C₂²×C₄	C₂²×C₃×C₄	1	12	yes	—
149	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
150	C₂²×C₄	C₂²×C₃×C₄	1	12	yes	—
151	C₂²×C₄	C₂²×C₃×C₄	1	3	yes	—
152	C₂²×C₄	C₃×#41	1	8	yes	—
153	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
154	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
155	C₂×C₈	—	2	24	no	—
156	C₂×C₈	—	2	24	no	—
157	C₂×C₈	C₂⁴×C₃	1	8	yes	—
158	C₂×C₈	C₂⁴×C₃	1	8	yes	—
159	C₄⋊C₄	C₂²×C₃×C₄	1	32	yes	—
160	C₂²⋊C₄	C₂²×C₃×C₄	1	32	yes	—
161	C₄²	C₂²×C₃×C₄	1	12	yes	—
162	C₄²	C₂²×C₃×C₄	1	12	yes	—
163	C₄²	C₂⁴×C₃	1	4	yes	—
164	C₄²	C₂⁴×C₃	1	4	yes	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	2	8	no	—
167	C₂²×C₃	—	4	24	no	—
168	C₂²×C₃	#433	1	16	yes	direct product factor
169	C₂²×C₃	C₂×#117	1	6	yes	—
170	C₂²×C₃	C₂×#131	1	12	yes	—
171	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	2	8	no	—
174	C₃×C₄	—	2	8	no	—
175	C₃×C₄	C₂×#117	1	6	yes	—
176	C₃×C₄	C₂²×#41	1	2	yes	—
177	C₂³	—	2	24	no	—
178	C₂³	—	2	12	no	—
179	C₂³	—	4	16	no	—
180	C₂³	C₃×#117	1	4	yes	—
181	C₂³	C₃×#131	1	8	yes	—
182	C₂³	C₂³×C₃×C₄	1	4	yes	—
183	C₂³	C₂³×C₃×C₄	1	3	yes	—
184	C₂×C₄	—	2	24	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	2	12	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	24	no	—
189	C₂×C₄	C₃×#117	1	12	yes	—
190	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
191	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
192	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
193	C₂×C₄	C₂³×C₃×C₄	1	3	yes	—
194	C₂×C₄	C₂×C₃×#41	1	12	yes	—
195	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
196	C₈	—	2	8	no	—
197	C₈	—	2	8	no	—
198	C₂×C₃	—	4	4	no	—
199	C₂×C₃	C₂×#433	1	12	yes	direct product factor
200	C₂×C₃	C₂²×#117	1	1	yes	characteristic
201	C₂×C₃	C₂²×#131	1	2	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	8	no	—
204	C₂²	—	4	24	no	—
205	C₂²	C₃×#433	1	16	yes	direct product factor
206	C₂²	C₂×C₃×#117	1	6	yes	—
207	C₂²	C₂×C₃×#131	1	12	yes	—
208	C₂²	C₂⁴×C₃×C₄	1	1	yes	characteristic
209	C₄	—	2	4	no	—
210	C₄	—	2	8	no	—
211	C₄	—	2	8	no	—
212	C₄	C₂×C₃×#117	1	6	yes	—
213	C₄	C₂²×C₃×#41	1	2	yes	—
214	C₃	C₂²×#433	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	4	4	no	—
216	C₂	C₂×C₃×#433	1	12	yes	direct product factor
217	C₂	C₂²×C₃×#117	1	1	yes	characteristic
218	C₂	C₂²×C₃×#131	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃×C₄	C₂³
commutator	C₂²	C₂⁴×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂⁴×C₃	C₂²×C₄
prime radical	C₂⁵×C₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#433	C₂	direct (N × H)
3. C₂²×C₃×C₄⋊C₈	C₂	semidirect
5. C₂²×C₃×#97	C₂	semidirect
6. C₂²×C₃×C₄×C₈	C₂	semidirect
7. C₂²×#433	C₃	direct (N × H)
17. C₃×#433	C₂²	direct (N × H)
18. C₂×C₃×C₄⋊C₈	C₂²	semidirect
20. C₂×C₃×#97	C₂²	semidirect
21. C₂×C₃×C₄×C₈	C₂²	semidirect
23. C₂×#433	C₂×C₃	direct (N × H)
24. C₂²×C₄⋊C₈	C₂×C₃	semidirect
26. C₂²×#97	C₂×C₃	semidirect
27. C₂²×C₄×C₈	C₂×C₃	semidirect
48. C₃×C₄⋊C₈	C₂³	semidirect
50. C₃×#97	C₂³	semidirect
51. C₃×C₄×C₈	C₂³	semidirect
61. #433	C₂²×C₃	direct (N × H)
62. C₂×C₄⋊C₈	C₂²×C₃	semidirect
64. C₂×#97	C₂²×C₃	semidirect
65. C₂×C₄×C₈	C₂²×C₃	semidirect
113. C₄⋊C₈	C₂³×C₃	semidirect
115. #97	C₂³×C₃	semidirect
116. C₄×C₈	C₂³×C₃	semidirect
168. C₂²×C₃	#433	direct (N × H)
199. C₂×C₃	C₂×#433	direct (N × H)
205. C₂²	C₃×#433	direct (N × H)
214. C₃	C₂²×#433	direct (N × H)
216. C₂	C₂×C₃×#433	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#117	2
C₂×C₃×#433	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×C₂²⋊C₈	2
C₂²×C₃×#97	2
C₂²×C₃×C₄×C₈	2
C₂²×#433	3

Subgroups — C22×C3×#433 #10678723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#433 #10678723