Subgroups — C₂×#69884 #10947845

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#812	C₂	1	2	yes	maximal
3	#69884	C₂	1	8	yes	maximal, direct product factor
4	C₂×#4466	C₂	1	4	yes	maximal, complemented
5	C₂×#2110	—	3	1	no	maximal, Sylow
6	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#117	C₄	1	1	yes	characteristic
9	C₃×#405	C₄	1	4	yes	—
10	C₃×#405	C₂²	1	4	yes	—
11	C₂×C₃×#97	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
13	C₂×#812	C₄	1	4	yes	complemented
14	C₂×#812	C₄	1	8	yes	complemented
15	C₂×#812	C₂²	1	4	yes	—
16	C₂×#812	C₂²	1	8	yes	—
17	C₂²×C₃⋊C₁₆	C₄	1	4	yes	complemented
18	#4466	C₂²	1	16	yes	complemented
19	C₂²×M₃₂	—	3	2	no	—
20	C₂×#405	D₃	1	1	yes	characteristic
21	#2110	—	3	8	no	—
22	C₂×#347	—	3	4	no	—
23	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×M₁₆	—	2	2	no	—
25	C₂×C₃×M₁₆	—	2	2	no	—
26	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₃×#117	—	2	2	no	—
36	C₃×#117	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
40	C₃×#97	C₂×C₄	1	4	yes	—
41	C₃×#97	C₂³	1	4	yes	—
42	C₃×C₄×C₈	C₂×C₄	1	8	yes	—
43	#812	C₂×C₄	1	8	yes	—
44	#812	C₂×C₄	1	16	yes	complemented
45	#812	C₂³	1	8	yes	—
46	C₂×C₃⋊C₁₆	C₂×C₄	1	8	yes	—
47	C₂×C₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
48	C₂²×M₁₆	Dic₃	1	1	yes	characteristic
49	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×#117	Dic₃	1	1	yes	characteristic
51	C₂×M₃₂	—	3	4	no	—
52	C₂×M₃₂	—	3	8	no	—
53	C₂×M₃₂	—	3	4	no	—
54	C₂×M₃₂	—	3	8	no	—
55	C₂²×C₁₆	—	3	4	no	—
56	#405	Dic₃	1	4	yes	—
57	#405	C₂×D₃	1	4	yes	—
58	C₂×#97	C₂×D₃	1	2	yes	—
59	C₂×C₄×C₈	Dic₃	1	2	yes	—
60	#347	—	3	16	no	—
61	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₃×C₄	—	2	2	no	—
63	C₂²×C₃×C₄	C₄²	1	2	yes	—
64	C₂²×C₃×C₄	C₄²	1	2	yes	—
65	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
66	C₂²×C₃×C₄	M₁₆	1	2	yes	—
67	C₂²×C₃×C₄	M₁₆	1	2	yes	—
68	C₂²×C₃×C₄	M₁₆	1	2	yes	—
69	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
71	C₃×M₁₆	—	2	8	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₄²	1	4	yes	—
75	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
76	C₂×C₃×C₈	C₄²	1	2	yes	—
77	C₂×C₃×C₈	C₄²	1	4	yes	—
78	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
81	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₂²⋊C₄	—	2	4	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	C₂²×C₄	1	4	yes	—
91	C₃⋊C₁₆	C₂²×C₄	1	16	yes	—
92	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
96	C₂²×C₈	C₄×D₃	1	2	yes	—
97	C₂²×C₈	C₄×D₃	1	4	yes	—
98	C₂²×C₈	C₄×D₃	1	2	yes	—
99	C₂²×C₈	C₂×Dic₃	1	2	yes	—
100	C₂²×C₈	C₂×Dic₃	1	2	yes	—
101	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₃	1	2	yes	—
104	#117	—	2	2	no	—
105	#117	C₂×Dic₃	1	4	yes	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₄²	C₂×Dic₃	1	2	yes	—
109	M₃₂	—	3	8	no	—
110	M₃₂	—	3	16	no	—
111	M₃₂	—	3	8	no	—
112	C₂×C₁₆	—	3	8	no	—
113	C₂×C₁₆	—	3	16	no	—
114	#97	C₂×Dic₃	1	4	yes	—
115	#97	C₂²×D₃	1	4	yes	—
116	C₄×C₈	C₂×Dic₃	1	8	yes	—
117	C₂³×C₃	—	4	2	no	—
118	C₂³×C₃	#97	1	2	yes	—
119	C₂³×C₃	#97	1	2	yes	—
120	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂³×C₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	4	4	no	—
124	C₂×C₃×C₄	#97	1	2	yes	—
125	C₂×C₃×C₄	#97	1	4	yes	—
126	C₂×C₃×C₄	#97	1	4	yes	—
127	C₂×C₃×C₄	#97	1	2	yes	—
128	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
129	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×M₁₆	1	4	yes	—
134	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
136	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
137	C₃×C₈	—	2	4	no	—
138	C₃×C₈	C₂×C₄²	1	2	yes	—
139	C₃×C₈	C₂×C₄²	1	2	yes	—
140	C₃×C₈	C₂×C₄²	1	2	yes	—
141	C₃×C₈	C₂³×C₄	1	2	yes	—
142	C₂⁴	#208	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	#208	1	1	yes	characteristic
145	C₂²×C₄	#208	1	2	yes	—
146	C₂²×C₄	#208	1	2	yes	—
147	C₂²×C₄	#208	1	2	yes	—
148	C₂²×C₄	C₄×Dic₃	1	2	yes	—
149	C₂²×C₄	C₄×Dic₃	1	2	yes	—
150	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
152	M₁₆	—	2	8	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₄×Dic₃	1	4	yes	—
156	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₃	1	2	yes	—
158	C₂×C₈	C₄×Dic₃	1	4	yes	—
159	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₃	1	4	yes	—
162	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₃	1	4	yes	—
164	C₂×C₈	C₂²×Dic₃	1	4	yes	—
165	C₂×C₈	C₂²×Dic₃	1	2	yes	—
166	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
167	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	C₂²×Dic₃	1	4	yes	—
172	C₁₆	—	3	16	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	4	no	—
175	C₂²×C₃	—	4	2	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	C₂×#97	1	1	yes	characteristic
178	C₂²×C₃	C₂×#97	1	4	yes	—
179	C₂²×C₃	C₂×#97	1	1	yes	characteristic
180	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
181	C₃×C₄	—	4	4	no	—
182	C₃×C₄	C₂×#97	1	1	yes	characteristic
183	C₃×C₄	C₂×#97	1	1	yes	characteristic
184	C₃×C₄	C₂×#97	1	4	yes	—
185	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
186	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
187	C₂³	—	4	2	no	—
188	C₂³	#803	1	2	yes	—
189	C₂³	#803	1	2	yes	—
190	C₂³	C₂×#208	1	1	yes	characteristic
191	C₂³	C₂×#208	1	2	yes	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	#803	1	2	yes	—
195	C₂×C₄	#803	1	4	yes	—
196	C₂×C₄	#803	1	4	yes	—
197	C₂×C₄	#803	1	2	yes	—
198	C₂×C₄	C₂×#208	1	1	yes	characteristic
199	C₂×C₄	C₂×#208	1	2	yes	—
200	C₂×C₄	C₂×#208	1	1	yes	characteristic
201	C₂×C₄	C₂×#208	1	1	yes	characteristic
202	C₂×C₄	C₂×#208	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₃	1	4	yes	—
204	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
206	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	C₂×C₄×Dic₃	1	2	yes	—
209	C₈	C₂×C₄×Dic₃	1	2	yes	—
210	C₈	C₂×C₄×Dic₃	1	2	yes	—
211	C₈	C₂²×C₄×D₃	1	2	yes	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	#2110	1	2	yes	complemented
216	C₂×C₃	C₂²×#97	1	1	yes	characteristic
217	C₂²	—	2	2	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂×#803	1	1	yes	characteristic
222	C₂²	C₂×#803	1	4	yes	—
223	C₂²	C₂×#803	1	1	yes	characteristic
224	C₂²	C₂²×#208	1	1	yes	characteristic
225	C₄	—	4	4	no	—
226	C₄	C₂×#803	1	1	yes	characteristic
227	C₄	C₂×#803	1	1	yes	characteristic
228	C₄	C₂×#803	1	4	yes	—
229	C₄	C₂²×#208	1	1	yes	characteristic
230	C₄	C₂²×C₄×Dic₃	1	1	yes	characteristic
231	C₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	#69884	1	2	yes	direct product factor
236	C₂	C₂²×#803	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₃
commutator	C₃×C₄	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×#405	C₂
socle	C₂²×C₃	C₂×#97
prime radical	C₂⁴×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#405	D₃
hypocenter	C₃	C₂×#2110
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #69884	C₂	direct (N × H)
4. C₂×#4466	C₂	semidirect
13. C₂×#812	C₄	semidirect
14. C₂×#812	C₄	semidirect
17. C₂²×C₃⋊C₁₆	C₄	semidirect
18. #4466	C₂²	semidirect
44. #812	C₂×C₄	semidirect
47. C₂×C₃⋊C₁₆	C₂×C₄	semidirect
215. C₂×C₃	#2110	semidirect
231. C₃	C₂×#2110	semidirect
235. C₂	#69884	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×#405	2
C₂²×#812	2
#69884	2
C₂×#4466	2
C₂×#2110	3

Subgroups — C2×#69884 #10947845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69884 #10947845