Subgroups — C₂×#83789 #11658885

Order 768.

2586 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#5819	C₂	1	2	yes	maximal
3	#83789	C₂	1	16	yes	maximal, direct product factor
4	C₂×#5125	C₂	1	6	yes	maximal, complemented
5	C₂²×#819	C₂	1	6	yes	maximal, complemented
6	C₂×#2916	—	3	1	no	maximal, Sylow
7	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×#131	C₄	1	8	yes	—
9	C₂×C₃×#131	C₄	1	8	yes	—
10	C₂×C₃×#131	C₂²	1	8	yes	—
11	C₂²×C₃×M₁₆	C₂²	1	3	yes	—
12	C₂³×C₃×C₈	C₂²	1	3	yes	—
13	#5819	—	2	8	no	—
14	#5819	C₂²	1	16	yes	—
15	C₂×#948	—	2	8	no	—
16	C₂²×#208	C₂²	1	12	yes	—
17	#5125	C₂²	1	48	yes	complemented
18	C₂×#819	C₂²	1	4	yes	—
19	C₂×#819	C₂²	1	12	yes	—
20	C₂×#819	C₂²	1	48	yes	complemented
21	C₂×#568	—	3	2	no	—
22	C₂²×#131	D₃	1	1	yes	characteristic
23	#2916	—	3	16	no	—
24	C₂×#430	—	3	6	no	—
25	C₂²×#108	—	3	6	no	—
26	C₂×C₃×#41	C₂×C₄	1	4	yes	—
27	C₂×C₃×#41	C₂×C₄	1	4	yes	—
28	C₂×C₃×#41	D₄	1	4	yes	—
29	C₂×C₃×#41	D₄	1	4	yes	—
30	C₂×C₃×#41	Q₈	1	4	yes	—
31	C₂×C₃×#41	C₂³	1	4	yes	—
32	C₂²×C₃×Q₈	Q₈	1	1	yes	characteristic
33	C₂²×C₃×D₄	Q₈	1	3	yes	—
34	C₂³×C₃×C₄	C₂³	1	3	yes	—
35	C₃×#131	C₂×C₄	1	64	yes	—
36	C₂×C₃×M₁₆	C₂×C₄	1	12	yes	—
37	C₂×C₃×M₁₆	C₂×C₄	1	12	yes	—
38	C₂×C₃×M₁₆	C₂³	1	12	yes	—
39	C₂²×C₃×C₈	C₂×C₄	1	12	yes	—
40	C₂²×C₃×C₈	C₂×C₄	1	12	yes	—
41	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	3	yes	—
43	C₂²×C₃×C₈	C₂³	1	12	yes	—
44	#948	—	2	64	no	—
45	C₂×#208	—	2	12	no	—
46	C₂×#208	—	2	24	no	—
47	C₂×#208	C₂³	1	48	yes	—
48	C₂×#208	C₂³	1	8	yes	—
49	C₂²×C₃⋊C₈	—	2	12	no	—
50	#819	C₂³	1	48	yes	—
51	#819	C₂³	1	16	yes	—
52	C₂²×#41	C₂×D₃	1	1	yes	characteristic
53	#568	—	3	16	no	—
54	#568	—	6	8	no	—
55	C₂×#131	—	6	8	no	—
56	C₂×#131	Dic₃	1	8	yes	—
57	C₂×#131	Dic₃	1	8	yes	—
58	C₂×#131	C₂×D₃	1	8	yes	—
59	C₂²×M₁₆	—	3	12	no	—
60	C₂²×M₁₆	C₂×D₃	1	3	yes	—
61	C₂³×C₈	C₂×D₃	1	3	yes	—
62	#430	—	3	48	no	—
63	C₂×#108	—	3	4	no	—
64	C₂×#108	—	3	12	no	—
65	C₂×#108	—	3	48	no	—
66	C₂⁴×C₃	C₂×Q₈	1	3	yes	—
67	C₃×#41	C₄⋊C₄	1	16	yes	—
68	C₃×#41	C₄⋊C₄	1	16	yes	—
69	C₃×#41	C₂²×C₄	1	16	yes	—
70	C₃×#41	C₂×D₄	1	16	yes	—
71	C₂×C₃×Q₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₃×Q₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₃×Q₈	C₂×Q₈	1	4	yes	—
74	C₂×C₃×D₄	C₄⋊C₄	1	12	yes	—
75	C₂×C₃×D₄	C₄⋊C₄	1	12	yes	—
76	C₂×C₃×D₄	C₂×Q₈	1	12	yes	—
77	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
78	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
79	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
80	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
81	C₂²×C₃×C₄	C₂×Q₈	1	3	yes	—
82	C₂²×C₃×C₄	C₂×Q₈	1	6	yes	—
83	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
85	C₃×M₁₆	C₂²×C₄	1	48	yes	—
86	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
89	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
90	C₂×C₃×C₈	C₂²×C₄	1	48	yes	—
91	C₂×C₃×C₈	C₂⁴	1	2	yes	—
92	C₂×C₃×C₈	C₂⁴	1	6	yes	—
93	#208	—	2	8	no	—
94	#208	—	2	48	no	—
95	#208	C₂⁴	1	16	yes	—
96	C₂×C₃⋊C₈	—	2	8	no	—
97	C₂×C₃⋊C₈	—	2	48	no	—
98	C₂×#41	Dic₆	1	4	yes	—
99	C₂×#41	D₁₂	1	4	yes	—
100	C₂×#41	D₁₂	1	4	yes	—
101	C₂×#41	C₂×Dic₃	1	4	yes	—
102	C₂×#41	C₂×Dic₃	1	4	yes	—
103	C₂×#41	C₂²×D₃	1	4	yes	—
104	C₂²×Q₈	Dic₆	1	1	yes	characteristic
105	C₂²×D₄	Dic₆	1	3	yes	—
106	C₂³×C₄	C₂²×D₃	1	3	yes	—
107	#131	—	6	64	no	—
108	#131	C₂×Dic₃	1	64	yes	—
109	C₂×M₁₆	—	3	8	no	—
110	C₂×M₁₆	—	3	48	no	—
111	C₂×M₁₆	—	6	12	no	—
112	C₂×M₁₆	—	6	24	no	—
113	C₂×M₁₆	C₂×Dic₃	1	12	yes	—
114	C₂×M₁₆	C₂×Dic₃	1	12	yes	—
115	C₂×M₁₆	C₂²×D₃	1	12	yes	—
116	C₂²×C₈	—	6	12	no	—
117	C₂²×C₈	C₂×Dic₃	1	12	yes	—
118	C₂²×C₈	C₂×Dic₃	1	12	yes	—
119	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₃	1	3	yes	—
121	C₂²×C₈	C₂²×D₃	1	12	yes	—
122	#108	—	3	16	no	—
123	#108	—	3	48	no	—
124	C₂³×C₃	—	4	6	no	—
125	C₂³×C₃	C₂×C₄⋊C₄	1	6	yes	—
126	C₂³×C₃	C₂×C₄⋊C₄	1	6	yes	—
127	C₂³×C₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂³×C₃	C₂²×Q₈	1	6	yes	—
129	C₃×Q₈	C₂×C₄⋊C₄	1	16	yes	—
130	C₃×D₄	C₂×C₄⋊C₄	1	48	yes	—
131	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
132	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
133	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
134	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
135	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
138	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
139	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
143	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
144	C₃×C₈	C₂³×C₄	1	4	yes	—
145	C₃×C₈	C₂³×C₄	1	12	yes	—
146	C₃⋊C₈	—	2	16	no	—
147	C₂⁴	C₂×Dic₆	1	3	yes	—
148	#41	C₄⋊Dic₃	1	16	yes	—
149	#41	C₄⋊Dic₃	1	16	yes	—
150	#41	C₂×D₁₂	1	16	yes	—
151	#41	C₂²×Dic₃	1	16	yes	—
152	C₂×Q₈	C₄⋊Dic₃	1	4	yes	—
153	C₂×Q₈	C₄⋊Dic₃	1	4	yes	—
154	C₂×Q₈	C₂×Dic₆	1	4	yes	—
155	C₂×D₄	C₄⋊Dic₃	1	12	yes	—
156	C₂×D₄	C₄⋊Dic₃	1	12	yes	—
157	C₂×D₄	C₂×Dic₆	1	12	yes	—
158	C₂²×C₄	C₂×Dic₆	1	3	yes	—
159	C₂²×C₄	C₂×Dic₆	1	6	yes	—
160	C₂²×C₄	C₂×D₁₂	1	6	yes	—
161	C₂²×C₄	C₂×D₁₂	1	6	yes	—
162	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
163	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
164	C₂²×C₄	C₂³×D₃	1	6	yes	—
165	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
166	M₁₆	—	3	16	no	—
167	M₁₆	—	6	8	no	—
168	M₁₆	—	6	48	no	—
169	M₁₆	C₂²×Dic₃	1	48	yes	—
170	C₂×C₈	—	6	48	no	—
171	C₂×C₈	—	6	8	no	—
172	C₂×C₈	C₂²×Dic₃	1	2	yes	—
173	C₂×C₈	C₂²×Dic₃	1	2	yes	—
174	C₂×C₈	C₂²×Dic₃	1	6	yes	—
175	C₂×C₈	C₂²×Dic₃	1	6	yes	—
176	C₂×C₈	C₂²×Dic₃	1	48	yes	—
177	C₂×C₈	C₂³×D₃	1	2	yes	—
178	C₂×C₈	C₂³×D₃	1	6	yes	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	12	no	—
181	C₂²×C₃	—	4	6	no	—
182	C₂²×C₃	—	4	6	no	—
183	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₃	C₂²×C₄⋊C₄	1	12	yes	—
185	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
187	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₃×C₄	C₂²×C₄⋊C₄	1	12	yes	—
190	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
191	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
192	C₂³	—	4	6	no	—
193	C₂³	C₂×C₄⋊Dic₃	1	6	yes	—
194	C₂³	C₂×C₄⋊Dic₃	1	6	yes	—
195	C₂³	C₂²×Dic₆	1	1	yes	characteristic
196	C₂³	C₂²×Dic₆	1	6	yes	—
197	Q₈	C₂×C₄⋊Dic₃	1	16	yes	—
198	D₄	C₂×C₄⋊Dic₃	1	48	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
203	C₂×C₄	C₂²×Dic₆	1	1	yes	characteristic
204	C₂×C₄	C₂²×Dic₆	1	6	yes	—
205	C₂×C₄	C₂²×D₁₂	1	12	yes	—
206	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
207	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
208	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
209	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×Dic₃	1	12	yes	—
211	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
212	C₈	—	6	16	no	—
213	C₈	C₂³×Dic₃	1	4	yes	—
214	C₈	C₂³×Dic₃	1	12	yes	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	12	no	—
218	C₂×C₃	#2916	1	2	yes	complemented
219	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	12	no	—
222	C₂²	—	4	6	no	—
223	C₂²	—	4	6	no	—
224	C₂²	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
225	C₂²	C₂²×C₄⋊Dic₃	1	12	yes	—
226	C₂²	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
227	C₂²	C₂³×Dic₆	1	1	yes	characteristic
228	C₄	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
229	C₄	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
230	C₄	C₂²×C₄⋊Dic₃	1	12	yes	—
231	C₄	C₂³×D₁₂	1	1	yes	characteristic
232	C₄	C₂⁴×Dic₃	1	1	yes	characteristic
233	C₃	C₂×#2916	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	12	no	—
237	C₂	#83789	1	2	yes	direct product factor
238	C₂	C₂³×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂
commutator	C₃×C₄	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×#131	C₂
socle	C₂²×C₃	C₂²×C₄⋊C₄
prime radical	C₂²×C₃×#41	C₂²
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂²×#131	D₃
hypocenter	C₃	C₂×#2916
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
3. #83789	C₂	direct (N × H)
4. C₂×#5125	C₂	semidirect
5. C₂²×#819	C₂	semidirect
17. #5125	C₂²	semidirect
20. C₂×#819	C₂²	semidirect
218. C₂×C₃	#2916	semidirect
233. C₃	C₂×#2916	semidirect
237. C₂	#83789	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×#131	2
C₂×#5819	2
#83789	2
C₂×#5125	2
C₂²×#819	2
C₂×#2916	3

Subgroups — C2×#83789 #11658885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#83789 #11658885