Subgroups — C₂×#78442 #11664792

Order 768.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#5470	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#5106	C₂	1	4	yes	maximal
4	#78442	C₂	1	16	yes	maximal, direct product factor
5	C₂×#4682	C₂	1	2	yes	maximal, complemented
6	C₂×#4683	C₂	1	4	yes	maximal, complemented
7	C₂×#4680	C₂	1	2	yes	maximal, complemented
8	C₂×#3160	—	3	1	no	maximal, Sylow
9	C₂³×C₃×C₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×#926	C₂²	1	4	yes	—
12	C₂×#924	—	2	4	no	—
13	#5470	—	2	8	no	—
14	#5470	C₂²	1	16	yes	—
15	C₂×#870	C₂²	1	4	yes	—
16	C₂×#869	C₂²	1	4	yes	—
17	C₂×C₄×Dic₆	—	2	4	no	—
18	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×C₈	C₂²	1	8	yes	—
22	#5106	C₂²	1	32	yes	—
23	C₂×#818	C₂²	1	8	yes	—
24	C₂×#817	C₂²	1	8	yes	—
25	#4683	C₂²	1	32	yes	complemented
26	#4682	C₂²	1	16	yes	complemented
27	#4680	C₂²	1	16	yes	complemented
28	C₂×#533	—	3	2	no	—
29	#3160	—	3	16	no	—
30	C₂×#500	—	3	2	no	—
31	C₂×#499	—	3	4	no	—
32	C₂×#498	—	3	2	no	—
33	C₂×#427	—	3	4	no	—
34	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
35	C₂³×C₃×C₄	D₄	1	2	yes	—
36	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂²×Dic₆	—	2	4	no	—
38	C₂²×C₃×C₈	Q₈	1	16	yes	complemented
39	C₂²×C₃×C₈	C₂³	1	2	yes	—
40	C₂²×C₃×C₈	C₂³	1	2	yes	—
41	C₂²×C₃×C₈	C₂³	1	8	yes	—
42	C₂×C₃×C₄²	D₄	1	2	yes	—
43	C₂×C₃×C₄²	D₄	1	4	yes	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂³	1	4	yes	—
47	C₂×#217	—	2	4	no	—
48	#926	—	2	8	no	—
49	#926	C₂³	1	16	yes	—
50	C₂×C₄⋊Dic₃	—	2	4	no	—
51	C₂×C₄⋊Dic₃	C₂³	1	8	yes	—
52	#924	—	2	32	no	—
53	C₂×Dic₃⋊C₄	—	2	8	no	—
54	C₂×C₄×Dic₃	—	2	4	no	—
55	#870	—	2	8	no	—
56	#870	C₂³	1	16	yes	—
57	#869	—	2	8	no	—
58	#869	C₂³	1	16	yes	—
59	C₄×Dic₆	—	2	32	no	—
60	C₃×C₄×C₈	C₂³	1	4	yes	—
61	C₃×C₄×C₈	C₂³	1	4	yes	—
62	C₃×C₄×C₈	C₂³	1	8	yes	—
63	#818	C₂³	1	32	yes	—
64	#817	C₂³	1	32	yes	—
65	C₂³×C₈	C₂×D₃	1	2	yes	—
66	#533	—	3	16	no	—
67	#533	—	6	8	no	—
68	C₂×C₄⋊Q₈	—	3	4	no	—
69	C₂×#125	—	3	4	no	—
70	C₂×C₂²⋊Q₈	—	6	4	no	—
71	C₂×C₄×Q₈	—	6	4	no	—
72	C₂×#117	—	3	4	no	—
73	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
74	#500	—	3	16	no	—
75	#499	—	3	32	no	—
76	#498	—	3	16	no	—
77	#427	—	3	32	no	—
78	C₂×#107	—	3	8	no	—
79	C₂×#106	—	3	8	no	—
80	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₃	1	8	yes	—
84	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	—	2	8	no	—
86	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
88	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
89	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×Q₈	1	8	yes	—
92	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
94	C₂²×Dic₃	—	2	8	no	—
95	C₂×Dic₆	—	2	16	no	—
96	C₂×Dic₆	—	4	16	no	—
97	C₂×C₃×C₈	C₂×Q₈	1	8	yes	—
98	C₂×C₃×C₈	C₂×Q₈	1	8	yes	—
99	C₂×C₃×C₈	C₂×Q₈	1	32	yes	complemented
100	C₂×C₃×C₈	C₂⁴	1	4	yes	—
101	C₂×C₃×C₈	C₂⁴	1	4	yes	—
102	C₃×C₄²	C₂×D₄	1	8	yes	—
103	C₃×C₄²	C₂×D₄	1	2	yes	—
104	C₃×C₄²	C₂×D₄	1	2	yes	—
105	C₃×C₄²	C₂⁴	1	2	yes	—
106	C₃×C₄²	C₂⁴	1	2	yes	—
107	#217	—	2	16	no	—
108	C₄⋊Dic₃	—	2	16	no	—
109	C₄⋊Dic₃	—	2	8	no	—
110	C₄⋊Dic₃	C₂⁴	1	16	yes	—
111	Dic₃⋊C₄	—	2	32	no	—
112	C₄×Dic₃	—	2	16	no	—
113	C₂²×Q₈	—	6	4	no	—
114	C₂³×C₄	D₁₂	1	2	yes	—
115	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂²×C₈	Dic₆	1	16	yes	complemented
117	C₂²×C₈	C₂²×D₃	1	2	yes	—
118	C₂²×C₈	C₂²×D₃	1	2	yes	—
119	C₂²×C₈	C₂²×D₃	1	8	yes	—
120	C₄⋊Q₈	—	3	16	no	—
121	C₄⋊Q₈	—	6	8	no	—
122	#125	—	3	16	no	—
123	#125	—	6	8	no	—
124	C₂²⋊Q₈	—	6	32	no	—
125	C₄×Q₈	—	6	32	no	—
126	#117	—	3	16	no	—
127	#117	—	6	8	no	—
128	C₂×C₄⋊C₄	—	3	8	no	—
129	C₂×C₄⋊C₄	—	6	8	no	—
130	C₂×C₄⋊C₄	—	6	4	no	—
131	C₂×C₂²⋊C₄	—	6	4	no	—
132	C₂×C₄²	—	6	4	no	—
133	C₂×C₄²	D₁₂	1	4	yes	—
134	C₂×C₄²	D₁₂	1	2	yes	—
135	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₄²	C₂²×D₃	1	4	yes	—
138	#107	—	3	32	no	—
139	#106	—	3	32	no	—
140	C₄×C₈	C₂²×D₃	1	4	yes	—
141	C₄×C₈	C₂²×D₃	1	8	yes	—
142	C₄×C₈	C₂²×D₃	1	4	yes	—
143	C₂³×C₃	—	2	4	no	—
144	C₂³×C₃	C₄⋊Q₈	1	4	yes	—
145	C₂³×C₃	C₂²×D₄	1	2	yes	—
146	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	16	no	—
149	C₂×C₃×C₄	C₄⋊Q₈	1	8	yes	—
150	C₂×C₃×C₄	C₄⋊Q₈	1	4	yes	—
151	C₂×C₃×C₄	#135	1	8	yes	—
152	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
153	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
154	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
156	C₂×C₃×C₄	C₂²×Q₈	1	4	yes	—
157	C₂×C₃×C₄	C₂²×Q₈	1	8	yes	—
158	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
159	C₂×Dic₃	—	2	16	no	—
160	C₂×Dic₃	—	4	16	no	—
161	Dic₆	—	4	32	no	—
162	C₃×C₈	C₂²×Q₈	1	8	yes	—
163	C₃×C₈	C₂²×Q₈	1	8	yes	—
164	C₂⁴	C₂×D₁₂	1	1	yes	characteristic
165	C₂×Q₈	—	6	16	no	—
166	C₂×Q₈	—	12	16	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	6	8	no	—
169	C₂²×C₄	C₂×Dic₆	1	8	yes	—
170	C₂²×C₄	C₂×D₁₂	1	2	yes	—
171	C₂²×C₄	C₂×D₁₂	1	8	yes	—
172	C₂²×C₄	C₂×D₁₂	1	2	yes	—
173	C₂²×C₄	C₂×D₁₂	1	2	yes	—
174	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
175	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×D₃	1	2	yes	—
177	C₂×C₈	C₂×Dic₆	1	8	yes	—
178	C₂×C₈	C₂×Dic₆	1	8	yes	—
179	C₂×C₈	C₂×Dic₆	1	32	yes	complemented
180	C₂×C₈	C₂³×D₃	1	4	yes	—
181	C₂×C₈	C₂³×D₃	1	4	yes	—
182	C₄⋊C₄	—	3	16	no	—
183	C₄⋊C₄	—	6	8	no	—
184	C₄⋊C₄	—	6	32	no	—
185	C₄⋊C₄	—	6	16	no	—
186	C₂²⋊C₄	—	6	16	no	—
187	C₄²	—	6	16	no	—
188	C₄²	C₂×D₁₂	1	8	yes	—
189	C₄²	C₂×D₁₂	1	2	yes	—
190	C₄²	C₂×D₁₂	1	2	yes	—
191	C₄²	C₂³×D₃	1	2	yes	—
192	C₄²	C₂³×D₃	1	2	yes	—
193	C₂²×C₃	—	2	8	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	C₂×C₄⋊Q₈	1	4	yes	—
196	C₂²×C₃	C₂×C₄⋊Q₈	1	2	yes	—
197	C₂²×C₃	C₂×#135	1	4	yes	—
198	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	C₂×C₄⋊Q₈	1	4	yes	—
201	C₃×C₄	C₂×C₄⋊Q₈	1	2	yes	—
202	C₃×C₄	C₂×#135	1	8	yes	—
203	C₃×C₄	C₂³×Q₈	1	2	yes	—
204	Dic₃	—	4	16	no	—
205	C₂³	—	2	4	no	—
206	C₂³	#869	1	4	yes	—
207	C₂³	C₂²×D₁₂	1	2	yes	—
208	C₂³	C₂²×D₁₂	1	1	yes	characteristic
209	Q₈	—	12	32	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	6	16	no	—
213	C₂×C₄	—	12	16	no	—
214	C₂×C₄	#869	1	8	yes	—
215	C₂×C₄	#869	1	4	yes	—
216	C₂×C₄	#904	1	8	yes	—
217	C₂×C₄	C₂²×Dic₆	1	4	yes	—
218	C₂×C₄	C₂²×Dic₆	1	8	yes	—
219	C₂×C₄	C₂²×D₁₂	1	4	yes	—
220	C₂×C₄	C₂²×D₁₂	1	2	yes	—
221	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₁₂	1	4	yes	—
223	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
224	C₈	C₂²×Dic₆	1	8	yes	—
225	C₈	C₂²×Dic₆	1	8	yes	—
226	C₂×C₃	—	2	4	no	—
227	C₂×C₃	#3160	1	4	yes	complemented
228	C₂×C₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
229	C₂×C₃	C₂²×#135	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	4	no	—
232	C₂²	C₂×#869	1	4	yes	—
233	C₂²	C₂×#869	1	2	yes	—
234	C₂²	C₂×#904	1	4	yes	—
235	C₂²	C₂³×D₁₂	1	1	yes	characteristic
236	C₄	—	2	4	no	—
237	C₄	—	12	16	no	—
238	C₄	C₂×#869	1	2	yes	—
239	C₄	C₂×#869	1	4	yes	—
240	C₄	C₂×#904	1	8	yes	—
241	C₄	C₂³×Dic₆	1	2	yes	—
242	C₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
243	C₂	—	2	4	no	—
244	C₂	#78442	1	4	yes	direct product factor
245	C₂	C₂²×#869	1	1	yes	characteristic
246	C₂	C₂²×#904	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle	C₂³×C₃	C₂²×D₄
prime radical	C₂⁴×C₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #78442	C₂	direct (N × H)
5. C₂×#4682	C₂	semidirect
6. C₂×#4683	C₂	semidirect
7. C₂×#4680	C₂	semidirect
25. #4683	C₂²	semidirect
26. #4682	C₂²	semidirect
27. #4680	C₂²	semidirect
38. C₂²×C₃×C₈	Q₈	semidirect
99. C₂×C₃×C₈	C₂×Q₈	semidirect
116. C₂²×C₈	Dic₆	semidirect
179. C₂×C₈	C₂×Dic₆	semidirect
227. C₂×C₃	#3160	semidirect
242. C₃	C₂×#3160	semidirect
244. C₂	#78442	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×#5470	2
C₂²×C₃×C₄×C₈	2
C₂×#5106	2
#78442	2
C₂×#4682	2
C₂×#4683	2
C₂×#4680	2
C₂×#3160	3

Subgroups — C2×#78442 #11664792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#78442 #11664792