Subgroups — C₃⋊C₈×H(5,𝔽₂) #11660846

Order 768.

2422 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#82520	C₂	1	6	yes	maximal, complemented
3	#41×C₃⋊C₈	C₂	1	6	yes	maximal, complemented
4	#82301	C₂	1	9	yes	maximal, complemented
5	C₂×D₄×C₃⋊C₈	C₂	1	9	yes	maximal, complemented
6	C₈×H(5,𝔽₂)	—	3	1	no	maximal, Sylow
7	C₂×C₃×H(5,𝔽₂)	C₄	1	1	yes	characteristic
8	C₃×#520	C₄	1	6	yes	—
9	C₃×#518	C₄	1	9	yes	—
10	C₃×C₄×#41	C₂²	1	6	yes	—
11	C₂×C₃×C₄×D₄	C₂²	1	9	yes	—
12	C₂³×C₃⋊C₈	C₂²	1	6	yes	complemented
13	C₂×#830	C₂²	1	18	yes	complemented
14	Q₈×C₃⋊C₈	C₂²	1	2	yes	complemented
15	Q₈×C₃⋊C₈	C₂²	1	6	yes	complemented
16	D₄×C₃⋊C₈	C₂²	1	18	yes	complemented
17	D₄×C₃⋊C₈	C₂²	1	18	yes	complemented
18	D₄×C₃⋊C₈	C₂²	1	36	yes	complemented
19	#4925	C₂²	1	18	yes	complemented
20	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	9	yes	complemented
21	C₂×C₄×C₃⋊C₈	C₂²	1	9	yes	complemented
22	C₄×H(5,𝔽₂)	D₃	1	1	yes	characteristic
23	#2988	—	3	6	no	—
24	#2972	—	3	9	no	—
25	C₈×#41	—	3	6	no	—
26	C₂×C₈×D₄	—	3	9	no	—
27	C₃×H(5,𝔽₂)	C₈	1	16	yes	complemented
28	C₂×C₃×#41	C₂×C₄	1	6	yes	—
29	C₂²×C₃×D₄	C₂×C₄	1	9	yes	—
30	C₂³×C₃×C₄	C₂³	1	6	yes	—
31	C₃×C₄×Q₈	C₂×C₄	1	6	yes	—
32	C₃×C₄×Q₈	C₂³	1	2	yes	—
33	C₃×C₄×D₄	C₂×C₄	1	36	yes	—
34	C₃×C₄×D₄	C₂×C₄	1	18	yes	—
35	C₃×C₄×D₄	C₂³	1	18	yes	—
36	C₃×#117	C₂×C₄	1	18	yes	—
37	C₂×C₃×C₄⋊C₄	C₂×C₄	1	9	yes	—
38	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	18	yes	—
39	C₂×C₃×C₄²	C₂³	1	9	yes	—
40	C₂²×C₃⋊C₈	—	4	12	no	—
41	C₂²×C₃⋊C₈	C₂³	1	9	yes	—
42	C₂²×C₃⋊C₈	C₂³	1	9	yes	—
43	C₂²×C₃⋊C₈	C₂³	1	18	yes	—
44	#830	C₂³	1	36	yes	—
45	C₄⋊(C₃⋊C₈)	C₂³	1	18	yes	—
46	C₄⋊(C₃⋊C₈)	C₂³	1	6	yes	—
47	C₄×C₃⋊C₈	C₂³	1	6	yes	—
48	C₄×C₃⋊C₈	C₂³	1	18	yes	—
49	C₂×H(5,𝔽₂)	Dic₃	1	1	yes	characteristic
50	C₂³×C₈	—	3	6	no	—
51	#520	Dic₃	1	6	yes	—
52	#518	Dic₃	1	9	yes	—
53	C₄×#41	C₂×D₃	1	6	yes	—
54	C₂×C₄×D₄	C₂×D₃	1	9	yes	—
55	C₈×Q₈	—	3	2	no	—
56	C₈×Q₈	—	3	6	no	—
57	C₈×D₄	—	3	18	no	—
58	C₈×D₄	—	3	36	no	—
59	C₈×D₄	—	3	18	no	—
60	#431	—	3	18	no	—
61	C₂×C₄⋊C₈	—	3	9	no	—
62	C₂×C₂²⋊C₈	—	3	18	no	—
63	C₂×C₄×C₈	—	3	9	no	—
64	C₂⁴×C₃	C₂²×C₄	1	6	yes	—
65	C₃×#41	C₂×C₈	1	48	yes	complemented
66	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
67	C₂×C₃×D₄	C₂×C₈	1	72	yes	complemented
68	C₂×C₃×D₄	C₂²×C₄	1	18	yes	—
69	C₂²×C₃×C₄	—	4	6	no	—
70	C₂²×C₃×C₄	—	4	6	no	—
71	C₂²×C₃×C₄	C₂²×C₄	1	9	yes	—
72	C₂²×C₃×C₄	C₂²×C₄	1	18	yes	—
73	C₂²×C₃×C₄	C₂²×C₄	1	9	yes	—
74	C₂²×C₃×C₄	C₂⁴	1	9	yes	—
75	C₃×C₄⋊C₄	C₂²×C₄	1	18	yes	—
76	C₃×C₄⋊C₄	C₂²×C₄	1	6	yes	—
77	C₃×C₂²⋊C₄	C₂²×C₄	1	36	yes	—
78	C₃×C₄²	C₂²×C₄	1	18	yes	—
79	C₃×C₄²	C₂⁴	1	6	yes	—
80	C₂×C₃⋊C₈	—	2	18	no	—
81	C₂×C₃⋊C₈	—	4	18	no	—
82	C₂×C₃⋊C₈	—	4	9	no	—
83	C₂×C₃⋊C₈	C₂⁴	1	6	yes	—
84	C₂×C₃⋊C₈	C₂⁴	1	9	yes	—
85	C₂×C₃⋊C₈	C₂⁴	1	1	yes	characteristic
86	H(5,𝔽₂)	C₃⋊C₈	1	16	yes	direct product factor
87	C₂×#41	C₂×Dic₃	1	6	yes	—
88	C₂²×D₄	C₂×Dic₃	1	9	yes	—
89	C₂³×C₄	C₂²×D₃	1	6	yes	—
90	C₂²×C₈	—	3	9	no	—
91	C₂²×C₈	—	3	18	no	—
92	C₂²×C₈	—	3	9	no	—
93	C₂²×C₈	—	12	12	no	—
94	C₄×Q₈	C₂×Dic₃	1	6	yes	—
95	C₄×Q₈	C₂²×D₃	1	2	yes	—
96	C₄×D₄	C₂×Dic₃	1	36	yes	—
97	C₄×D₄	C₂×Dic₃	1	18	yes	—
98	C₄×D₄	C₂²×D₃	1	18	yes	—
99	#117	C₂×Dic₃	1	18	yes	—
100	C₂×C₄⋊C₄	C₂×Dic₃	1	9	yes	—
101	C₂×C₂²⋊C₄	C₂×Dic₃	1	18	yes	—
102	C₂×C₄²	C₂²×D₃	1	9	yes	—
103	C₄⋊C₈	—	3	18	no	—
104	C₄⋊C₈	—	3	6	no	—
105	C₂²⋊C₈	—	3	36	no	—
106	C₄×C₈	—	3	6	no	—
107	C₄×C₈	—	3	18	no	—
108	C₂³×C₃	—	4	6	no	—
109	C₂³×C₃	—	4	6	no	—
110	C₂³×C₃	C₂²×C₈	1	24	yes	complemented
111	C₂³×C₃	C₂³×C₄	1	9	yes	—
112	C₃×Q₈	C₂²×C₈	1	8	yes	complemented
113	C₃×D₄	C₂²×C₈	1	72	yes	complemented
114	C₂×C₃×C₄	—	2	9	no	—
115	C₂×C₃×C₄	—	2	9	no	—
116	C₂×C₃×C₄	—	4	18	no	—
117	C₂×C₃×C₄	—	4	9	no	—
118	C₂×C₃×C₄	C₂²×C₈	1	36	yes	complemented
119	C₂×C₃×C₄	C₂³×C₄	1	6	yes	—
120	C₂×C₃×C₄	C₂³×C₄	1	9	yes	—
121	C₂×C₃×C₄	C₂³×C₄	1	6	yes	—
122	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
123	C₃⋊C₈	—	2	9	no	—
124	C₃⋊C₈	—	2	6	no	—
125	C₃⋊C₈	H(5,𝔽₂)	1	2	yes	direct product factor
126	C₂⁴	C₂²×Dic₃	1	6	yes	—
127	#41	C₂×C₃⋊C₈	1	48	yes	complemented
128	C₂×Q₈	C₂²×Dic₃	1	2	yes	—
129	C₂×D₄	C₂×C₃⋊C₈	1	72	yes	complemented
130	C₂×D₄	C₂²×Dic₃	1	18	yes	—
131	C₂²×C₄	—	4	6	no	—
132	C₂²×C₄	—	4	6	no	—
133	C₂²×C₄	C₂²×Dic₃	1	9	yes	—
134	C₂²×C₄	C₂²×Dic₃	1	18	yes	—
135	C₂²×C₄	C₂²×Dic₃	1	9	yes	—
136	C₂²×C₄	C₂³×D₃	1	9	yes	—
137	C₂×C₈	—	3	1	no	—
138	C₂×C₈	—	3	6	no	—
139	C₂×C₈	—	3	9	no	—
140	C₂×C₈	—	6	18	no	—
141	C₂×C₈	—	12	18	no	—
142	C₂×C₈	—	12	9	no	—
143	C₄⋊C₄	C₂²×Dic₃	1	18	yes	—
144	C₄⋊C₄	C₂²×Dic₃	1	6	yes	—
145	C₂²⋊C₄	C₂²×Dic₃	1	36	yes	—
146	C₄²	C₂²×Dic₃	1	18	yes	—
147	C₄²	C₂³×D₃	1	6	yes	—
148	C₂²×C₃	—	2	9	no	—
149	C₂²×C₃	—	2	9	no	—
150	C₂²×C₃	—	4	24	no	—
151	C₂²×C₃	C₂³×C₈	1	18	yes	—
152	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
153	C₃×C₄	—	2	9	no	—
154	C₃×C₄	—	2	6	no	—
155	C₃×C₄	C₂³×C₈	1	12	yes	—
156	C₃×C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
157	C₃×C₄	#585	1	1	yes	characteristic
158	C₂³	—	4	6	no	—
159	C₂³	—	4	6	no	—
160	C₂³	C₂²×C₃⋊C₈	1	24	yes	complemented
161	C₂³	C₂³×Dic₃	1	9	yes	—
162	Q₈	C₂²×C₃⋊C₈	1	8	yes	complemented
163	D₄	C₂²×C₃⋊C₈	1	72	yes	complemented
164	C₂×C₄	—	2	9	no	—
165	C₂×C₄	—	2	9	no	—
166	C₂×C₄	—	4	18	no	—
167	C₂×C₄	—	4	9	no	—
168	C₂×C₄	C₂²×C₃⋊C₈	1	36	yes	complemented
169	C₂×C₄	C₂³×Dic₃	1	6	yes	—
170	C₂×C₄	C₂³×Dic₃	1	9	yes	—
171	C₂×C₄	C₂³×Dic₃	1	6	yes	—
172	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
173	C₈	—	3	2	no	—
174	C₈	—	6	9	no	—
175	C₈	—	6	6	no	—
176	C₂×C₃	—	2	18	no	—
177	C₂×C₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
179	C₂×C₃	#3573	1	1	yes	characteristic
180	C₂²	—	2	9	no	—
181	C₂²	—	2	9	no	—
182	C₂²	—	4	24	no	—
183	C₂²	C₂³×C₃⋊C₈	1	18	yes	—
184	C₂²	C₂⁴×Dic₃	1	1	yes	characteristic
185	C₄	—	2	6	no	—
186	C₄	—	2	9	no	—
187	C₄	C₂³×C₃⋊C₈	1	12	yes	—
188	C₄	D₃×H(5,𝔽₂)	1	1	yes	characteristic
189	C₄	#5966	1	1	yes	characteristic
190	C₃	C₈×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
191	C₂	—	2	18	no	—
192	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic
193	C₂	Dic₃×H(5,𝔽₂)	1	1	yes	characteristic
194	C₂	#87296	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₃
commutator	C₂×C₃	C₂⁴×C₈
Fitting	C₃×C₄×H(5,𝔽₂)	C₂
socle	C₂²×C₃	C₂⁴×C₄
prime radical	C₂×C₃×H(5,𝔽₂)	C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₄×H(5,𝔽₂)	D₃
hypocenter	C₃	C₈×H(5,𝔽₂)
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #82520	C₂	semidirect
3. #41×C₃⋊C₈	C₂	semidirect
4. #82301	C₂	semidirect
5. C₂×D₄×C₃⋊C₈	C₂	semidirect
12. C₂³×C₃⋊C₈	C₂²	semidirect
13. C₂×#830	C₂²	semidirect
14. Q₈×C₃⋊C₈	C₂²	semidirect
15. Q₈×C₃⋊C₈	C₂²	semidirect
16. D₄×C₃⋊C₈	C₂²	semidirect
17. D₄×C₃⋊C₈	C₂²	semidirect
18. D₄×C₃⋊C₈	C₂²	semidirect
19. #4925	C₂²	semidirect
20. C₂×C₄⋊(C₃⋊C₈)	C₂²	semidirect
21. C₂×C₄×C₃⋊C₈	C₂²	semidirect
27. C₃×H(5,𝔽₂)	C₈	semidirect
65. C₃×#41	C₂×C₈	semidirect
67. C₂×C₃×D₄	C₂×C₈	semidirect
86. H(5,𝔽₂)	C₃⋊C₈	direct (N × H)
110. C₂³×C₃	C₂²×C₈	semidirect
112. C₃×Q₈	C₂²×C₈	semidirect
113. C₃×D₄	C₂²×C₈	semidirect
118. C₂×C₃×C₄	C₂²×C₈	semidirect
125. C₃⋊C₈	H(5,𝔽₂)	direct (N × H)
127. #41	C₂×C₃⋊C₈	semidirect
129. C₂×D₄	C₂×C₃⋊C₈	semidirect
160. C₂³	C₂²×C₃⋊C₈	semidirect
162. Q₈	C₂²×C₃⋊C₈	semidirect
163. D₄	C₂²×C₃⋊C₈	semidirect
168. C₂×C₄	C₂²×C₃⋊C₈	semidirect
190. C₃	C₈×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×H(5,𝔽₂)	2
#82520	2
#41×C₃⋊C₈	2
#82301	2
C₂×D₄×C₃⋊C₈	2
C₈×H(5,𝔽₂)	3

Subgroups — C3⋊C8×H(5,𝔽2) #11660846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊C₈×H(5,𝔽₂) #11660846