Subgroups — C₃²×C₈×D₈ #11768624

Order 1152.

588 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₈×D₄	C₂	1	2	yes	maximal, complemented
3	C₃²×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×D₈	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×D₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	—	2	2	no	—
9	C₃²×C₄×D₄	C₂²	1	2	yes	—
10	C₃²×#107	C₄	1	1	yes	characteristic
11	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃²×#102	C₄	1	2	yes	—
13	C₃²×C₂²⋊C₈	—	2	2	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₈	C₂×C₃	1	4	yes	—
18	C₃×C₈×D₄	C₂×C₃	1	8	yes	complemented
19	C₃×#335	C₂×C₃	1	4	yes	complemented
20	C₃×D₄⋊C₈	C₂×C₃	1	8	yes	complemented
21	C₃×C₈²	C₂×C₃	1	4	yes	complemented
22	C₂×C₃²×D₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃²×C₄	—	2	2	no	—
24	C₃²×D₈	C₈	1	4	yes	direct product factor
25	C₂×C₃²×C₈	—	2	1	no	—
26	C₂×C₃²×C₈	—	2	2	no	—
27	C₂×C₃²×C₈	—	4	4	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
32	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₃²×C₂²⋊C₄	—	2	2	no	—
34	C₃²×C₄²	C₂³	1	1	yes	characteristic
35	C₈×D₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃×D₈	C₃×C₄	1	4	yes	—
37	C₂²×C₃×C₈	—	2	8	no	—
38	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
39	C₃×#107	C₃×C₄	1	4	yes	—
40	C₃×C₄⋊C₈	C₂²×C₃	1	8	yes	—
41	C₃×#102	C₃×C₄	1	8	yes	—
42	C₃×C₂²⋊C₈	—	2	8	no	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
45	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
46	C₂³×C₃²	—	2	2	no	—
47	C₃²×D₄	—	2	2	no	—
48	C₃²×D₄	C₂×C₈	1	4	yes	complemented
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	—	4	2	no	—
51	C₂×C₃²×C₄	—	4	2	no	—
52	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	#41	1	1	yes	characteristic
55	C₃²×C₈	—	2	2	no	—
56	C₃²×C₈	—	2	1	no	—
57	C₃²×C₈	—	2	1	no	—
58	C₃²×C₈	—	4	2	no	—
59	C₃²×C₈	C₂×C₈	1	2	yes	complemented
60	C₃²×C₈	D₈	1	2	yes	direct product factor
61	C₄×D₈	C₂×C₃²	1	1	yes	characteristic
62	C₈×D₄	C₂×C₃²	1	2	yes	complemented
63	#335	C₂×C₃²	1	1	yes	characteristic, complemented
64	D₄⋊C₈	C₂×C₃²	1	2	yes	complemented
65	C₈²	C₂×C₃²	1	1	yes	characteristic, complemented
66	C₂×C₃×D₄	C₂×C₃×C₄	1	8	yes	—
67	C₂²×C₃×C₄	—	2	8	no	—
68	C₃×D₈	C₃×C₈	1	16	yes	direct product factor
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	—	2	8	no	—
71	C₂×C₃×C₈	—	4	16	no	—
72	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
74	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
75	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
76	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
77	C₃×C₂²⋊C₄	—	2	8	no	—
78	C₃×C₄²	C₂³×C₃	1	4	yes	—
79	C₂²×C₃²	—	2	4	no	—
80	C₂²×C₃²	—	4	2	no	—
81	C₂²×C₃²	—	4	2	no	—
82	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
83	C₃²×C₄	—	2	1	no	—
84	C₃²×C₄	—	4	2	no	—
85	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
86	C₃²×C₄	#131	1	1	yes	characteristic
87	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
88	C₃²×C₄	#135	1	1	yes	characteristic
89	C₂×D₈	C₃²×C₄	1	1	yes	characteristic
90	C₂²×C₈	—	2	2	no	—
91	C₄×D₄	C₂²×C₃²	1	2	yes	—
92	#107	C₃²×C₄	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₃²	1	2	yes	—
94	#102	C₃²×C₄	1	2	yes	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
97	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
98	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
99	C₂³×C₃	—	2	8	no	—
100	C₃×D₄	—	2	8	no	—
101	C₃×D₄	C₂×C₃×C₈	1	16	yes	complemented
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	4	8	no	—
104	C₂×C₃×C₄	—	4	8	no	—
105	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
106	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
107	C₂×C₃×C₄	C₃×#41	1	4	yes	—
108	C₃×C₈	—	2	8	no	—
109	C₃×C₈	—	2	4	no	—
110	C₃×C₈	—	2	4	no	—
111	C₃×C₈	—	4	8	no	—
112	C₃×C₈	C₂×C₃×C₈	1	8	yes	complemented
113	C₃×C₈	C₃×D₈	1	8	yes	direct product factor
114	C₂×C₃²	—	4	4	no	—
115	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
116	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
117	C₂×C₃²	#443	1	1	yes	characteristic
118	C₂×D₄	C₂×C₃²×C₄	1	2	yes	—
119	C₂²×C₄	—	2	2	no	—
120	D₈	C₃²×C₈	1	4	yes	direct product factor
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	4	no	—
124	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
126	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
127	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
128	C₄⋊C₄	C₂×C₃²×C₄	1	2	yes	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₄²	C₂³×C₃²	1	1	yes	characteristic
131	C₂²×C₃	—	2	16	no	—
132	C₂²×C₃	—	4	8	no	—
133	C₂²×C₃	—	4	8	no	—
134	C₂²×C₃	C₃×C₄×D₄	1	4	yes	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	—	4	8	no	—
137	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
138	C₃×C₄	C₃×#131	1	4	yes	—
139	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
140	C₃×C₄	C₃×#135	1	4	yes	—
141	C₃²	C₈×D₈	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³	—	2	2	no	—
143	D₄	—	2	2	no	—
144	D₄	C₂×C₃²×C₈	1	4	yes	complemented
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
149	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
150	C₂×C₄	C₃²×#41	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	2	no	—
155	C₈	C₂×C₃²×C₈	1	2	yes	complemented
156	C₈	C₃²×D₈	1	2	yes	direct product factor
157	C₂×C₃	—	4	16	no	—
158	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
159	C₂×C₃	C₃×C₄×D₈	1	4	yes	—
160	C₂×C₃	C₃×#443	1	4	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	C₃²×C₄×D₄	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	4	2	no	—
167	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
168	C₄	C₃²×#131	1	1	yes	characteristic
169	C₄	C₂×C₃²×D₈	1	1	yes	characteristic
170	C₄	C₃²×#135	1	1	yes	characteristic
171	C₃	C₃×C₈×D₈	1	4	yes	direct product factor
172	C₂	—	4	4	no	—
173	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
174	C₂	C₃²×C₄×D₈	1	1	yes	characteristic
175	C₂	C₃²×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₈	D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₄²	C₂³×C₃²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃²×D₈	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₈×D₄	C₂	semidirect
3. C₃²×#335	C₂	semidirect
4. C₃²×D₄⋊C₈	C₂	semidirect
5. C₃²×C₈²	C₂	semidirect
6. C₃×C₈×D₈	C₃	direct (N × H)
18. C₃×C₈×D₄	C₂×C₃	semidirect
19. C₃×#335	C₂×C₃	semidirect
20. C₃×D₄⋊C₈	C₂×C₃	semidirect
21. C₃×C₈²	C₂×C₃	semidirect
24. C₃²×D₈	C₈	direct (N × H)
35. C₈×D₈	C₃²	direct (N × H)
48. C₃²×D₄	C₂×C₈	semidirect
59. C₃²×C₈	C₂×C₈	semidirect
60. C₃²×C₈	D₈	direct (N × H)
62. C₈×D₄	C₂×C₃²	semidirect
63. #335	C₂×C₃²	semidirect
64. D₄⋊C₈	C₂×C₃²	semidirect
65. C₈²	C₂×C₃²	semidirect
68. C₃×D₈	C₃×C₈	direct (N × H)
101. C₃×D₄	C₂×C₃×C₈	semidirect
112. C₃×C₈	C₂×C₃×C₈	semidirect
113. C₃×C₈	C₃×D₈	direct (N × H)
120. D₈	C₃²×C₈	direct (N × H)
141. C₃²	C₈×D₈	direct (N × H)
144. D₄	C₂×C₃²×C₈	semidirect
155. C₈	C₂×C₃²×C₈	semidirect
156. C₈	C₃²×D₈	direct (N × H)
171. C₃	C₃×C₈×D₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₈	2
C₃²×C₈×D₄	2
C₃²×#335	2
C₃²×D₄⋊C₈	2
C₃²×C₈²	2
C₃×C₈×D₈	3

Subgroups — C32×C8×D8 #11768624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₈×D₈ #11768624