Subgroups — C₂×#10589415 #11795955

Order 1152.

816 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	#10589415	C₂	1	4	yes	maximal, direct product factor
4	C₂×#4549	—	3	4	no	maximal
5	C₂³×C₃²×C₄	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₈	—	2	4	no	—
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₄	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	2	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
13	C₂³×C₃×C₈	D₃	1	4	yes	—
14	C₂²×C₃⋊C₁₆	—	3	8	no	—
15	#4549	—	3	16	no	—
16	C₂⁴×C₃²	C₈	1	1	yes	characteristic
17	C₂²×C₃²×C₄	—	2	2	no	—
18	C₂²×C₃²×C₄	—	2	2	no	—
19	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
20	C₂²×C₃²×C₄	C₈	1	2	yes	—
21	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	—	2	4	no	—
25	C₂×C₃²×C₈	—	2	4	no	—
26	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	4	yes	—
29	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
30	C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
31	C₂×C₂²⋊C₁₆	—	9	1	no	Sylow
32	C₂³×C₃×C₄	Dic₃	1	4	yes	—
33	C₂²×C₃×C₈	—	2	16	no	—
34	C₂²×C₃×C₈	Dic₃	1	4	yes	—
35	C₂²×C₃×C₈	Dic₃	1	8	yes	—
36	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
37	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
38	C₂×C₃⋊C₁₆	—	3	16	no	—
39	C₂×C₃⋊C₁₆	—	6	16	no	—
40	C₂³×C₃²	—	2	2	no	—
41	C₂³×C₃²	—	2	2	no	—
42	C₂³×C₃²	C₁₆	1	4	yes	complemented
43	C₂³×C₃²	C₂×C₈	1	2	yes	—
44	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	—	2	2	no	—
46	C₂×C₃²×C₄	—	2	4	no	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	—	2	2	no	—
49	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
50	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
51	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃²×C₄	M₁₆	1	2	yes	—
53	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₃²×C₈	—	2	2	no	—
55	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
56	C₃²×C₈	C₂×D₄	1	2	yes	—
57	C₂³×C₈	C₃⋊D₃	1	1	yes	characteristic
58	C₂²×C₁₆	—	9	2	no	—
59	C₂²⋊C₁₆	—	9	4	no	—
60	C₂⁴×C₃	C₃⋊C₈	1	4	yes	—
61	C₂²×C₃×C₄	—	2	8	no	—
62	C₂²×C₃×C₄	—	2	8	no	—
63	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
64	C₂²×C₃×C₄	C₃⋊C₈	1	8	yes	—
65	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
67	C₂×C₃×C₈	—	2	8	no	—
68	C₂×C₃×C₈	—	2	16	no	—
69	C₂×C₃×C₈	—	2	16	no	—
70	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
71	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
72	C₂×C₃×C₈	#67	1	16	yes	—
73	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
74	C₃⋊C₁₆	—	6	16	no	—
75	C₂²×C₃²	—	2	8	no	—
76	C₂²×C₃²	—	2	2	no	—
77	C₂²×C₃²	—	2	2	no	—
78	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
79	C₂²×C₃²	C₂×C₁₆	1	2	yes	—
80	C₂²×C₃²	C₂×C₁₆	1	4	yes	complemented
81	C₂²×C₃²	M₃₂	1	2	yes	—
82	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
83	C₃²×C₄	—	2	2	no	—
84	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
85	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
87	C₂³×C₄	C₃⋊Dic₃	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	C₃⋊Dic₃	1	1	yes	characteristic
90	C₂²×C₈	C₃⋊Dic₃	1	2	yes	—
91	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
92	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
93	C₂×C₁₆	—	9	4	no	—
94	C₂×C₁₆	—	18	4	no	—
95	C₂³×C₃	—	2	8	no	—
96	C₂³×C₃	—	2	8	no	—
97	C₂³×C₃	C₃⋊C₁₆	1	16	yes	complemented
98	C₂³×C₃	C₂×C₃⋊C₈	1	8	yes	—
99	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
100	C₂×C₃×C₄	—	2	8	no	—
101	C₂×C₃×C₄	—	2	16	no	—
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	2	8	no	—
104	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
105	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
106	C₂×C₃×C₄	#208	1	8	yes	—
107	C₂×C₃×C₄	#217	1	8	yes	—
108	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
109	C₃×C₈	—	2	8	no	—
110	C₃×C₈	#217	1	8	yes	—
111	C₃×C₈	C₂×#67	1	8	yes	—
112	C₂×C₃²	—	2	4	no	—
113	C₂×C₃²	C₂²⋊C₁₆	1	4	yes	complemented
114	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
116	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
117	C₂⁴	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
121	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
122	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
123	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
128	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
129	C₂×C₈	#638	1	4	yes	—
130	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
131	C₁₆	—	18	4	no	—
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	—	2	32	no	—
134	C₂²×C₃	—	2	8	no	—
135	C₂²×C₃	C₂×C₃⋊C₁₆	1	8	yes	—
136	C₂²×C₃	C₂×C₃⋊C₁₆	1	16	yes	complemented
137	C₂²×C₃	#812	1	8	yes	—
138	C₂²×C₃	#830	1	8	yes	—
139	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
140	C₃×C₄	—	2	8	no	—
141	C₃×C₄	#830	1	8	yes	—
142	C₃×C₄	C₂×#208	1	4	yes	—
143	C₃×C₄	C₂×#217	1	4	yes	—
144	C₃²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	C₃⋊(C₃⋊C₁₆)	1	4	yes	complemented
148	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
149	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
155	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
156	C₂×C₄	#3748	1	2	yes	—
157	C₂×C₄	#3757	1	2	yes	—
158	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	#3757	1	2	yes	—
161	C₈	C₂×#638	1	2	yes	—
162	C₂×C₃	—	2	16	no	—
163	C₂×C₃	#4549	1	16	yes	complemented
164	C₂×C₃	C₂²×C₃⋊C₁₆	1	4	yes	—
165	C₂×C₃	C₂×#812	1	4	yes	—
166	C₂×C₃	C₂×#830	1	4	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	2	yes	—
171	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	4	yes	complemented
172	C₂²	#63639	1	2	yes	—
173	C₂²	#63657	1	2	yes	—
174	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	#63657	1	2	yes	—
177	C₄	C₂×#3748	1	1	yes	characteristic
178	C₄	C₂×#3757	1	1	yes	characteristic
179	C₃	C₂×#4549	1	4	yes	complemented
180	C₂	—	2	4	no	—
181	C₂	#10589415	1	4	yes	direct product factor
182	C₂	C₂²×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
183	C₂	C₂×#63639	1	1	yes	characteristic
184	C₂	C₂×#63657	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×C₈
prime radical	C₂⁴×C₃²	C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂³×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
3. #10589415	C₂	direct (N × H)
12. C₂×C₃⋊(C₃⋊C₁₆)	C₂²	semidirect
42. C₂³×C₃²	C₁₆	semidirect
80. C₂²×C₃²	C₂×C₁₆	semidirect
97. C₂³×C₃	C₃⋊C₁₆	semidirect
113. C₂×C₃²	C₂²⋊C₁₆	semidirect
136. C₂²×C₃	C₂×C₃⋊C₁₆	semidirect
144. C₃²	C₂×C₂²⋊C₁₆	semidirect
147. C₂³	C₃⋊(C₃⋊C₁₆)	semidirect
163. C₂×C₃	#4549	semidirect
171. C₂²	C₂×C₃⋊(C₃⋊C₁₆)	semidirect
179. C₃	C₂×#4549	semidirect
181. C₂	#10589415	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
#10589415	2
C₂×#4549	3

Subgroups — C2×#10589415 #11795955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589415 #11795955