Subgroups — C₂×#11771322 #72245957

Order 2304.

906 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10589327	C₂	1	1	yes	maximal, characteristic
2	C₂×#10589327	C₂	1	1	yes	maximal, characteristic
3	#11771322	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×#67442	—	3	4	no	maximal
6	C₂×C₃²×C₄×C₈	C₄	1	2	yes	—
7	C₂×C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	—
10	C₃²×#322	C₄	1	2	yes	—
11	C₃²×#322	C₂²	1	2	yes	—
12	#10589327	C₄	1	2	yes	—
13	#10589327	C₄	1	2	yes	—
14	#10589327	C₂²	1	2	yes	—
15	#10589327	C₂²	1	2	yes	—
16	C₂×C₃×#322	D₃	1	4	yes	—
17	C₂×#4461	—	3	4	no	—
18	C₂×#4461	—	3	4	no	—
19	#67442	—	3	16	no	—
20	C₂²×C₃²×C₈	—	2	2	no	—
21	C₂²×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃²×C₄×C₈	—	2	2	no	—
25	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₄×C₈	C₂×C₄	1	4	yes	—
28	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
31	C₂×C₃⋊(C₃⋊C₁₆)	C₈	1	8	yes	complemented
32	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	4	yes	—
33	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	4	yes	—
34	C₂×#1313	—	9	1	no	Sylow
35	C₂×C₃×C₄×C₈	Dic₃	1	8	yes	—
36	C₂×C₃×C₄×C₈	C₂×D₃	1	4	yes	—
37	C₂²×C₃⋊C₁₆	—	3	8	no	—
38	C₂²×C₃⋊C₁₆	—	3	8	no	—
39	C₃×#322	Dic₃	1	8	yes	—
40	C₃×#322	C₂×D₃	1	8	yes	—
41	#4461	—	3	8	no	—
42	#4461	—	3	8	no	—
43	#4461	—	3	8	no	—
44	#4461	—	3	8	no	—
45	C₂²×C₃²×C₄	C₄²	1	1	yes	characteristic
46	C₂²×C₃²×C₄	M₁₆	1	2	yes	—
47	C₂×C₃²×C₈	—	2	4	no	—
48	C₂×C₃²×C₈	—	4	4	no	—
49	C₂×C₃²×C₈	C₄²	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₄²	1	1	yes	characteristic
51	C₂×C₃²×C₈	C₂×C₈	1	4	yes	—
52	C₂×C₃²×C₈	M₁₆	1	4	yes	—
53	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₃²×C₄²	C₄²	1	1	yes	characteristic
56	C₃²×C₄²	M₁₆	1	2	yes	—
57	C₃²×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
59	C₃⋊(C₃⋊C₁₆)	C₂×C₈	1	8	yes	complemented
60	C₂×#346	—	9	1	no	—
61	C₂×#346	—	9	1	no	—
62	C₂×#322	C₃⋊D₃	1	1	yes	characteristic
63	#1313	—	9	4	no	—
64	C₂²×C₃×C₈	—	2	8	no	—
65	C₂²×C₃×C₈	C₄×D₃	1	4	yes	—
66	C₂²×C₃×C₈	C₄×D₃	1	4	yes	—
67	C₂×C₃×C₄²	C₂×Dic₃	1	4	yes	—
68	C₃×C₄×C₈	—	2	8	no	—
69	C₃×C₄×C₈	C₄×D₃	1	4	yes	—
70	C₃×C₄×C₈	C₄×D₃	1	4	yes	—
71	C₃×C₄×C₈	C₂×Dic₃	1	16	yes	—
72	C₃×C₄×C₈	C₂×Dic₃	1	4	yes	—
73	C₃×C₄×C₈	C₂²×D₃	1	4	yes	—
74	C₂×C₃⋊C₁₆	—	3	32	no	—
75	C₂×C₃⋊C₁₆	—	3	16	no	—
76	C₂×C₃⋊C₁₆	—	3	16	no	—
77	C₂×C₃⋊C₁₆	—	6	16	no	—
78	C₂³×C₃²	#97	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	C₄×C₈	1	2	yes	—
81	C₂×C₃²×C₄	#97	1	2	yes	—
82	C₂×C₃²×C₄	#97	1	1	yes	characteristic
83	C₂×C₃²×C₄	#97	1	2	yes	—
84	C₂×C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×M₁₆	1	2	yes	—
86	C₃²×C₈	—	4	4	no	—
87	C₃²×C₈	C₄×C₈	1	2	yes	—
88	C₃²×C₈	#97	1	2	yes	—
89	C₃²×C₈	C₂²×C₈	1	2	yes	—
90	C₃²×C₈	C₂×M₁₆	1	2	yes	—
91	C₂²×C₁₆	—	9	2	no	—
92	C₂²×C₁₆	—	9	2	no	—
93	C₂×C₄×C₈	C₃⋊Dic₃	1	2	yes	—
94	C₂×C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
95	#346	—	9	2	no	—
96	#346	—	9	2	no	—
97	#346	—	9	2	no	—
98	#346	—	9	2	no	—
99	#322	C₃⋊Dic₃	1	2	yes	—
100	#322	C₂×C₃⋊D₃	1	2	yes	—
101	C₂²×C₃×C₄	#208	1	8	yes	—
102	C₂²×C₃×C₄	C₄×Dic₃	1	4	yes	—
103	C₂×C₃×C₈	—	2	16	no	—
104	C₂×C₃×C₈	—	4	16	no	—
105	C₂×C₃×C₈	D₃×C₈	1	16	yes	—
106	C₂×C₃×C₈	ΓL(1,𝔽₂₅)	1	16	yes	—
107	C₂×C₃×C₈	C₄×Dic₃	1	4	yes	—
108	C₂×C₃×C₈	C₄×Dic₃	1	4	yes	—
109	C₂×C₃×C₈	C₂×C₄×D₃	1	4	yes	—
110	C₂×C₃×C₈	C₂×C₄×D₃	1	4	yes	—
111	C₃×C₄²	#208	1	8	yes	—
112	C₃×C₄²	C₄×Dic₃	1	4	yes	—
113	C₃×C₄²	C₂²×Dic₃	1	4	yes	—
114	C₃⋊C₁₆	—	3	32	no	—
115	C₃⋊C₁₆	—	6	16	no	—
116	C₂²×C₃²	#322	1	2	yes	—
117	C₂²×C₃²	#347	1	4	yes	—
118	C₂²×C₃²	C₂×#97	1	1	yes	characteristic
119	C₃²×C₄	—	2	2	no	—
120	C₃²×C₄	—	2	2	no	—
121	C₃²×C₄	#322	1	2	yes	—
122	C₃²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
123	C₃²×C₄	C₂×#97	1	1	yes	characteristic
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
126	C₂²×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
127	C₂×C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
128	C₂×C₁₆	—	9	8	no	—
129	C₂×C₁₆	—	9	4	no	—
130	C₂×C₁₆	—	9	4	no	—
131	C₂×C₁₆	—	18	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
134	C₄×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
135	C₄×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
136	C₄×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
137	C₄×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
138	C₂³×C₃	#803	1	4	yes	—
139	C₂×C₃×C₄	—	2	16	no	—
140	C₂×C₃×C₄	#803	1	8	yes	—
141	C₂×C₃×C₄	#803	1	4	yes	—
142	C₂×C₃×C₄	C₈×Dic₃	1	8	yes	—
143	C₂×C₃×C₄	#815	1	8	yes	—
144	C₂×C₃×C₄	C₂×#208	1	8	yes	—
145	C₂×C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
146	C₃×C₈	—	4	16	no	—
147	C₃×C₈	C₈×Dic₃	1	8	yes	—
148	C₃×C₈	#815	1	8	yes	—
149	C₃×C₈	C₂×D₃×C₈	1	8	yes	—
150	C₃×C₈	C₂×ΓL(1,𝔽₂₅)	1	8	yes	—
151	C₂×C₃²	#1313	1	4	yes	complemented
152	C₂×C₃²	C₂×#322	1	1	yes	characteristic
153	C₂×C₃²	C₂×#347	1	2	yes	—
154	C₂²×C₄	#3748	1	2	yes	—
155	C₂²×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	C₈×C₃⋊D₃	1	4	yes	—
159	C₂×C₈	#3743	1	4	yes	—
160	C₂×C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
161	C₂×C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
164	C₄²	#3748	1	2	yes	—
165	C₄²	C₄×C₃⋊Dic₃	1	1	yes	characteristic
166	C₄²	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
167	C₁₆	—	9	8	no	—
168	C₁₆	—	18	4	no	—
169	C₂²×C₃	#4454	1	8	yes	—
170	C₂²×C₃	#4466	1	16	yes	—
171	C₂²×C₃	C₂×#803	1	4	yes	—
172	C₃×C₄	—	2	8	no	—
173	C₃×C₄	—	2	8	no	—
174	C₃×C₄	#4454	1	8	yes	—
175	C₃×C₄	C₂×C₈×Dic₃	1	4	yes	—
176	C₃×C₄	C₂×#815	1	4	yes	—
177	C₃²	C₂×#1313	1	1	yes	characteristic, complemented, Sylow
178	C₂³	#63630	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	#63630	1	2	yes	—
181	C₂×C₄	#63630	1	1	yes	characteristic
182	C₂×C₄	C₈×C₃⋊Dic₃	1	2	yes	—
183	C₂×C₄	#63642	1	2	yes	—
184	C₂×C₄	C₂×#3748	1	2	yes	—
185	C₂×C₄	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
186	C₈	—	4	4	no	—
187	C₈	C₈×C₃⋊Dic₃	1	2	yes	—
188	C₈	#63642	1	2	yes	—
189	C₈	C₂×C₈×C₃⋊D₃	1	2	yes	—
190	C₈	C₂×#3743	1	2	yes	—
191	C₂×C₃	#67442	1	16	yes	complemented
192	C₂×C₃	C₂×#4454	1	4	yes	—
193	C₂×C₃	C₂×#4466	1	8	yes	—
194	C₂²	#10589320	1	2	yes	—
195	C₂²	#10589332	1	4	yes	—
196	C₂²	C₂×#63630	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	#10589320	1	2	yes	—
200	C₄	C₂×C₈×C₃⋊Dic₃	1	1	yes	characteristic
201	C₄	C₂×#63642	1	1	yes	characteristic
202	C₃	C₂×#67442	1	4	yes	complemented
203	C₂	#11771322	1	4	yes	direct product factor
204	C₂	C₂×#10589320	1	1	yes	characteristic
205	C₂	C₂×#10589332	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄×C₃⋊Dic₃
commutator	C₃²×C₄	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#322	C₂
socle, prime radical	C₂³×C₃²	#97
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	C₂×#322	C₃⋊D₃
hypocenter	C₃²	C₂×#1313
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #11771322	C₂	direct (N × H)
31. C₂×C₃⋊(C₃⋊C₁₆)	C₈	semidirect
59. C₃⋊(C₃⋊C₁₆)	C₂×C₈	semidirect
151. C₂×C₃²	#1313	semidirect
177. C₃²	C₂×#1313	semidirect
191. C₂×C₃	#67442	semidirect
202. C₃	C₂×#67442	semidirect
203. C₂	#11771322	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×#10589327	2
#11771322	2
C₂×C₃²×#322	2
C₂×#67442	3

Subgroups — C2×#11771322 #72245957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11771322 #72245957