Subgroups — C₂×#11772193 #72255495

Order 2304.

666 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10589329	C₂	1	2	yes	maximal
2	#11772193	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃²×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#67514	—	3	4	no	maximal
5	C₂²×C₃²×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×#63639	—	2	2	no	—
9	C₃²×#346	C₄	1	2	yes	—
10	C₃²×#346	C₂²	1	2	yes	—
11	#10589329	—	2	2	no	—
12	#10589329	C₂²	1	4	yes	—
13	C₂×C₃×#346	D₃	1	4	yes	—
14	C₂×#4463	—	3	8	no	—
15	#67514	—	3	16	no	—
16	C₂²×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃²×C₄²	Q₈	1	1	yes	characteristic
19	C₂×C₃²×C₁₆	—	2	1	no	—
20	C₂×C₃²×C₁₆	—	2	1	no	—
21	C₂×C₃²×C₁₆	C₈	1	2	yes	—
22	C₂×C₃²×C₁₆	C₈	1	2	yes	—
23	C₂×C₃²×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₃²×C₁₆	C₂×C₄	1	2	yes	—
25	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄×C₈	D₄	1	1	yes	characteristic
27	C₃²×C₄×C₈	Q₈	1	1	yes	characteristic
28	C₃²×C₄×C₈	C₂³	1	1	yes	characteristic
29	#63639	—	2	4	no	—
30	#63639	—	4	2	no	—
31	C₂×C₃⋊(C₃⋊C₁₆)	—	4	2	no	—
32	C₂×#1369	—	9	1	no	Sylow
33	C₂²×C₃×C₁₆	Dic₃	1	4	yes	—
34	C₂²×C₃×C₁₆	Dic₃	1	4	yes	—
35	C₂×C₃×C₄×C₈	C₂×D₃	1	4	yes	—
36	C₂×#812	—	6	8	no	—
37	C₃×#346	Dic₃	1	8	yes	—
38	C₃×#346	C₂×D₃	1	8	yes	—
39	#4463	—	3	16	no	—
40	#4463	—	6	8	no	—
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃²×C₈	M₁₆	1	1	yes	characteristic
48	C₂×C₃²×C₈	M₁₆	1	1	yes	characteristic
49	C₂×C₃²×C₈	SD₁₆	1	2	yes	—
50	C₂×C₃²×C₈	SD₁₆	1	2	yes	—
51	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₃²×C₄²	—	2	1	no	—
54	C₃²×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₃²×C₄²	C₂×Q₈	1	1	yes	characteristic
56	C₃²×C₁₆	—	2	2	no	—
57	C₃²×C₁₆	C₂×C₈	1	4	yes	—
58	C₃⋊(C₃⋊C₁₆)	—	4	4	no	—
59	C₂×#364	—	9	2	no	—
60	C₂×#346	C₃⋊D₃	1	1	yes	characteristic
61	#1369	—	9	4	no	—
62	C₂²×C₃×C₈	D₁₂	1	4	yes	—
63	C₂²×C₃×C₈	C₂×Dic₃	1	4	yes	—
64	C₂×C₃×C₄²	Dic₆	1	4	yes	—
65	C₂×C₃×C₁₆	—	2	4	no	—
66	C₂×C₃×C₁₆	—	2	4	no	—
67	C₂×C₃×C₁₆	C₃⋊C₈	1	8	yes	—
68	C₂×C₃×C₁₆	C₃⋊C₈	1	8	yes	—
69	C₂×C₃×C₁₆	C₂×Dic₃	1	8	yes	—
70	C₂×C₃×C₁₆	C₂×Dic₃	1	8	yes	—
71	C₃×C₄×C₈	Dic₆	1	4	yes	—
72	C₃×C₄×C₈	D₁₂	1	4	yes	—
73	C₃×C₄×C₈	C₂×Dic₃	1	4	yes	—
74	C₃×C₄×C₈	C₂²×D₃	1	4	yes	—
75	#812	—	6	16	no	—
76	#812	—	12	8	no	—
77	C₂×C₃⋊C₁₆	—	12	8	no	—
78	C₂³×C₃²	#108	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	2	1	no	—
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	—	4	2	no	—
82	C₂×C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	#106	1	1	yes	characteristic
85	C₂×C₃²×C₄	#106	1	1	yes	characteristic
86	C₂×C₃²×C₄	#108	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₃²×C₈	C₄⋊C₈	1	1	yes	characteristic
89	C₃²×C₈	C₄⋊C₈	1	1	yes	characteristic
90	C₃²×C₈	#106	1	2	yes	—
91	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
92	C₃²×C₈	C₂×M₁₆	1	1	yes	characteristic
93	C₃²×C₈	C₂×SD₁₆	1	2	yes	—
94	C₂×M₃₂	—	18	2	no	—
95	C₂²×C₁₆	C₃⋊Dic₃	1	1	yes	characteristic
96	C₂²×C₁₆	C₃⋊Dic₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
98	#364	—	9	4	no	—
99	#364	—	18	2	no	—
100	#346	C₃⋊Dic₃	1	2	yes	—
101	#346	C₂×C₃⋊D₃	1	2	yes	—
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
104	C₂×C₃×C₈	#204	1	8	yes	—
105	C₂×C₃×C₈	#204	1	8	yes	—
106	C₂×C₃×C₈	C₂×C₃⋊C₈	1	4	yes	—
107	C₂×C₃×C₈	C₂×C₃⋊C₈	1	4	yes	—
108	C₂×C₃×C₈	#208	1	4	yes	—
109	C₂×C₃×C₈	#208	1	4	yes	—
110	C₂×C₃×C₈	C₄⋊Dic₃	1	4	yes	—
111	C₂×C₃×C₈	C₄⋊Dic₃	1	4	yes	—
112	C₂×C₃×C₈	C₂×D₁₂	1	4	yes	—
113	C₂×C₃×C₈	C₂²×Dic₃	1	4	yes	—
114	C₃×C₄²	—	2	4	no	—
115	C₃×C₄²	C₄⋊Dic₃	1	4	yes	—
116	C₃×C₄²	C₂×Dic₆	1	4	yes	—
117	C₃×C₁₆	—	2	8	no	—
118	C₃×C₁₆	C₂×C₃⋊C₈	1	16	yes	—
119	C₃⋊C₁₆	—	12	16	no	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	#334	1	1	yes	characteristic
122	C₂²×C₃²	#334	1	1	yes	characteristic
123	C₂²×C₃²	C₂×#108	1	1	yes	characteristic
124	C₃²×C₄	—	4	1	no	—
125	C₃²×C₄	—	4	1	no	—
126	C₃²×C₄	#334	1	1	yes	characteristic
127	C₃²×C₄	#334	1	1	yes	characteristic
128	C₃²×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₃²×C₄	C₂×#106	1	1	yes	characteristic
130	C₂²×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
132	C₂×C₄²	#634	1	1	yes	characteristic
133	M₃₂	—	18	4	no	—
134	M₃₂	—	36	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	36	2	no	—
138	C₂×C₁₆	C₃⋊(C₃⋊C₈)	1	2	yes	—
139	C₂×C₁₆	C₃⋊(C₃⋊C₈)	1	2	yes	—
140	C₂×C₁₆	C₂×C₃⋊Dic₃	1	2	yes	—
141	C₂×C₁₆	C₂×C₃⋊Dic₃	1	2	yes	—
142	C₄×C₈	#634	1	1	yes	characteristic
143	C₄×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₄×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
145	C₄×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
146	C₂³×C₃	#819	1	4	yes	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	4	8	no	—
150	C₂×C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
151	C₂×C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
152	C₂×C₃×C₄	#817	1	4	yes	—
153	C₂×C₃×C₄	#817	1	4	yes	—
154	C₂×C₃×C₄	#819	1	4	yes	—
155	C₂×C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
156	C₃×C₈	C₄⋊(C₃⋊C₈)	1	4	yes	—
157	C₃×C₈	C₄⋊(C₃⋊C₈)	1	4	yes	—
158	C₃×C₈	#817	1	8	yes	—
159	C₃×C₈	C₂×#204	1	8	yes	—
160	C₃×C₈	C₂²×C₃⋊C₈	1	4	yes	—
161	C₃×C₈	C₂×#208	1	4	yes	—
162	C₂×C₃²	—	2	1	no	—
163	C₂×C₃²	—	2	1	no	—
164	C₂×C₃²	#1369	1	2	yes	complemented
165	C₂×C₃²	C₂×#334	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
168	C₂×C₈	#3744	1	2	yes	—
169	C₂×C₈	#3744	1	2	yes	—
170	C₂×C₈	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
171	C₂×C₈	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
172	C₂×C₈	#3748	1	1	yes	characteristic
173	C₂×C₈	#3748	1	1	yes	characteristic
174	C₂×C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
175	C₂×C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
178	C₄²	—	2	1	no	—
179	C₄²	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
180	C₄²	C₂×#634	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	36	4	no	—
183	C₁₆	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
184	C₂²×C₃	—	2	8	no	—
185	C₂²×C₃	#4457	1	4	yes	—
186	C₂²×C₃	#4457	1	4	yes	—
187	C₂²×C₃	C₂×#819	1	4	yes	—
188	C₃×C₄	—	4	4	no	—
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	#4457	1	4	yes	—
191	C₃×C₄	#4457	1	4	yes	—
192	C₃×C₄	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
193	C₃×C₄	C₂×#817	1	4	yes	—
194	C₃²	C₂×#1369	1	1	yes	characteristic, complemented, Sylow
195	C₂³	#63646	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
200	C₂×C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
201	C₂×C₄	#63644	1	1	yes	characteristic
202	C₂×C₄	#63644	1	1	yes	characteristic
203	C₂×C₄	#63646	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
205	C₈	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
206	C₈	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
207	C₈	#63644	1	2	yes	—
208	C₈	C₂×#3744	1	2	yes	—
209	C₈	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
210	C₈	C₂×#3748	1	1	yes	characteristic
211	C₂×C₃	—	2	4	no	—
212	C₂×C₃	—	2	4	no	—
213	C₂×C₃	#67514	1	8	yes	complemented
214	C₂×C₃	C₂×#4457	1	4	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	#10589323	1	1	yes	characteristic
217	C₂²	#10589323	1	1	yes	characteristic
218	C₂²	C₂×#63646	1	1	yes	characteristic
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	#10589323	1	1	yes	characteristic
222	C₄	#10589323	1	1	yes	characteristic
223	C₄	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
224	C₄	C₂×#63644	1	1	yes	characteristic
225	C₃	C₂×#67514	1	4	yes	complemented
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	#11772193	1	2	yes	direct product factor
229	C₂	C₂×#10589323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#63644
commutator	C₃²×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#346	C₂
socle	C₂²×C₃²	#334
prime radical	C₂³×C₃²	#108
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	C₂×#346	C₃⋊D₃
hypocenter	C₃²	C₂×#1369
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #11772193	C₂	direct (N × H)
164. C₂×C₃²	#1369	semidirect
194. C₃²	C₂×#1369	semidirect
213. C₂×C₃	#67514	semidirect
225. C₃	C₂×#67514	semidirect
228. C₂	#11772193	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×#10589329	2
#11772193	2
C₂×C₃²×#346	2
C₂×#67514	3

Subgroups — C2×#11772193 #72255495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11772193 #72255495