Subgroups — C₂×#10589361 #11796875

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3745	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63639	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589361	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4495	—	3	4	no	maximal
6	C₂²×C₄⋊(C₃⋊D₃)	—	2	1	no	—
7	C₂×C₃²×M₁₆	—	2	1	no	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3745	—	2	1	no	—
10	C₂×#3745	—	2	1	no	—
11	C₂×#3745	—	2	1	no	—
12	C₂×#3745	C₄	1	2	yes	—
13	C₂×#3745	C₄	1	2	yes	—
14	C₂×#3745	C₂²	1	2	yes	—
15	C₃²×#108	—	2	1	no	—
16	C₃²×#108	C₂²	1	2	yes	complemented
17	#63639	—	2	1	no	—
18	#63639	C₂²	1	2	yes	complemented
19	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
20	C₂²×D₂₄	—	3	4	no	—
21	C₂×C₃×#108	D₃	1	4	yes	complemented
22	C₂×#812	—	3	4	no	—
23	#4495	—	3	16	no	—
24	C₂³×C₃⋊D₃	—	4	1	no	—
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
27	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
28	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
29	C₂×C₄⋊(C₃⋊D₃)	—	4	1	no	—
30	C₂×C₄⋊(C₃⋊D₃)	—	4	1	no	—
31	C₂×C₄⋊(C₃⋊D₃)	—	4	1	no	—
32	C₃²×M₁₆	—	2	2	no	—
33	C₃²×M₁₆	—	4	1	no	—
34	C₂×C₃²×C₈	—	4	1	no	—
35	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	#3745	—	2	2	no	—
42	#3745	—	2	2	no	—
43	#3745	—	2	2	no	—
44	#3745	C₂×C₄	1	4	yes	—
45	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
46	C₂×#361	—	9	1	no	Sylow
47	C₂²×D₁₂	—	6	4	no	—
48	C₂×C₃×M₁₆	—	2	4	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
50	C₂×D₂₄	—	3	8	no	—
51	C₂×D₂₄	—	3	8	no	—
52	C₂×D₂₄	—	3	8	no	—
53	C₂×D₂₄	—	6	4	no	—
54	C₂×D₂₄	—	6	4	no	—
55	C₂×D₂₄	—	6	4	no	—
56	C₃×#108	—	2	4	no	—
57	C₃×#108	C₂×D₃	1	8	yes	complemented
58	#812	—	3	8	no	—
59	#812	—	6	4	no	—
60	C₂×C₃⋊C₁₆	—	6	4	no	—
61	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
62	C₂²×C₃⋊D₃	—	4	2	no	—
63	C₂²×C₃⋊D₃	—	4	2	no	—
64	C₂²×C₃⋊D₃	—	4	2	no	—
65	C₂²×C₃⋊D₃	—	8	4	no	—
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₄⋊(C₃⋊D₃)	—	2	4	no	—
73	C₄⋊(C₃⋊D₃)	—	4	2	no	—
74	C₄⋊(C₃⋊D₃)	—	4	2	no	—
75	C₄⋊(C₃⋊D₃)	—	4	2	no	—
76	C₃²×C₈	—	4	2	no	—
77	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₈	—	9	1	no	—
82	C₂×M₃₂	—	9	1	no	—
83	C₂×#108	C₃⋊D₃	1	1	yes	characteristic, complemented
84	#361	—	9	4	no	—
85	C₂³×D₃	—	12	4	no	—
86	C₂²×C₃×C₄	#67	1	4	yes	—
87	C₂×D₁₂	—	6	8	no	—
88	C₂×D₁₂	—	6	8	no	—
89	C₂×D₁₂	—	6	8	no	—
90	C₂×D₁₂	—	12	4	no	—
91	C₂×D₁₂	—	12	4	no	—
92	C₂×D₁₂	—	12	4	no	—
93	C₃×M₁₆	—	2	8	no	—
94	C₃×M₁₆	—	4	4	no	—
95	C₂×C₃×C₈	—	4	4	no	—
96	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
97	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
98	C₂×C₃×C₈	D₁₂	1	4	yes	—
99	C₂×C₃×C₈	D₁₂	1	4	yes	—
100	C₂×C₃×C₈	#67	1	4	yes	—
101	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
102	D₂₄	—	3	16	no	—
103	D₂₄	—	6	8	no	—
104	D₂₄	—	6	8	no	—
105	D₂₄	—	6	8	no	—
106	C₃⋊C₁₆	—	6	8	no	—
107	C₂²×C₃²	—	2	2	no	—
108	C₂²×C₃²	#102	1	1	yes	characteristic
109	C₂²×C₃²	#102	1	1	yes	characteristic
110	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₃⋊D₃	—	4	4	no	—
112	C₂×C₃⋊D₃	—	8	4	no	—
113	C₂×C₃⋊D₃	—	8	4	no	—
114	C₂×C₃⋊D₃	—	8	4	no	—
115	C₃²×C₄	#102	1	1	yes	characteristic
116	C₃²×C₄	#102	1	1	yes	characteristic
117	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂²×D₄	—	18	1	no	—
120	C₂×D₈	—	9	2	no	—
121	C₂×D₈	—	9	2	no	—
122	C₂×D₈	—	9	2	no	—
123	C₂×D₈	—	18	1	no	—
124	C₂×D₈	—	18	1	no	—
125	C₂×D₈	—	18	1	no	—
126	C₂×M₁₆	—	2	1	no	—
127	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
128	M₃₂	—	9	2	no	—
129	M₃₂	—	18	1	no	—
130	C₂×C₁₆	—	18	1	no	—
131	#108	—	2	1	no	—
132	#108	C₂×C₃⋊D₃	1	2	yes	complemented
133	C₂³×C₃	#215	1	4	yes	—
134	C₂²×D₃	—	12	8	no	—
135	C₂²×D₃	—	12	8	no	—
136	C₂²×D₃	—	12	8	no	—
137	C₂²×D₃	—	24	16	no	—
138	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
139	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
140	C₂×C₃×C₄	#213	1	4	yes	—
141	C₂×C₃×C₄	#213	1	4	yes	—
142	C₂×C₃×C₄	#215	1	4	yes	—
143	C₂×C₃×C₄	C₂×#67	1	4	yes	—
144	D₁₂	—	6	16	no	—
145	D₁₂	—	12	8	no	—
146	D₁₂	—	12	8	no	—
147	D₁₂	—	12	8	no	—
148	C₃×C₈	—	4	8	no	—
149	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
150	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
151	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
152	C₃×C₈	C₂×D₁₂	1	4	yes	—
153	C₂×C₃²	—	2	1	no	—
154	C₂×C₃²	—	2	1	no	—
155	C₂×C₃²	#361	1	2	yes	complemented
156	C₂×C₃²	C₂×#102	1	1	yes	characteristic
157	C₃⋊D₃	—	8	4	no	—
158	C₂⁴	—	36	1	no	—
159	C₂×D₄	—	18	2	no	—
160	C₂×D₄	—	18	2	no	—
161	C₂×D₄	—	18	2	no	—
162	C₂×D₄	—	36	1	no	—
163	C₂×D₄	—	36	1	no	—
164	C₂×D₄	—	36	1	no	—
165	C₂²×C₄	#638	1	1	yes	characteristic
166	D₈	—	9	4	no	—
167	D₈	—	18	2	no	—
168	D₈	—	18	2	no	—
169	D₈	—	18	2	no	—
170	M₁₆	—	2	2	no	—
171	M₁₆	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
174	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
176	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
177	C₂×C₈	#638	1	1	yes	characteristic
178	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
179	C₁₆	—	18	2	no	—
180	C₂²×C₃	—	2	8	no	—
181	C₂²×C₃	#809	1	4	yes	—
182	C₂²×C₃	#809	1	4	yes	—
183	C₂²×C₃	C₂×#215	1	4	yes	—
184	C₂×D₃	—	12	16	no	—
185	C₂×D₃	—	24	16	no	—
186	C₂×D₃	—	24	16	no	—
187	C₂×D₃	—	24	16	no	—
188	C₃×C₄	#809	1	4	yes	—
189	C₃×C₄	#809	1	4	yes	—
190	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
191	C₃×C₄	C₂×#213	1	4	yes	—
192	C₃²	C₂×#361	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	36	2	no	—
194	C₂³	—	36	2	no	—
195	C₂³	—	36	2	no	—
196	C₂³	—	72	4	no	—
197	C₂³	#3755	1	1	yes	characteristic
198	D₄	—	18	4	no	—
199	D₄	—	36	2	no	—
200	D₄	—	36	2	no	—
201	D₄	—	36	2	no	—
202	C₂×C₄	#3752	1	1	yes	characteristic
203	C₂×C₄	#3752	1	1	yes	characteristic
204	C₂×C₄	#3753	1	1	yes	characteristic
205	C₂×C₄	#3753	1	1	yes	characteristic
206	C₂×C₄	#3755	1	1	yes	characteristic
207	C₂×C₄	C₂×#638	1	1	yes	characteristic
208	C₈	—	4	2	no	—
209	C₈	#3752	1	1	yes	characteristic
210	C₈	#3752	1	1	yes	characteristic
211	C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
212	C₈	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
213	C₂×C₃	—	2	4	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	#4495	1	8	yes	complemented
216	C₂×C₃	C₂×#809	1	4	yes	—
217	D₃	—	24	16	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	36	4	no	—
220	C₂²	—	72	4	no	—
221	C₂²	—	72	4	no	—
222	C₂²	—	72	4	no	—
223	C₂²	#63636	1	1	yes	characteristic
224	C₂²	#63636	1	1	yes	characteristic
225	C₂²	C₂×#3755	1	1	yes	characteristic
226	C₄	#63636	1	1	yes	characteristic
227	C₄	#63636	1	1	yes	characteristic
228	C₄	C₂×#3752	1	1	yes	characteristic
229	C₄	C₂×#3753	1	1	yes	characteristic
230	C₃	C₂×#4495	1	4	yes	complemented
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	—	72	4	no	—
234	C₂	#10589361	1	2	yes	direct product factor
235	C₂	C₂×#63636	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63636
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	#102
prime radical	C₂²×#3745	C₂
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×#108	C₃⋊D₃
hypocenter	C₃²	C₂×#361
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#108	C₂	semidirect
3. C₂×#63639	C₂	semidirect
4. #10589361	C₂	direct (N × H)
16. C₃²×#108	C₂²	semidirect
18. #63639	C₂²	semidirect
21. C₂×C₃×#108	D₃	semidirect
57. C₃×#108	C₂×D₃	semidirect
83. C₂×#108	C₃⋊D₃	semidirect
132. #108	C₂×C₃⋊D₃	semidirect
155. C₂×C₃²	#361	semidirect
192. C₃²	C₂×#361	semidirect
215. C₂×C₃	#4495	semidirect
230. C₃	C₂×#4495	semidirect
234. C₂	#10589361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#3745	2
C₂×C₃²×#108	2
C₂×#63639	2
#10589361	2
C₂×#4495	3

Subgroups — C2×#10589361 #11796875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589361 #11796875