Subgroups — C₂×#10588174 #11796903

Order 1152.

324 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×#63371	C₂	1	1	yes	maximal, characteristic
4	#10588174	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4496	—	3	1	no	maximal
6	C₂²×Dic₁₈	—	2	1	no	—
7	C₂×C₉×M₁₆	—	2	1	no	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₆	—	2	1	no	—
10	C₂×Dic₃₆	—	2	1	no	—
11	C₂×Dic₃₆	—	2	1	no	—
12	C₂×Dic₃₆	C₄	1	2	yes	—
13	C₂×Dic₃₆	C₄	1	2	yes	—
14	C₂×Dic₃₆	C₂²	1	2	yes	—
15	C₉×#108	—	2	1	no	—
16	C₉×#108	C₂²	1	2	yes	—
17	#63371	—	2	1	no	—
18	#63371	C₂²	1	2	yes	—
19	C₂×C₉⋊C₁₆	—	2	1	no	—
20	C₂²×Dic₁₂	—	3	1	no	—
21	C₂×C₃×#108	D₃	1	1	yes	characteristic
22	C₂×#812	—	3	1	no	—
23	#4496	—	3	4	no	—
24	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂²×Dic₉	—	4	1	no	—
26	C₂×Dic₁₈	—	2	2	no	—
27	C₂×Dic₁₈	—	2	2	no	—
28	C₂×Dic₁₈	—	2	2	no	—
29	C₂×Dic₁₈	—	4	1	no	—
30	C₂×Dic₁₈	—	4	1	no	—
31	C₂×Dic₁₈	—	4	1	no	—
32	C₉×M₁₆	—	2	2	no	—
33	C₉×M₁₆	—	4	1	no	—
34	C₂×C₈×C₉	—	4	1	no	—
35	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₉	D₄	1	1	yes	characteristic
38	C₂×C₈×C₉	D₄	1	1	yes	characteristic
39	C₂×C₈×C₉	D₄	1	1	yes	characteristic
40	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
41	Dic₃₆	—	2	2	no	—
42	Dic₃₆	—	2	2	no	—
43	Dic₃₆	—	2	2	no	—
44	Dic₃₆	C₂×C₄	1	4	yes	—
45	C₉⋊C₁₆	—	2	2	no	—
46	C₂×#362	—	9	1	no	Sylow
47	C₂²×Dic₆	—	6	1	no	—
48	C₂×C₃×M₁₆	—	2	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×Dic₁₂	—	3	2	no	—
51	C₂×Dic₁₂	—	3	2	no	—
52	C₂×Dic₁₂	—	3	2	no	—
53	C₂×Dic₁₂	—	6	1	no	—
54	C₂×Dic₁₂	—	6	1	no	—
55	C₂×Dic₁₂	—	6	1	no	—
56	C₃×#108	—	2	1	no	—
57	C₃×#108	C₂×D₃	1	2	yes	—
58	#812	—	3	2	no	—
59	#812	—	6	1	no	—
60	C₂×C₃⋊C₁₆	—	6	1	no	—
61	C₂³×C₉	SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₉	D₈	1	1	yes	characteristic
65	C₂×C₄×C₉	D₈	1	1	yes	characteristic
66	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₉	—	4	2	no	—
69	C₂×Dic₉	—	4	2	no	—
70	C₂×Dic₉	—	4	2	no	—
71	Dic₁₈	—	2	4	no	—
72	Dic₁₈	—	4	2	no	—
73	Dic₁₈	—	4	2	no	—
74	Dic₁₈	—	4	2	no	—
75	C₈×C₉	—	4	2	no	—
76	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×Q₁₆	—	9	1	no	—
81	C₂×M₃₂	—	9	1	no	—
82	C₂×#108	D₉	1	1	yes	characteristic
83	#362	—	9	4	no	—
84	C₂²×C₃×C₄	#67	1	1	yes	characteristic
85	C₂²×Dic₃	—	12	1	no	—
86	C₂×Dic₆	—	6	2	no	—
87	C₂×Dic₆	—	6	2	no	—
88	C₂×Dic₆	—	6	2	no	—
89	C₂×Dic₆	—	12	1	no	—
90	C₂×Dic₆	—	12	1	no	—
91	C₂×Dic₆	—	12	1	no	—
92	C₃×M₁₆	—	2	2	no	—
93	C₃×M₁₆	—	4	1	no	—
94	C₂×C₃×C₈	—	4	1	no	—
95	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
98	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
99	C₂×C₃×C₈	#67	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
101	Dic₁₂	—	3	4	no	—
102	Dic₁₂	—	6	2	no	—
103	Dic₁₂	—	6	2	no	—
104	Dic₁₂	—	6	2	no	—
105	C₃⋊C₁₆	—	6	2	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	#102	1	1	yes	characteristic
108	C₂²×C₉	#102	1	1	yes	characteristic
109	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₄×C₉	#102	1	1	yes	characteristic
111	C₄×C₉	#102	1	1	yes	characteristic
112	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₄×C₉	C₂×D₈	1	1	yes	characteristic
114	Dic₉	—	4	4	no	—
115	C₂²×Q₈	—	18	1	no	—
116	C₂×Q₁₆	—	9	2	no	—
117	C₂×Q₁₆	—	9	2	no	—
118	C₂×Q₁₆	—	9	2	no	—
119	C₂×Q₁₆	—	18	1	no	—
120	C₂×Q₁₆	—	18	1	no	—
121	C₂×Q₁₆	—	18	1	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
124	M₃₂	—	9	2	no	—
125	M₃₂	—	18	1	no	—
126	C₂×C₁₆	—	18	1	no	—
127	#108	—	2	1	no	—
128	#108	C₂×D₉	1	2	yes	—
129	C₂³×C₃	#215	1	1	yes	characteristic
130	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	#213	1	1	yes	characteristic
133	C₂×C₃×C₄	#213	1	1	yes	characteristic
134	C₂×C₃×C₄	#215	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
136	C₂×Dic₃	—	12	2	no	—
137	C₂×Dic₃	—	12	2	no	—
138	C₂×Dic₃	—	12	2	no	—
139	Dic₆	—	6	4	no	—
140	Dic₆	—	12	2	no	—
141	Dic₆	—	12	2	no	—
142	Dic₆	—	12	2	no	—
143	C₃×C₈	—	4	2	no	—
144	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
145	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
146	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
147	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₉	—	2	1	no	—
149	C₂×C₉	—	2	1	no	—
150	C₂×C₉	#362	1	2	yes	complemented
151	C₂×C₉	C₂×#102	1	1	yes	characteristic
152	C₂×Q₈	—	18	2	no	—
153	C₂×Q₈	—	18	2	no	—
154	C₂×Q₈	—	18	2	no	—
155	C₂×Q₈	—	36	1	no	—
156	C₂×Q₈	—	36	1	no	—
157	C₂×Q₈	—	36	1	no	—
158	C₂²×C₄	—	36	1	no	—
159	C₂²×C₄	#611	1	1	yes	characteristic
160	Q₁₆	—	9	4	no	—
161	Q₁₆	—	18	2	no	—
162	Q₁₆	—	18	2	no	—
163	Q₁₆	—	18	2	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	C₄×D₉	1	1	yes	characteristic
168	C₂×C₈	C₄×D₉	1	1	yes	characteristic
169	C₂×C₈	D₃₆	1	1	yes	characteristic
170	C₂×C₈	D₃₆	1	1	yes	characteristic
171	C₂×C₈	#611	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
173	C₁₆	—	18	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	#809	1	1	yes	characteristic
176	C₂²×C₃	#809	1	1	yes	characteristic
177	C₂²×C₃	C₂×#215	1	1	yes	characteristic
178	C₃×C₄	#809	1	1	yes	characteristic
179	C₃×C₄	#809	1	1	yes	characteristic
180	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₃×C₄	C₂×#213	1	1	yes	characteristic
182	Dic₃	—	12	4	no	—
183	C₉	C₂×#362	1	1	yes	characteristic, complemented, Sylow
184	C₂³	#3675	1	1	yes	characteristic
185	Q₈	—	18	4	no	—
186	Q₈	—	36	2	no	—
187	Q₈	—	36	2	no	—
188	Q₈	—	36	2	no	—
189	C₂×C₄	—	36	2	no	—
190	C₂×C₄	—	36	2	no	—
191	C₂×C₄	—	36	2	no	—
192	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
193	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
194	C₂×C₄	#3673	1	1	yes	characteristic
195	C₂×C₄	#3673	1	1	yes	characteristic
196	C₂×C₄	#3675	1	1	yes	characteristic
197	C₂×C₄	C₂×#611	1	1	yes	characteristic
198	C₈	—	4	2	no	—
199	C₈	(C₂×D₉)⋊C₄	1	1	yes	characteristic
200	C₈	(C₂×D₉)⋊C₄	1	1	yes	characteristic
201	C₈	C₂×C₄×D₉	1	1	yes	characteristic
202	C₈	C₂×D₃₆	1	1	yes	characteristic
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	#4496	1	2	yes	—
206	C₂×C₃	C₂×#809	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	#63369	1	1	yes	characteristic
209	C₂²	#63369	1	1	yes	characteristic
210	C₂²	C₂×#3675	1	1	yes	characteristic
211	C₄	—	36	4	no	—
212	C₄	#63369	1	1	yes	characteristic
213	C₄	#63369	1	1	yes	characteristic
214	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
215	C₄	C₂×#3673	1	1	yes	characteristic
216	C₃	C₂×#4496	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	#10588174	1	2	yes	direct product factor
220	C₂	C₂×#63369	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63369
commutator	C₈×C₉	C₂²×C₄
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₂×C₉×#108	C₂
socle	C₂²×C₃	#809
prime radical	C₂³×C₃	#215
cosocle	C₂×C₈×C₉	C₂³
hypercenter	C₂×#108	D₉
hypocenter	C₉	C₂×#362
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
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
4. #10588174	C₂	direct (N × H)
150. C₂×C₉	#362	semidirect
183. C₉	C₂×#362	semidirect
219. C₂	#10588174	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×Dic₃₆	2
C₂×C₉×#108	2
C₂×#63371	2
#10588174	2
C₂×#4496	3

Subgroups — C2×#10588174 #11796903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588174 #11796903