Subgroups — C₂×#10588773 #11873359

Order 1152.

792 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#63436	C₂	1	1	yes	maximal, characteristic
3	#10588773	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63369	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#5088	—	3	1	no	maximal
7	C₂²×Q₈×C₉	—	2	1	no	—
8	C₂²×D₃₆	—	2	1	no	—
9	C₂²×D₃₆	—	2	1	no	—
10	C₂²×D₃₆	C₂²	1	2	yes	—
11	C₉×C₄⋊Q₈	—	2	2	no	—
12	C₉×C₄⋊Q₈	C₂²	1	4	yes	complemented
13	C₂×C₉×C₄⋊C₄	—	2	1	no	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
17	#63436	—	2	2	no	—
18	#63436	C₂²	1	4	yes	—
19	#63369	C₂²	1	16	yes	complemented
20	C₄×C₉⋊C₈	C₂²	1	4	yes	complemented
21	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
22	C₂×#874	—	3	1	no	—
23	#5088	—	3	8	no	—
24	C₂×#809	—	3	4	no	—
25	C₂×C₄×C₃⋊C₈	—	3	1	no	—
26	C₂³×D₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	4	no	—
28	C₂×Q₈×C₉	—	4	4	no	—
29	C₂²×C₄×C₉	—	2	2	no	—
30	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×D₃₆	—	2	8	no	—
34	C₂×D₃₆	—	2	2	no	—
35	C₂×D₃₆	—	2	4	no	—
36	C₂×D₃₆	—	2	4	no	—
37	C₂×D₃₆	—	4	4	no	—
38	C₂×D₃₆	—	4	4	no	—
39	C₂×D₃₆	C₂³	1	4	yes	—
40	C₉×C₄⋊C₄	—	2	4	no	—
41	C₉×C₄⋊C₄	—	2	2	no	—
42	C₉×C₄⋊C₄	C₂³	1	4	yes	—
43	C₄²×C₉	D₄	1	2	yes	—
44	C₄²×C₉	C₂³	1	2	yes	—
45	C₂×C₉⋊C₈	—	2	4	no	—
46	C₂×C₉⋊C₈	C₂³	1	4	yes	—
47	C₂×#486	—	9	1	no	Sylow
48	C₂²×C₃×Q₈	—	2	1	no	—
49	C₂²×D₁₂	—	3	2	no	—
50	C₂²×D₁₂	—	6	1	no	—
51	C₂²×D₁₂	—	6	1	no	—
52	C₃×C₄⋊Q₈	—	2	2	no	—
53	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	—
54	C₂×C₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃⋊C₈	—	3	2	no	—
58	#874	—	3	4	no	—
59	#874	—	6	2	no	—
60	#809	—	3	16	no	—
61	C₄×C₃⋊C₈	—	3	4	no	—
62	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×D₉	—	2	4	no	—
64	C₂²×D₉	—	4	4	no	—
65	C₂²×D₉	—	4	4	no	—
66	C₂²×D₉	—	4	4	no	—
67	Q₈×C₉	—	4	8	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	4	4	no	—
70	C₂×C₄×C₉	D₈	1	4	yes	—
71	C₂×C₄×C₉	SD₁₆	1	4	yes	—
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₉	#41	1	4	yes	—
76	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
77	D₃₆	—	2	8	no	—
78	D₃₆	—	4	4	no	—
79	D₃₆	—	4	8	no	—
80	D₃₆	—	4	8	no	—
81	C₉⋊C₈	—	2	4	no	—
82	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic, complemented
83	C₂×#127	—	9	1	no	—
84	#486	—	9	8	no	—
85	C₂×#102	—	9	4	no	—
86	C₂×C₄×C₈	—	9	1	no	—
87	C₂³×D₃	—	6	2	no	—
88	C₂×C₃×Q₈	—	2	4	no	—
89	C₂×C₃×Q₈	—	4	4	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	#67	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂×D₁₂	—	3	4	no	—
95	C₂×D₁₂	—	6	8	no	—
96	C₂×D₁₂	—	6	4	no	—
97	C₂×D₁₂	—	6	4	no	—
98	C₂×D₁₂	—	6	2	no	—
99	C₂×D₁₂	—	12	4	no	—
100	C₂×D₁₂	—	12	4	no	—
101	C₃×C₄⋊C₄	—	2	4	no	—
102	C₃×C₄⋊C₄	—	2	2	no	—
103	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
104	C₃×C₄²	#67	1	2	yes	—
105	C₃×C₄²	C₂²×D₃	1	2	yes	—
106	C₂×C₃⋊C₈	—	3	4	no	—
107	C₂×C₃⋊C₈	—	6	4	no	—
108	C₂²×C₉	#124	1	2	yes	—
109	C₂²×C₉	C₂×D₈	1	2	yes	—
110	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
111	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₉	—	4	4	no	—
113	C₂×D₉	—	4	4	no	—
114	C₂×D₉	—	4	4	no	—
115	C₂×D₉	—	8	8	no	—
116	C₄×C₉	—	4	4	no	—
117	C₄×C₉	#124	1	2	yes	—
118	C₄×C₉	C₂×D₈	1	4	yes	—
119	C₄×C₉	C₂×SD₁₆	1	4	yes	—
120	C₄×C₉	C₂×#41	1	2	yes	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂²×D₄	—	9	2	no	—
123	C₂²×D₄	—	18	1	no	—
124	C₂²×D₄	—	18	1	no	—
125	C₂²×C₈	—	9	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×D₉	1	4	yes	complemented
128	#127	—	9	4	no	—
129	#127	—	18	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
132	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
133	#102	—	9	16	no	—
134	C₄×C₈	—	9	4	no	—
135	C₂³×C₃	C₂×#67	1	1	yes	characteristic
136	C₂²×D₃	—	6	4	no	—
137	C₂²×D₃	—	12	4	no	—
138	C₂²×D₃	—	12	4	no	—
139	C₂²×D₃	—	12	4	no	—
140	C₃×Q₈	—	4	8	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	4	4	no	—
143	C₂×C₃×C₄	#213	1	4	yes	—
144	C₂×C₃×C₄	#215	1	4	yes	—
145	C₂×C₃×C₄	#239	1	4	yes	—
146	C₂×C₃×C₄	C₂×#67	1	2	yes	—
147	C₂×C₃×C₄	C₂×#67	1	2	yes	—
148	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
150	D₁₂	—	6	8	no	—
151	D₁₂	—	12	8	no	—
152	D₁₂	—	12	8	no	—
153	D₁₂	—	12	4	no	—
154	C₃⋊C₈	—	6	4	no	—
155	C₂×C₉	#486	1	4	yes	complemented
156	C₂×C₉	C₂×#124	1	1	yes	characteristic
157	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
158	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
159	D₉	—	8	4	no	—
160	C₂⁴	—	18	2	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂×D₄	—	9	4	no	—
164	C₂×D₄	—	18	8	no	—
165	C₂×D₄	—	18	4	no	—
166	C₂×D₄	—	18	4	no	—
167	C₂×D₄	—	18	2	no	—
168	C₂×D₄	—	36	4	no	—
169	C₂×D₄	—	36	4	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	#611	1	1	yes	characteristic
172	C₂²×C₄	#611	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
174	C₂×C₈	—	9	4	no	—
175	C₂×C₈	—	18	4	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂²×D₉	1	4	yes	—
179	C₄²	#611	1	2	yes	—
180	C₄²	C₂²×D₉	1	2	yes	—
181	C₂²×C₃	C₂×#213	1	2	yes	—
182	C₂²×C₃	C₂×#215	1	2	yes	—
183	C₂²×C₃	#947	1	2	yes	—
184	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
185	C₂×D₃	—	12	4	no	—
186	C₂×D₃	—	12	4	no	—
187	C₂×D₃	—	12	4	no	—
188	C₂×D₃	—	24	8	no	—
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	C₂×#213	1	4	yes	—
191	C₃×C₄	C₂×#215	1	4	yes	—
192	C₃×C₄	#947	1	2	yes	—
193	C₃×C₄	C₂×#239	1	2	yes	—
194	C₉	C₂×#486	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	18	4	no	—
196	C₂³	—	36	4	no	—
197	C₂³	—	36	4	no	—
198	C₂³	—	36	4	no	—
199	C₂³	C₂×#611	1	1	yes	characteristic
200	Q₈	—	4	8	no	—
201	D₄	—	18	8	no	—
202	D₄	—	36	8	no	—
203	D₄	—	36	8	no	—
204	D₄	—	36	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	#3673	1	4	yes	—
208	C₂×C₄	#3675	1	4	yes	—
209	C₂×C₄	#3701	1	4	yes	—
210	C₂×C₄	C₂×#611	1	2	yes	—
211	C₂×C₄	C₂×#611	1	2	yes	—
212	C₂×C₄	C₂×#611	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
214	C₈	—	18	4	no	—
215	C₂×C₃	#5088	1	4	yes	—
216	C₂×C₃	C₂²×#213	1	1	yes	characteristic
217	C₂×C₃	C₂²×#215	1	1	yes	characteristic
218	C₂×C₃	C₂×#947	1	1	yes	characteristic
219	D₃	—	24	4	no	—
220	C₂²	—	36	4	no	—
221	C₂²	—	36	4	no	—
222	C₂²	—	36	4	no	—
223	C₂²	—	72	8	no	—
224	C₂²	C₂×#3673	1	2	yes	—
225	C₂²	C₂×#3675	1	2	yes	—
226	C₂²	#63509	1	2	yes	—
227	C₂²	C₂²×#611	1	1	yes	characteristic
228	C₄	—	4	4	no	—
229	C₄	C₂×#3673	1	4	yes	—
230	C₄	C₂×#3675	1	4	yes	—
231	C₄	#63509	1	2	yes	—
232	C₄	C₂×#3701	1	2	yes	—
233	C₃	C₂×#5088	1	1	yes	characteristic
234	C₂	—	72	4	no	—
235	C₂	#10588773	1	4	yes	direct product factor
236	C₂	C₂²×#3673	1	1	yes	characteristic
237	C₂	C₂²×#3675	1	1	yes	characteristic
238	C₂	C₂×#63509	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
socle	C₂³×C₃	C₂×#67
prime radical	C₂×#63436	C₂
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	C₂×#486
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×C₄⋊Q₈	C₂	semidirect
3. #10588773	C₂	direct (N × H)
4. C₂×#63369	C₂	semidirect
5. C₂×C₄×C₉⋊C₈	C₂	semidirect
12. C₉×C₄⋊Q₈	C₂²	semidirect
19. #63369	C₂²	semidirect
20. C₄×C₉⋊C₈	C₂²	semidirect
82. C₂×C₄⋊Q₈	D₉	semidirect
127. C₄⋊Q₈	C₂×D₉	semidirect
155. C₂×C₉	#486	semidirect
194. C₉	C₂×#486	semidirect
235. C₂	#10588773	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×C₉×C₄⋊Q₈	2
C₂×#63436	2
#10588773	2
C₂×#63369	2
C₂×C₄×C₉⋊C₈	2
C₂×#5088	3

Subgroups — C2×#10588773 #11873359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588773 #11873359