Subgroups — C₂×#10588788 #11866779

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63486	C₂	1	2	yes	maximal
3	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588788	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63378	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63376	C₂	1	2	yes	maximal, complemented
7	C₂×#5116	—	3	1	no	maximal
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₈	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	—	2	4	no	—
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	4	yes	—
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×#3676	—	2	2	no	—
15	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
16	#63486	—	2	4	no	—
17	#63486	C₂²	1	8	yes	—
18	C₂×Dic₉⋊C₄	—	2	2	no	—
19	C₂×Dic₃₆	—	2	4	no	—
20	C₂×Dic₃₆	C₂²	1	4	yes	complemented
21	#63378	C₂²	1	4	yes	complemented
22	#63376	C₂²	1	8	yes	complemented
23	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
24	C₂×#924	—	3	2	no	—
25	C₂²×Dic₁₂	—	3	1	no	—
26	#5116	—	3	8	no	—
27	C₂×#818	—	3	1	no	—
28	C₂×#816	—	3	2	no	—
29	C₂⁴×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	D₄	1	2	yes	—
34	C₂²×C₄×C₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₉	—	2	2	no	—
37	C₂²×Dic₉	—	2	2	no	—
38	C₂×Dic₁₈	—	2	4	no	—
39	C₂×Dic₁₈	—	2	4	no	—
40	C₂×Dic₁₈	—	2	2	no	—
41	C₂×Dic₁₈	C₂³	1	4	yes	—
42	C₂×C₈×C₉	—	2	2	no	—
43	C₂×C₈×C₉	—	2	8	no	—
44	C₂×C₈×C₉	D₄	1	4	yes	—
45	C₂×C₈×C₉	C₂³	1	2	yes	—
46	C₂×C₈×C₉	C₂³	1	2	yes	—
47	#3676	—	2	8	no	—
48	C₄⋊Dic₉	—	2	2	no	—
49	C₄⋊Dic₉	C₂³	1	4	yes	—
50	Dic₉⋊C₄	—	2	8	no	—
51	Dic₃₆	—	2	8	no	—
52	C₂×#467	—	9	1	no	Sylow
53	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂²×Dic₆	—	3	2	no	—
55	C₂²×C₃×C₈	—	2	4	no	—
56	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
58	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×#217	—	6	2	no	—
60	C₂×C₄⋊Dic₃	—	3	2	no	—
61	#924	—	3	8	no	—
62	#924	—	6	4	no	—
63	C₂×Dic₃⋊C₄	—	6	2	no	—
64	C₂×Dic₁₂	—	3	4	no	—
65	C₂×Dic₁₂	—	6	4	no	—
66	#818	—	3	4	no	—
67	#816	—	3	8	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	—	2	2	no	—
70	C₂³×C₉	Q₁₆	1	4	yes	complemented
71	C₂³×C₉	C₂×D₄	1	2	yes	—
72	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₉	#41	1	2	yes	—
81	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₉	—	2	4	no	—
83	C₂×Dic₉	—	2	4	no	—
84	C₂×Dic₉	—	4	4	no	—
85	C₂×Dic₉	—	4	4	no	—
86	Dic₁₈	—	2	4	no	—
87	Dic₁₈	—	2	4	no	—
88	Dic₁₈	—	4	4	no	—
89	C₈×C₉	—	2	2	no	—
90	C₈×C₉	C₂×D₄	1	4	yes	—
91	C₂²×Q₁₆	—	9	1	no	—
92	C₂³×C₈	D₉	1	1	yes	characteristic
93	C₂×C₂²⋊Q₈	—	9	2	no	—
94	#467	—	9	8	no	—
95	C₂×#107	—	9	1	no	—
96	C₂×#103	—	9	2	no	—
97	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
101	C₂²×C₃×C₄	D₁₂	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
103	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂²×Dic₃	—	6	2	no	—
105	C₂²×Dic₃	—	6	2	no	—
106	C₂×Dic₆	—	3	4	no	—
107	C₂×Dic₆	—	6	4	no	—
108	C₂×Dic₆	—	6	4	no	—
109	C₂×Dic₆	—	6	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	8	no	—
112	C₂×C₃×C₈	#67	1	4	yes	—
113	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
114	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
115	#217	—	6	8	no	—
116	C₄⋊Dic₃	—	3	4	no	—
117	C₄⋊Dic₃	—	6	2	no	—
118	Dic₃⋊C₄	—	6	8	no	—
119	Dic₁₂	—	6	8	no	—
120	C₂²×C₉	—	2	2	no	—
121	C₂²×C₉	—	2	2	no	—
122	C₂²×C₉	—	2	8	no	—
123	C₂²×C₉	#121	1	2	yes	—
124	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
125	C₂²×C₉	C₂×Q₁₆	1	4	yes	complemented
126	C₂²×C₉	#135	1	2	yes	—
127	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	#121	1	2	yes	—
130	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₉	C₂×#41	1	1	yes	characteristic
132	Dic₉	—	4	4	no	—
133	Dic₉	—	4	4	no	—
134	C₂²×Q₈	—	9	2	no	—
135	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
136	C₂×Q₁₆	—	9	4	no	—
137	C₂×Q₁₆	—	18	4	no	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
140	C₂²×C₈	C₂×D₉	1	4	yes	—
141	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
142	C₂²⋊Q₈	—	9	8	no	—
143	C₂²⋊Q₈	—	18	4	no	—
144	C₂×C₄⋊C₄	—	9	2	no	—
145	C₂×C₄⋊C₄	—	18	2	no	—
146	C₂×C₂²⋊C₄	—	18	2	no	—
147	#107	—	9	4	no	—
148	#103	—	9	8	no	—
149	C₂³×C₃	—	2	2	no	—
150	C₂³×C₃	—	2	2	no	—
151	C₂³×C₃	Dic₁₂	1	4	yes	—
152	C₂³×C₃	C₂×D₁₂	1	2	yes	—
153	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
159	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
160	C₂×C₃×C₄	#235	1	2	yes	—
161	C₂×C₃×C₄	C₂×#67	1	2	yes	—
162	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
163	C₂×Dic₃	—	6	4	no	—
164	C₂×Dic₃	—	6	4	no	—
165	C₂×Dic₃	—	12	4	no	—
166	C₂×Dic₃	—	12	4	no	—
167	Dic₆	—	6	4	no	—
168	Dic₆	—	6	4	no	—
169	Dic₆	—	12	4	no	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	C₂×#67	1	4	yes	—
172	C₂×C₉	—	2	4	no	—
173	C₂×C₉	#467	1	4	yes	complemented
174	C₂×C₉	C₂×#121	1	1	yes	characteristic
175	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
176	C₂×C₉	C₂×#135	1	1	yes	characteristic
177	C₂⁴	D₃₆	1	1	yes	characteristic
178	C₂×Q₈	—	9	4	no	—
179	C₂×Q₈	—	18	2	no	—
180	C₂×Q₈	—	18	4	no	—
181	C₂×Q₈	—	18	4	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	18	2	no	—
185	C₂²×C₄	—	18	2	no	—
186	C₂²×C₄	D₃₆	1	2	yes	—
187	C₂²×C₄	D₃₆	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₉	1	2	yes	—
189	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
190	Q₁₆	—	18	8	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	#611	1	4	yes	—
194	C₂×C₈	C₂²×D₉	1	2	yes	—
195	C₂×C₈	C₂²×D₉	1	2	yes	—
196	C₄⋊C₄	—	9	4	no	—
197	C₄⋊C₄	—	18	2	no	—
198	C₄⋊C₄	—	18	8	no	—
199	C₂²⋊C₄	—	18	8	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	8	no	—
203	C₂²×C₃	#904	1	2	yes	—
204	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
205	C₂²×C₃	C₂×Dic₁₂	1	4	yes	—
206	C₂²×C₃	#930	1	2	yes	—
207	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	#930	1	2	yes	—
210	C₃×C₄	C₂×#235	1	1	yes	characteristic
211	C₃×C₄	C₂²×#67	1	1	yes	characteristic
212	Dic₃	—	12	4	no	—
213	Dic₃	—	12	4	no	—
214	C₉	C₂×#467	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	Dic₃₆	1	4	yes	complemented
218	C₂³	C₂×D₃₆	1	2	yes	—
219	C₂³	C₂×D₃₆	1	1	yes	characteristic
220	Q₈	—	18	4	no	—
221	Q₈	—	18	4	no	—
222	Q₈	—	36	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	18	4	no	—
228	C₂×C₄	—	18	4	no	—
229	C₂×C₄	—	36	4	no	—
230	C₂×C₄	—	36	4	no	—
231	C₂×C₄	C₂×D₃₆	1	2	yes	—
232	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
233	C₂×C₄	#3697	1	2	yes	—
234	C₂×C₄	C₂×#611	1	2	yes	—
235	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	C₂×#611	1	4	yes	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	#5116	1	4	yes	—
240	C₂×C₃	C₂×#904	1	1	yes	characteristic
241	C₂×C₃	C₂²×Dic₁₂	1	1	yes	characteristic
242	C₂×C₃	C₂×#930	1	1	yes	characteristic
243	C₂²	—	2	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	C₂×Dic₃₆	1	2	yes	—
247	C₂²	C₂×Dic₃₆	1	4	yes	complemented
248	C₂²	#63467	1	2	yes	—
249	C₂²	#63492	1	2	yes	—
250	C₂²	C₂²×D₃₆	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	36	4	no	—
253	C₄	—	36	4	no	—
254	C₄	#63492	1	2	yes	—
255	C₄	C₂×#3697	1	1	yes	characteristic
256	C₄	C₂²×#611	1	1	yes	characteristic
257	C₃	C₂×#5116	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	#10588788	1	4	yes	direct product factor
260	C₂	C₂²×Dic₃₆	1	1	yes	characteristic
261	C₂	C₂×#63467	1	1	yes	characteristic
262	C₂	C₂×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₂×D₁₂
prime radical	C₂⁴×C₃	D₁₂
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	C₂×#467
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×Dic₃₆	C₂	semidirect
4. #10588788	C₂	direct (N × H)
5. C₂×#63378	C₂	semidirect
6. C₂×#63376	C₂	semidirect
20. C₂×Dic₃₆	C₂²	semidirect
21. #63378	C₂²	semidirect
22. #63376	C₂²	semidirect
70. C₂³×C₉	Q₁₆	semidirect
125. C₂²×C₉	C₂×Q₁₆	semidirect
173. C₂×C₉	#467	semidirect
214. C₉	C₂×#467	semidirect
217. C₂³	Dic₃₆	semidirect
247. C₂²	C₂×Dic₃₆	semidirect
259. C₂	#10588788	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×#63486	2
C₂²×Dic₃₆	2
#10588788	2
C₂×#63378	2
C₂×#63376	2
C₂×#5116	3

Subgroups — C2×#10588788 #11866779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588788 #11866779