Subgroups — C₂×#10588767 #11868381

Order 1152.

624 conjugacy classes of subgroups.

The subgroup lattice has 249 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
2	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
3	#10588767	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63368	C₂	1	2	yes	maximal
5	C₂×#63367	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₂×#5094	—	3	1	no	maximal
8	C₂²×Q₈×C₉	—	2	1	no	—
9	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
10	C₉×C₄⋊Q₈	—	2	2	no	—
11	C₉×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂×C₉×C₄⋊C₄	—	2	1	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₉⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₉	—	2	1	no	—
18	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
19	C₄×Dic₁₈	—	2	2	no	—
20	C₄×Dic₁₈	C₂²	1	4	yes	—
21	#63368	C₂²	1	8	yes	—
22	#63367	C₂²	1	8	yes	—
23	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
24	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
25	C₂×C₄×Dic₆	—	3	1	no	—
26	#5094	—	3	8	no	—
27	C₂×#810	—	3	2	no	—
28	C₂×#808	—	3	2	no	—
29	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
30	C₂×Q₈×C₉	—	2	4	no	—
31	C₂×Q₈×C₉	—	4	4	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₉	—	2	1	no	—
37	C₂²×Dic₉	—	2	1	no	—
38	C₂×Dic₁₈	—	2	1	no	—
39	C₂×Dic₁₈	—	2	2	no	—
40	C₂×Dic₁₈	Q₈	1	4	yes	complemented
41	C₂×Dic₁₈	C₂³	1	2	yes	—
42	C₉×C₄⋊C₄	—	2	4	no	—
43	C₉×C₄⋊C₄	—	2	2	no	—
44	C₉×C₄⋊C₄	C₂³	1	4	yes	—
45	C₄²×C₉	D₄	1	2	yes	—
46	C₄²×C₉	C₂³	1	2	yes	—
47	C₄⋊Dic₉	—	2	1	no	—
48	C₄⋊Dic₉	C₂³	1	2	yes	—
49	Dic₉⋊C₄	—	2	4	no	—
50	C₄×Dic₉	—	2	4	no	—
51	C₂×C₉⋊C₈	—	2	4	no	—
52	C₂×C₉⋊C₈	C₂³	1	4	yes	—
53	C₂×#475	—	9	1	no	Sylow
54	C₂²×C₃×Q₈	—	2	1	no	—
55	C₂²×Dic₆	—	3	1	no	—
56	C₃×C₄⋊Q₈	—	2	2	no	—
57	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	—
58	C₂×C₃×C₄⋊C₄	—	2	1	no	—
59	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
60	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	3	1	no	—
62	C₂×Dic₃⋊C₄	—	6	1	no	—
63	C₂×C₄×Dic₃	—	6	1	no	—
64	C₂²×C₃⋊C₈	—	3	2	no	—
65	C₄×Dic₆	—	3	4	no	—
66	C₄×Dic₆	—	6	2	no	—
67	#810	—	3	8	no	—
68	#808	—	3	8	no	—
69	C₄⋊(C₃⋊C₈)	—	3	4	no	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₉	—	4	8	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	4	no	—
74	C₂×C₄×C₉	—	4	4	no	—
75	C₂×C₄×C₉	SD₁₆	1	4	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₉	C₂×Q₈	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	2	no	—
84	C₂×Dic₉	—	2	2	no	—
85	C₂×Dic₉	—	4	2	no	—
86	Dic₁₈	—	2	4	no	—
87	Dic₁₈	—	2	2	no	—
88	Dic₁₈	C₂×Q₈	1	4	yes	complemented
89	C₉⋊C₈	—	2	4	no	—
90	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
91	C₂×C₄×Q₈	—	9	1	no	—
92	#475	—	9	8	no	—
93	C₂×#106	—	9	2	no	—
94	C₂×C₄⋊C₈	—	9	1	no	—
95	C₂×#103	—	9	2	no	—
96	C₂×C₃×Q₈	—	2	4	no	—
97	C₂×C₃×Q₈	—	4	4	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	#67	1	1	yes	characteristic
100	C₂²×C₃×C₄	#67	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×Dic₃	—	6	1	no	—
103	C₂²×Dic₃	—	6	1	no	—
104	C₂×Dic₆	—	3	4	no	—
105	C₂×Dic₆	—	3	2	no	—
106	C₂×Dic₆	—	6	2	no	—
107	C₂×Dic₆	—	6	1	no	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
111	C₃×C₄²	#67	1	2	yes	—
112	C₃×C₄²	C₂²×D₃	1	2	yes	—
113	C₄⋊Dic₃	—	3	2	no	—
114	C₄⋊Dic₃	—	6	1	no	—
115	Dic₃⋊C₄	—	6	4	no	—
116	C₄×Dic₃	—	6	4	no	—
117	C₂×C₃⋊C₈	—	3	4	no	—
118	C₂×C₃⋊C₈	—	6	4	no	—
119	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
120	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
121	C₂²×C₉	#137	1	2	yes	—
122	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	—	4	4	no	—
125	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
126	C₄×C₉	C₂×SD₁₆	1	4	yes	—
127	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
128	C₄×C₉	C₂×#41	1	1	yes	characteristic
129	Dic₉	—	2	4	no	—
130	Dic₉	—	4	2	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂²×Q₈	—	9	1	no	—
133	C₂²×C₈	—	9	2	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	C₂×D₉	1	4	yes	—
136	C₄×Q₈	—	9	4	no	—
137	C₄×Q₈	—	18	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	9	1	no	—
140	C₂×C₄⋊C₄	—	18	1	no	—
141	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
142	C₂×C₄²	—	18	1	no	—
143	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
144	#106	—	9	8	no	—
145	C₄⋊C₈	—	9	4	no	—
146	#103	—	9	8	no	—
147	C₂³×C₃	C₂×#67	1	1	yes	characteristic
148	C₃×Q₈	—	4	8	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	#214	1	4	yes	—
153	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
154	C₂×C₃×C₄	#239	1	2	yes	—
155	C₂×C₃×C₄	C₂×#67	1	2	yes	—
156	C₂×C₃×C₄	C₂×#67	1	2	yes	—
157	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
159	C₂×Dic₃	—	6	4	no	—
160	C₂×Dic₃	—	6	2	no	—
161	C₂×Dic₃	—	6	2	no	—
162	C₂×Dic₃	—	12	2	no	—
163	Dic₆	—	3	4	no	—
164	Dic₆	—	6	4	no	—
165	Dic₆	—	6	2	no	—
166	C₃⋊C₈	—	6	4	no	—
167	C₂×C₉	#475	1	4	yes	complemented
168	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
170	C₂×C₉	C₂×#137	1	1	yes	characteristic
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	9	4	no	—
174	C₂×Q₈	—	9	2	no	—
175	C₂×Q₈	—	18	2	no	—
176	C₂×Q₈	—	18	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	18	1	no	—
179	C₂²×C₄	—	18	1	no	—
180	C₂²×C₄	#611	1	1	yes	characteristic
181	C₂²×C₄	#611	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
183	C₂×C₈	—	9	4	no	—
184	C₂×C₈	—	18	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	9	2	no	—
188	C₄⋊C₄	—	18	1	no	—
189	C₄⋊C₄	—	18	4	no	—
190	C₄⋊C₄	C₂²×D₉	1	4	yes	—
191	C₄²	—	18	4	no	—
192	C₄²	#611	1	2	yes	—
193	C₄²	C₂²×D₉	1	2	yes	—
194	C₂²×C₃	C₂×#214	1	2	yes	—
195	C₂²×C₃	#942	1	2	yes	—
196	C₂²×C₃	#946	1	2	yes	—
197	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	4	4	no	—
200	C₃×C₄	C₂×#214	1	4	yes	—
201	C₃×C₄	#946	1	2	yes	—
202	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
203	C₃×C₄	C₂×#239	1	1	yes	characteristic
204	Dic₃	—	6	4	no	—
205	Dic₃	—	12	2	no	—
206	C₉	C₂×#475	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×#611	1	1	yes	characteristic
208	Q₈	—	4	8	no	—
209	Q₈	—	9	4	no	—
210	Q₈	—	18	4	no	—
211	Q₈	—	18	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	18	4	no	—
216	C₂×C₄	—	18	2	no	—
217	C₂×C₄	—	18	2	no	—
218	C₂×C₄	—	36	2	no	—
219	C₂×C₄	#3672	1	4	yes	—
220	C₂×C₄	Q₈×D₉	1	2	yes	—
221	C₂×C₄	#3701	1	2	yes	—
222	C₂×C₄	C₂×#611	1	2	yes	—
223	C₂×C₄	C₂×#611	1	2	yes	—
224	C₂×C₄	C₂×#611	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
226	C₈	—	18	4	no	—
227	C₂×C₃	#5094	1	4	yes	—
228	C₂×C₃	C₂²×#214	1	1	yes	characteristic
229	C₂×C₃	C₂×#942	1	1	yes	characteristic
230	C₂×C₃	C₂×#946	1	1	yes	characteristic
231	C₂²	C₂×#3672	1	2	yes	—
232	C₂²	#63505	1	2	yes	—
233	C₂²	#63508	1	2	yes	—
234	C₂²	C₂²×#611	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	4	4	no	—
237	C₄	—	18	4	no	—
238	C₄	—	36	2	no	—
239	C₄	C₂×#3672	1	4	yes	—
240	C₄	#63508	1	2	yes	—
241	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
242	C₄	C₂×#3701	1	1	yes	characteristic
243	C₃	C₂×#5094	1	1	yes	characteristic
244	C₂	#10588767	1	4	yes	direct product factor
245	C₂	C₂²×#3672	1	1	yes	characteristic
246	C₂	C₂×#63505	1	1	yes	characteristic
247	C₂	C₂×#63508	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, prime radical	C₂³×C₃	C₂×#67
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	C₂×#475
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
3. #10588767	C₂	direct (N × H)
40. C₂×Dic₁₈	Q₈	semidirect
88. Dic₁₈	C₂×Q₈	semidirect
167. C₂×C₉	#475	semidirect
206. C₉	C₂×#475	semidirect
244. C₂	#10588767	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×C₄×Dic₁₈	2
#10588767	2
C₂×#63368	2
C₂×#63367	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×#5094	3

Subgroups — C2×#10588767 #11868381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588767 #11868381