Subgroups — #11906266

Order 1152.

1446 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#10589914	C₂	1	2	yes	maximal
3	#10589672	C₂	1	4	yes	maximal, complemented
4	#10589671	C₂	1	8	yes	maximal, complemented
5	Q₈×Dic₁₈	C₂	1	4	yes	maximal, complemented
6	#10589602	C₂	1	6	yes	maximal
7	#10589572	C₂	1	6	yes	maximal
8	#85803	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₉	C₂²	1	4	yes	—
11	C₄×Q₈×C₉	C₂²	1	4	yes	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂×C₄²×C₉	C₂²	1	3	yes	—
14	Q₈×Dic₉	C₂²	1	8	yes	—
15	Q₈×Dic₉	C₂²	1	8	yes	—
16	#63488	C₂²	1	12	yes	—
17	#63486	C₂²	1	4	yes	—
18	#63486	C₂²	1	12	yes	—
19	#63452	C₂²	1	24	yes	—
20	#63450	C₂²	1	24	yes	—
21	#63432	C₂²	1	12	yes	—
22	#63431	C₂²	1	6	yes	—
23	#63431	C₂²	1	6	yes	—
24	C₄×Dic₁₈	C₂²	1	12	yes	—
25	C₄×Dic₁₈	C₂²	1	12	yes	—
26	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
27	#5812	—	3	2	no	—
28	#5570	—	3	4	no	—
29	#5569	—	3	8	no	—
30	Q₈×Dic₆	—	3	4	no	—
31	#5500	—	3	6	no	—
32	#5470	—	3	6	no	—
33	C₂×Q₈×C₉	Q₈	1	8	yes	complemented
34	C₂×Q₈×C₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂³	1	3	yes	—
37	C₂²×C₄×C₉	C₂³	1	3	yes	—
38	C₂×Dic₁₈	C₂³	1	4	yes	—
39	C₂×Dic₁₈	C₂³	1	12	yes	—
40	C₉×C₄⋊C₄	C₂³	1	12	yes	—
41	C₄²×C₉	C₂³	1	6	yes	—
42	C₄²×C₉	C₂³	1	6	yes	—
43	#3676	C₂³	1	8	yes	—
44	C₄⋊Dic₉	C₂³	1	12	yes	—
45	C₄⋊Dic₉	C₂³	1	12	yes	—
46	C₄⋊Dic₉	C₂³	1	4	yes	—
47	C₄⋊Dic₉	C₂³	1	12	yes	—
48	Dic₉⋊C₄	C₂³	1	8	yes	—
49	Dic₉⋊C₄	C₂³	1	24	yes	—
50	C₄×Dic₉	C₂³	1	24	yes	—
51	#3502	—	9	1	no	Sylow
52	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
53	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
54	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
55	C₂×C₃×C₄⋊C₄	C₂×D₃	1	3	yes	—
56	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
57	Q₈×Dic₃	—	3	8	no	—
58	Q₈×Dic₃	—	3	8	no	—
59	#926	—	3	12	no	—
60	#924	—	3	4	no	—
61	#924	—	3	12	no	—
62	#890	—	3	24	no	—
63	#888	—	3	24	no	—
64	#870	—	3	12	no	—
65	#869	—	3	6	no	—
66	#869	—	3	6	no	—
67	C₄×Dic₆	—	3	12	no	—
68	C₄×Dic₆	—	3	12	no	—
69	C₂³×C₉	C₂⁴	1	1	yes	characteristic
70	Q₈×C₉	C₂×Q₈	1	16	yes	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	4	3	no	—
73	C₂×C₄×C₉	C₂×Q₈	1	12	yes	—
74	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂⁴	1	6	yes	—
76	C₂×C₄×C₉	C₂⁴	1	6	yes	—
77	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₉	C₂⁴	1	8	yes	—
79	C₂×Dic₉	C₂⁴	1	8	yes	—
80	Dic₁₈	—	2	8	no	—
81	Dic₁₈	—	4	12	no	—
82	Q₈²	—	9	4	no	—
83	#557	—	9	8	no	—
84	#557	—	9	4	no	—
85	#544	—	9	6	no	—
86	#533	—	9	6	no	—
87	#519	—	9	2	no	—
88	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
89	C₂×C₃×Q₈	Dic₆	1	8	yes	—
90	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
91	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
93	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
94	C₂×Dic₆	—	3	4	no	—
95	C₂×Dic₆	—	3	12	no	—
96	C₃×C₄⋊C₄	C₂²×D₃	1	12	yes	—
97	C₃×C₄²	C₂²×D₃	1	6	yes	—
98	C₃×C₄²	C₂²×D₃	1	6	yes	—
99	#217	—	3	8	no	—
100	C₄⋊Dic₃	—	3	4	no	—
101	C₄⋊Dic₃	—	3	12	no	—
102	C₄⋊Dic₃	—	3	12	no	—
103	C₄⋊Dic₃	—	3	12	no	—
104	Dic₃⋊C₄	—	3	8	no	—
105	Dic₃⋊C₄	—	3	24	no	—
106	C₄×Dic₃	—	3	24	no	—
107	C₂²×C₉	—	2	1	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	C₂²×Q₈	1	2	yes	—
110	C₂²×C₉	C₂⁵	1	1	yes	characteristic
111	C₄×C₉	—	2	1	no	—
112	C₄×C₉	—	2	6	no	—
113	C₄×C₉	C₂²×Q₈	1	12	yes	—
114	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
115	Dic₉	—	2	8	no	—
116	Dic₉	—	2	8	no	—
117	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
118	C₄⋊Q₈	—	9	24	no	—
119	C₄⋊Q₈	—	9	6	no	—
120	C₄⋊Q₈	—	9	6	no	—
121	#125	—	9	24	no	—
122	#125	—	9	12	no	—
123	C₂²⋊Q₈	—	9	4	no	—
124	C₂²⋊Q₈	—	9	12	no	—
125	C₄×Q₈	—	9	8	no	—
126	C₄×Q₈	—	9	8	no	—
127	C₄×Q₈	—	9	12	no	—
128	C₄×Q₈	—	9	12	no	—
129	C₄×Q₈	C₂×D₉	1	4	yes	—
130	C₄×Q₈	C₂×D₉	1	4	yes	—
131	#117	—	9	12	no	—
132	C₂×C₄⋊C₄	C₂×D₉	1	3	yes	—
133	C₂×C₄²	C₂×D₉	1	3	yes	—
134	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
135	C₃×Q₈	C₂×Dic₆	1	16	yes	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	4	3	no	—
138	C₂×C₃×C₄	C₂×Dic₆	1	12	yes	—
139	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂³×D₃	1	6	yes	—
141	C₂×C₃×C₄	C₂³×D₃	1	6	yes	—
142	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
143	C₂×Dic₃	—	3	8	no	—
144	C₂×Dic₃	—	3	8	no	—
145	Dic₆	—	6	8	no	—
146	Dic₆	—	12	12	no	—
147	C₂×C₉	—	2	2	no	—
148	C₂×C₉	C₂³×Q₈	1	1	yes	characteristic
149	C₂×C₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×C₉	#585	1	1	yes	characteristic
151	C₂×Q₈	—	9	4	no	—
152	C₂×Q₈	—	9	12	no	—
153	C₂×Q₈	Dic₁₈	1	8	yes	complemented
154	C₂×Q₈	C₂²×D₉	1	4	yes	—
155	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
156	C₂²×C₄	C₂²×D₉	1	3	yes	—
157	C₂²×C₄	C₂²×D₉	1	3	yes	—
158	C₄⋊C₄	—	9	12	no	—
159	C₄⋊C₄	—	9	12	no	—
160	C₄⋊C₄	—	9	8	no	—
161	C₄⋊C₄	—	9	4	no	—
162	C₄⋊C₄	—	9	12	no	—
163	C₄⋊C₄	—	9	24	no	—
164	C₄⋊C₄	C₂²×D₉	1	12	yes	—
165	C₂²⋊C₄	—	9	8	no	—
166	C₄²	—	9	24	no	—
167	C₄²	C₂²×D₉	1	6	yes	—
168	C₄²	C₂²×D₉	1	6	yes	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	C₂²×Dic₆	1	2	yes	—
172	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	6	no	—
175	C₃×C₄	C₂²×Dic₆	1	12	yes	—
176	C₃×C₄	#1007	1	2	yes	—
177	Dic₃	—	6	8	no	—
178	Dic₃	—	6	8	no	—
179	C₉	#3502	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂³×D₉	1	1	yes	characteristic
181	Q₈	—	18	8	no	—
182	Q₈	—	36	12	no	—
183	Q₈	C₂×Dic₁₈	1	16	yes	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	3	no	—
186	C₂×C₄	—	9	8	no	—
187	C₂×C₄	—	9	8	no	—
188	C₂×C₄	C₂×Dic₁₈	1	12	yes	—
189	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
190	C₂×C₄	C₂³×D₉	1	6	yes	—
191	C₂×C₄	C₂³×D₉	1	6	yes	—
192	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
193	C₂×C₃	—	2	2	no	—
194	C₂×C₃	C₂³×Dic₆	1	1	yes	characteristic
195	C₂×C₃	C₂×#1007	1	1	yes	characteristic
196	C₂×C₃	#5964	1	1	yes	characteristic
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₂²×Dic₁₈	1	2	yes	—
200	C₂²	C₂⁴×D₉	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	6	no	—
203	C₄	—	18	8	no	—
204	C₄	—	18	8	no	—
205	C₄	C₂²×Dic₁₈	1	12	yes	—
206	C₄	#63713	1	2	yes	—
207	C₃	#85803	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₂³×Dic₁₈	1	1	yes	characteristic
210	C₂	C₂×#63713	1	1	yes	characteristic
211	C₂	#10593139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁴×D₉
commutator, cosocle	C₂²×C₉	C₂⁵
Frattini, socle	C₂²×C₃	C₂⁴×D₃
Fitting	C₂×C₄×Q₈×C₉	C₂
prime radical	C₂³×C₃	C₂³×D₃
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#3502
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #10589672	C₂	semidirect
4. #10589671	C₂	semidirect
5. Q₈×Dic₁₈	C₂	semidirect
33. C₂×Q₈×C₉	Q₈	semidirect
153. C₂×Q₈	Dic₁₈	semidirect
179. C₉	#3502	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
#10589914	2
#10589672	2
#10589671	2
Q₈×Dic₁₈	2
#10589602	2
#10589572	2
#85803	3