Subgroups — #11802924

Order 1152.

900 conjugacy classes of subgroups.

The subgroup lattice has 195 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	2	yes	maximal
3	#10588892	C₂	1	4	yes	maximal
4	#10588664	C₂	1	8	yes	maximal, complemented
5	#82156	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	2	yes	—
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₄	1	8	yes	complemented
9	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×#3676	C₂²	1	4	yes	—
11	#63488	—	2	8	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₉	C₂²	1	8	yes	—
14	C₂×C₄×Dic₉	C₂²	1	4	yes	—
15	C₉⋊#95	C₂²	1	8	yes	—
16	C₂²×C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₂×#926	—	3	2	no	—
18	#5212	—	3	4	no	—
19	#4984	—	3	8	no	—
20	C₂⁴×C₉	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₉	—	4	4	no	—
22	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₉	D₄	1	4	yes	—
25	C₂²×C₄×C₉	Q₈	1	4	yes	—
26	C₂²×C₄×C₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₉	C₂³	1	2	yes	—
28	C₂²×Dic₉	C₂³	1	8	yes	—
29	C₉×C₄⋊C₄	C₂×C₄	1	16	yes	—
30	#3676	—	2	8	no	—
31	C₄⋊Dic₉	—	2	8	no	—
32	C₄⋊Dic₉	—	4	8	no	—
33	C₄×Dic₉	—	2	8	no	—
34	#2316	—	9	1	no	Sylow
35	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
36	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂×C₃×C₄⋊C₄	Dic₃	1	8	yes	—
38	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
39	C₂×#217	—	3	4	no	—
40	#926	—	6	8	no	—
41	C₂×C₄⋊Dic₃	—	3	4	no	—
42	C₂×C₄⋊Dic₃	—	3	8	no	—
43	C₂×C₄×Dic₃	—	3	4	no	—
44	C₃⋊#95	—	3	8	no	—
45	C₂³×C₉	—	2	2	no	—
46	C₂³×C₉	—	2	2	no	—
47	C₂³×C₉	C₂²×C₄	1	2	yes	—
48	C₂³×C₉	C₂×D₄	1	2	yes	—
49	C₂³×C₉	C₂×Q₈	1	2	yes	—
50	C₂³×C₉	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₉	—	2	8	no	—
52	C₂×C₄×C₉	—	4	8	no	—
53	C₂×C₄×C₉	C₄⋊C₄	1	16	yes	complemented
54	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
58	C₂×Dic₉	—	2	8	no	—
59	C₂×Dic₉	—	2	8	no	—
60	C₂×Dic₉	—	2	8	no	—
61	C₂×#117	—	9	2	no	—
62	C₂²×C₄⋊C₄	D₉	1	1	yes	characteristic
63	#389	—	9	8	no	—
64	#380	—	9	4	no	—
65	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	4	4	no	—
67	C₂²×C₃×C₄	Dic₆	1	4	yes	—
68	C₂²×C₃×C₄	D₁₂	1	4	yes	—
69	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
70	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
71	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
72	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
73	C₂²×Dic₃	—	3	8	no	—
74	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
75	#217	—	6	8	no	—
76	C₄⋊Dic₃	—	6	8	no	—
77	C₄⋊Dic₃	—	12	8	no	—
78	C₄×Dic₃	—	6	8	no	—
79	C₂²×C₉	—	2	2	no	—
80	C₂²×C₉	—	2	2	no	—
81	C₂²×C₉	—	2	4	no	—
82	C₂²×C₉	—	2	4	no	—
83	C₂²×C₉	C₂×C₄⋊C₄	1	4	yes	—
84	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
87	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
88	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
89	C₄×C₉	—	2	8	no	—
90	C₄×C₉	C₂×C₄⋊C₄	1	8	yes	—
91	Dic₉	—	4	8	no	—
92	C₂³×C₄	C₂×D₉	1	2	yes	—
93	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
94	#117	—	18	8	no	—
95	C₂×C₄⋊C₄	—	9	4	no	—
96	C₂×C₄⋊C₄	—	9	8	no	—
97	C₂×C₄⋊C₄	Dic₉	1	8	yes	complemented
98	C₂×C₄⋊C₄	C₂×D₉	1	4	yes	—
99	C₂×C₂²⋊C₄	—	9	4	no	—
100	C₂×C₄²	—	9	4	no	—
101	#95	—	9	8	no	—
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	C₂×Dic₆	1	2	yes	—
105	C₂³×C₃	C₂×D₁₂	1	2	yes	—
106	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
107	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	8	no	—
109	C₂×C₃×C₄	—	4	8	no	—
110	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
111	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
112	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
113	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
114	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
115	C₂×Dic₃	—	6	8	no	—
116	C₂×Dic₃	—	6	8	no	—
117	C₂×Dic₃	—	6	8	no	—
118	C₂×C₉	—	2	4	no	—
119	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₉	#518	1	1	yes	characteristic
121	C₂×C₉	#519	1	1	yes	characteristic
122	C₂×C₉	#537	1	2	yes	—
123	C₂×C₉	#544	1	2	yes	—
124	C₂⁴	C₂²×D₉	1	1	yes	characteristic
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	—	9	8	no	—
127	C₂²×C₄	Dic₁₈	1	4	yes	—
128	C₂²×C₄	D₃₆	1	4	yes	—
129	C₂²×C₄	C₂×Dic₉	1	8	yes	—
130	C₂²×C₄	C₂×Dic₉	1	4	yes	—
131	C₂²×C₄	C₂²×D₉	1	4	yes	—
132	C₂²×C₄	C₂²×D₉	1	2	yes	—
133	C₄⋊C₄	—	18	8	no	—
134	C₄⋊C₄	—	36	8	no	—
135	C₄⋊C₄	C₂×Dic₉	1	16	yes	—
136	C₂²⋊C₄	—	18	8	no	—
137	C₄²	—	18	8	no	—
138	C₂²×C₃	—	2	4	no	—
139	C₂²×C₃	—	2	4	no	—
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
143	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
144	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
145	C₂²×C₃	#1004	1	2	yes	—
146	C₂²×C₃	#1007	1	2	yes	—
147	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
148	C₃×C₄	—	2	8	no	—
149	C₃×C₄	C₂×C₄⋊Dic₃	1	8	yes	—
150	Dic₃	—	12	8	no	—
151	C₉	#2316	1	1	yes	characteristic, complemented, Sylow
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	C₂×Dic₁₈	1	2	yes	—
155	C₂³	C₂×D₃₆	1	2	yes	—
156	C₂³	C₂²×Dic₉	1	2	yes	—
157	C₂³	C₂³×D₉	1	1	yes	characteristic
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	—	18	8	no	—
161	C₂×C₄	—	18	8	no	—
162	C₂×C₄	—	18	8	no	—
163	C₂×C₄	C₄⋊Dic₉	1	16	yes	complemented
164	C₂×C₄	C₂×Dic₁₈	1	4	yes	—
165	C₂×C₄	C₂×D₃₆	1	4	yes	—
166	C₂×C₄	C₂²×Dic₉	1	8	yes	—
167	C₂×C₄	C₂²×Dic₉	1	4	yes	—
168	C₂×C₃	—	2	4	no	—
169	C₂×C₃	#5500	1	2	yes	—
170	C₂×C₃	#5510	1	2	yes	—
171	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
172	C₂×C₃	#5798	1	1	yes	characteristic
173	C₂×C₃	#5812	1	1	yes	characteristic
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	C₂×C₄⋊Dic₉	1	4	yes	—
179	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
180	C₂²	C₂²×D₃₆	1	1	yes	characteristic
181	C₂²	#63710	1	2	yes	—
182	C₂²	#63713	1	2	yes	—
183	C₂²	C₂³×Dic₉	1	1	yes	characteristic
184	C₄	—	2	8	no	—
185	C₄	—	36	8	no	—
186	C₄	C₂×C₄⋊Dic₉	1	8	yes	—
187	C₃	#82156	1	1	yes	characteristic
188	C₂	—	2	4	no	—
189	C₂	#10589602	1	2	yes	—
190	C₂	#10589612	1	2	yes	—
191	C₂	C₂²×C₄⋊Dic₉	1	1	yes	characteristic
192	C₂	#10589900	1	1	yes	characteristic
193	C₂	#10589914	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle	C₂³×C₃	C₂³×D₃
Fitting	C₂²×C₉×C₄⋊C₄	C₂
prime radical	C₂⁴×C₃	C₂²×D₃
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₉
hypocenter	C₉	#2316
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #10588664	C₂	semidirect
8. C₂×C₉×C₄⋊C₄	C₄	semidirect
53. C₂×C₄×C₉	C₄⋊C₄	semidirect
97. C₂×C₄⋊C₄	Dic₉	semidirect
151. C₉	#2316	semidirect
163. C₂×C₄	C₄⋊Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂×#63488	2
#10588892	2
#10588664	2
#82156	3