Subgroups — C₂²×#63381 #11858608

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#63381	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#822	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₉×M₁₆	—	2	3	no	—
8	C₂×C₉×M₁₆	C₂²	1	6	yes	—
9	C₂²×C₈×C₉	—	2	1	no	—
10	C₂×#3667	—	2	3	no	—
11	C₂×#3667	C₂²	1	6	yes	—
12	C₂²×C₉⋊C₈	—	2	1	no	—
13	C₂²×C₉⋊C₈	C₄	1	6	yes	—
14	C₂²×C₉⋊C₈	C₄	1	2	yes	—
15	C₂²×C₉⋊C₈	C₂²	1	6	yes	—
16	#63381	C₂²	1	16	yes	direct product factor
17	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
18	C₂²×#208	—	3	1	no	—
19	C₂³×C₃⋊C₈	—	3	1	no	—
20	C₂×#822	—	3	12	no	—
21	C₂⁴×C₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	D₄	1	3	yes	—
26	C₂²×C₄×C₉	Q₈	1	3	yes	—
27	C₂²×C₄×C₉	C₂³	1	3	yes	—
28	C₉×M₁₆	—	2	6	no	—
29	C₉×M₁₆	C₂³	1	4	yes	—
30	C₂×C₈×C₉	—	2	6	no	—
31	#3667	—	2	6	no	—
32	#3667	C₂³	1	4	yes	—
33	C₂×C₉⋊C₈	—	2	6	no	—
34	C₂×C₉⋊C₈	C₂×C₄	1	12	yes	—
35	C₂×C₉⋊C₈	C₂×C₄	1	12	yes	—
36	C₂×C₉⋊C₈	C₂³	1	4	yes	—
37	C₂²×#108	—	9	1	no	Sylow
38	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×M₁₆	—	2	3	no	—
40	C₂×C₃×M₁₆	C₂×D₃	1	6	yes	—
41	C₂²×C₃×C₈	—	2	1	no	—
42	C₂×#208	—	3	6	no	—
43	C₂×#208	—	6	3	no	—
44	C₂²×C₃⋊C₈	—	3	6	no	—
45	C₂²×C₃⋊C₈	—	3	2	no	—
46	C₂²×C₃⋊C₈	—	3	6	no	—
47	C₂²×C₃⋊C₈	—	6	1	no	—
48	#822	—	3	16	no	—
49	C₂³×C₉	—	2	4	no	—
50	C₂³×C₉	C₄⋊C₄	1	3	yes	—
51	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂³×C₉	C₂×Q₈	1	3	yes	—
53	C₂×C₄×C₉	C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₉	C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₉	C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
57	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
58	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
59	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
60	C₂×C₄×C₉	C₂×Q₈	1	3	yes	—
61	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
62	C₈×C₉	—	2	4	no	—
63	C₉⋊C₈	—	2	4	no	—
64	C₉⋊C₈	C₂²×C₄	1	8	yes	—
65	C₂²×M₁₆	—	9	1	no	—
66	C₂²×M₁₆	D₉	1	1	yes	characteristic
67	C₂³×C₈	—	9	1	no	—
68	C₂×#108	—	9	12	no	—
69	C₂⁴×C₃	Dic₆	1	1	yes	characteristic
70	C₂²×C₃×C₄	Dic₆	1	3	yes	—
71	C₂²×C₃×C₄	C₄×D₃	1	3	yes	—
72	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
73	C₂²×C₃×C₄	#67	1	3	yes	—
74	C₂²×C₃×C₄	#67	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
76	C₃×M₁₆	—	2	6	no	—
77	C₃×M₁₆	C₂²×D₃	1	4	yes	—
78	C₂×C₃×C₈	—	2	6	no	—
79	#208	—	3	4	no	—
80	#208	—	6	6	no	—
81	C₂×C₃⋊C₈	—	3	12	no	—
82	C₂×C₃⋊C₈	—	3	12	no	—
83	C₂×C₃⋊C₈	—	3	4	no	—
84	C₂×C₃⋊C₈	—	6	6	no	—
85	C₂²×C₉	—	2	6	no	—
86	C₂²×C₉	—	2	6	no	—
87	C₂²×C₉	#108	1	4	yes	complemented
88	C₂²×C₉	C₂×C₄⋊C₄	1	3	yes	—
89	C₂²×C₉	C₂×C₄⋊C₄	1	3	yes	—
90	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
91	C₄×C₉	C₂×C₄⋊C₄	1	3	yes	—
92	C₄×C₉	C₂×C₄⋊C₄	1	3	yes	—
93	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
95	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
96	C₂×M₁₆	—	2	3	no	—
97	C₂×M₁₆	—	9	6	no	—
98	C₂×M₁₆	—	18	3	no	—
99	C₂×M₁₆	C₂×D₉	1	6	yes	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	9	2	no	—
102	C₂²×C₈	—	9	6	no	—
103	C₂²×C₈	—	9	6	no	—
104	C₂²×C₈	—	18	1	no	—
105	#108	—	9	16	no	—
106	C₂³×C₃	—	2	4	no	—
107	C₂³×C₃	Dic₃⋊C₄	1	3	yes	—
108	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
109	C₂³×C₃	C₂×Dic₆	1	3	yes	—
110	C₂×C₃×C₄	Dic₃⋊C₄	1	6	yes	—
111	C₂×C₃×C₄	Dic₃⋊C₄	1	3	yes	—
112	C₂×C₃×C₄	Dic₃⋊C₄	1	3	yes	—
113	C₂×C₃×C₄	C₂×Dic₆	1	3	yes	—
114	C₂×C₃×C₄	C₂×C₄×D₃	1	3	yes	—
115	C₂×C₃×C₄	C₂×C₄×D₃	1	3	yes	—
116	C₂×C₃×C₄	C₂×#67	1	3	yes	—
117	C₂×C₃×C₄	C₂×#67	1	3	yes	—
118	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
119	C₃×C₈	—	2	4	no	—
120	C₃⋊C₈	—	3	8	no	—
121	C₃⋊C₈	—	6	4	no	—
122	C₂×C₉	—	2	3	no	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	C₂×#108	1	6	yes	complemented
125	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂⁴	Dic₁₈	1	1	yes	characteristic
127	C₂²×C₄	Dic₁₈	1	3	yes	—
128	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₉	1	3	yes	—
130	C₂²×C₄	#611	1	3	yes	—
131	C₂²×C₄	#611	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₉	1	3	yes	—
133	M₁₆	—	2	6	no	—
134	M₁₆	—	9	4	no	—
135	M₁₆	—	18	6	no	—
136	M₁₆	C₂²×D₉	1	4	yes	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	—	9	12	no	—
139	C₂×C₈	—	9	12	no	—
140	C₂×C₈	—	9	4	no	—
141	C₂×C₈	—	18	6	no	—
142	C₂²×C₃	—	2	6	no	—
143	C₂²×C₃	—	2	6	no	—
144	C₂²×C₃	#822	1	4	yes	—
145	C₂²×C₃	C₂×Dic₃⋊C₄	1	3	yes	—
146	C₂²×C₃	C₂×Dic₃⋊C₄	1	3	yes	—
147	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
148	C₃×C₄	C₂×Dic₃⋊C₄	1	3	yes	—
149	C₃×C₄	C₂×Dic₃⋊C₄	1	3	yes	—
150	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
151	C₃×C₄	C₂²×#67	1	1	yes	characteristic
152	C₉	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
153	C₂³	—	2	4	no	—
154	C₂³	Dic₉⋊C₄	1	3	yes	—
155	C₂³	Dic₉⋊C₄	1	1	yes	characteristic
156	C₂³	C₂×Dic₁₈	1	3	yes	—
157	C₂×C₄	Dic₉⋊C₄	1	6	yes	—
158	C₂×C₄	Dic₉⋊C₄	1	3	yes	—
159	C₂×C₄	Dic₉⋊C₄	1	3	yes	—
160	C₂×C₄	C₂×Dic₁₈	1	3	yes	—
161	C₂×C₄	C₂×C₄×D₉	1	3	yes	—
162	C₂×C₄	C₂×C₄×D₉	1	3	yes	—
163	C₂×C₄	C₂×#611	1	3	yes	—
164	C₂×C₄	C₂×#611	1	3	yes	—
165	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	—	9	8	no	—
168	C₈	—	18	4	no	—
169	C₂×C₃	—	2	3	no	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	C₂×#822	1	6	yes	—
172	C₂×C₃	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
173	C₂²	—	2	6	no	—
174	C₂²	—	2	6	no	—
175	C₂²	#63381	1	4	yes	direct product factor
176	C₂²	C₂×Dic₉⋊C₄	1	3	yes	—
177	C₂²	C₂×Dic₉⋊C₄	1	3	yes	—
178	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
179	C₄	C₂×Dic₉⋊C₄	1	3	yes	—
180	C₄	C₂×Dic₉⋊C₄	1	3	yes	—
181	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
182	C₄	C₂²×#611	1	1	yes	characteristic
183	C₃	C₂²×#822	1	1	yes	characteristic
184	C₂	—	2	1	no	—
185	C₂	—	2	3	no	—
186	C₂	C₂×#63381	1	6	yes	direct product factor
187	C₂	C₂²×Dic₉⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#611
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂³×C₃	Dic₃⋊C₄
prime radical	C₂⁴×C₃	Dic₆
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	C₂²×#108
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×#63381	C₂	direct (N × H)
16. #63381	C₂²	direct (N × H)
87. C₂²×C₉	#108	semidirect
124. C₂×C₉	C₂×#108	semidirect
152. C₉	C₂²×#108	semidirect
175. C₂²	#63381	direct (N × H)
186. C₂	C₂×#63381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×M₁₆	2
C₂²×#3667	2
C₂³×C₉⋊C₈	2
C₂×#63381	2
C₂²×#822	3

Subgroups — C22×#63381 #11858608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63381 #11858608