Subgroups — #11790613

Order 1152.

498 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	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63395	C₂	1	4	yes	maximal
3	#10588615	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#71785	—	3	1	no	maximal
6	C₂²×Q₈×C₉	C₂²	1	2	yes	—
7	C₉×C₄⋊Q₈	—	2	2	no	—
8	C₉×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₉×C₄⋊C₄	—	2	2	no	—
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	—	2	2	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
14	#63395	—	2	8	no	—
15	C₄×C₉⋊C₈	—	2	2	no	—
16	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
17	C₂×#835	—	3	4	no	—
18	#4935	—	3	1	no	—
19	C₂×C₄×C₃⋊C₈	—	3	1	no	—
20	C₂×Q₈×C₉	—	2	2	no	—
21	C₂×Q₈×C₉	—	2	4	no	—
22	C₂×Q₈×C₉	—	2	4	no	—
23	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₉	—	2	2	no	—
28	C₉×C₄⋊C₄	—	2	8	no	—
29	C₄²×C₉	C₂×C₄	1	2	yes	—
30	C₄²×C₉	D₄	1	2	yes	—
31	C₄⋊Dic₉	—	2	4	no	—
32	C₄⋊Dic₉	—	4	4	no	—
33	C₂×C₉⋊C₈	—	2	2	no	—
34	C₂×C₉⋊C₈	—	2	2	no	—
35	C₂×C₉⋊C₈	D₄	1	4	yes	—
36	#1956	—	9	1	no	Sylow
37	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
38	C₃×C₄⋊Q₈	—	2	2	no	—
39	C₃×C₄⋊Q₈	Dic₃	1	4	yes	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄⋊Dic₃	—	3	2	no	—
43	C₂×C₄⋊Dic₃	—	6	2	no	—
44	C₂²×C₃⋊C₈	—	3	2	no	—
45	#835	—	6	8	no	—
46	C₄×C₃⋊C₈	—	6	2	no	—
47	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₉	—	2	8	no	—
49	Q₈×C₉	—	4	4	no	—
50	C₂×C₄×C₉	—	2	4	no	—
51	C₂×C₄×C₉	—	4	2	no	—
52	C₂×C₄×C₉	—	4	2	no	—
53	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₉	SD₁₆	1	4	yes	—
55	C₂×C₄×C₉	Q₁₆	1	4	yes	—
56	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉	#41	1	2	yes	—
60	C₂×Dic₉	—	4	2	no	—
61	C₂×Dic₉	—	4	4	no	—
62	C₉⋊C₈	—	2	4	no	—
63	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
64	C₂×#103	—	9	4	no	—
65	C₂×C₄×C₈	—	9	1	no	—
66	#384	—	9	1	no	—
67	C₂×C₃×Q₈	—	2	2	no	—
68	C₂×C₃×Q₈	—	2	4	no	—
69	C₂×C₃×Q₈	—	2	4	no	—
70	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	#67	1	2	yes	—
73	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂²×Dic₃	—	6	2	no	—
75	C₃×C₄⋊C₄	—	2	8	no	—
76	C₃×C₄²	C₂×Dic₃	1	2	yes	—
77	C₃×C₄²	#67	1	2	yes	—
78	C₄⋊Dic₃	—	6	4	no	—
79	C₄⋊Dic₃	—	12	4	no	—
80	C₂×C₃⋊C₈	—	3	4	no	—
81	C₂×C₃⋊C₈	—	6	2	no	—
82	C₂×C₃⋊C₈	—	6	2	no	—
83	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₉	#124	1	1	yes	characteristic
85	C₂²×C₉	#127	1	1	yes	characteristic
86	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
87	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
88	C₄×C₉	—	4	4	no	—
89	C₄×C₉	#103	1	8	yes	complemented
90	C₄×C₉	C₄×D₄	1	2	yes	—
91	C₄×C₉	#121	1	2	yes	—
92	Dic₉	—	8	2	no	—
93	C₂²×Q₈	C₂×D₉	1	2	yes	—
94	C₂²×C₈	—	9	2	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₄⋊Q₈	Dic₉	1	4	yes	complemented
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	9	2	no	—
99	C₂×C₄⋊C₄	—	18	2	no	—
100	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
101	#103	—	18	8	no	—
102	C₄×C₈	—	18	2	no	—
103	C₂³×C₃	C₂×#67	1	1	yes	characteristic
104	C₃×Q₈	—	2	8	no	—
105	C₃×Q₈	—	4	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	4	2	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	#215	1	4	yes	—
110	C₂×C₃×C₄	#216	1	4	yes	—
111	C₂×C₃×C₄	#217	1	4	yes	—
112	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
113	C₂×C₃×C₄	#237	1	2	yes	—
114	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
116	C₂×Dic₃	—	12	2	no	—
117	C₂×Dic₃	—	12	4	no	—
118	C₃⋊C₈	—	6	4	no	—
119	C₂×C₉	#390	1	1	yes	characteristic
120	C₂×C₉	C₂×#103	1	2	yes	—
121	C₂×C₉	#487	1	2	yes	—
122	C₂×C₉	#492	1	1	yes	characteristic
123	C₂×C₉	#494	1	1	yes	characteristic
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	2	4	no	—
127	C₂×Q₈	C₂×Dic₉	1	4	yes	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	18	2	no	—
130	C₂²×C₄	#611	1	2	yes	—
131	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
132	C₂×C₈	—	9	4	no	—
133	C₂×C₈	—	18	2	no	—
134	C₂×C₈	—	18	2	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	18	4	no	—
137	C₄⋊C₄	—	36	4	no	—
138	C₄²	C₂×Dic₉	1	2	yes	—
139	C₄²	#611	1	2	yes	—
140	C₂²×C₃	#936	1	1	yes	characteristic
141	C₂²×C₃	#940	1	1	yes	characteristic
142	C₂²×C₃	C₂×#215	1	2	yes	—
143	C₂²×C₃	C₂×#216	1	2	yes	—
144	C₂²×C₃	C₂×#217	1	1	yes	characteristic
145	C₃×C₄	—	4	4	no	—
146	C₃×C₄	#835	1	8	yes	—
147	C₃×C₄	D₄×Dic₃	1	2	yes	—
148	C₃×C₄	#938	1	2	yes	—
149	Dic₃	—	24	2	no	—
150	C₉	#1956	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂×#611	1	1	yes	characteristic
152	Q₈	—	2	8	no	—
153	Q₈	—	4	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	36	2	no	—
158	C₂×C₄	—	36	4	no	—
159	C₂×C₄	#3674	1	4	yes	—
160	C₂×C₄	#3675	1	4	yes	—
161	C₂×C₄	#3676	1	4	yes	—
162	C₂×C₄	D₄×D₉	1	2	yes	—
163	C₂×C₄	#3699	1	2	yes	—
164	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
165	C₂×C₄	C₂×#611	1	1	yes	characteristic
166	C₈	—	18	4	no	—
167	C₂×C₃	#5079	1	2	yes	—
168	C₂×C₃	#5085	1	1	yes	characteristic
169	C₂×C₃	#5092	1	1	yes	characteristic
170	C₂×C₃	#5221	1	1	yes	characteristic
171	C₂×C₃	C₂×#835	1	2	yes	—
172	C₂²	#63498	1	1	yes	characteristic
173	C₂²	#63502	1	1	yes	characteristic
174	C₂²	C₂×#3674	1	2	yes	—
175	C₂²	C₂×#3675	1	2	yes	—
176	C₂²	C₂×#3676	1	1	yes	characteristic
177	C₄	—	4	4	no	—
178	C₄	—	72	2	no	—
179	C₄	#63395	1	8	yes	complemented
180	C₄	D₄×Dic₉	1	2	yes	—
181	C₄	#63500	1	2	yes	—
182	C₃	#71785	1	1	yes	characteristic
183	C₂	#10588759	1	2	yes	—
184	C₂	#10588765	1	1	yes	characteristic
185	C₂	#10588770	1	1	yes	characteristic
186	C₂	#10588901	1	1	yes	characteristic
187	C₂	C₂×#63395	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	#1956
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
8. C₉×C₄⋊Q₈	C₄	semidirect
89. C₄×C₉	#103	semidirect
96. C₄⋊Q₈	Dic₉	semidirect
150. C₉	#1956	semidirect
179. C₄	#63395	semidirect

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×#63395	2
#10588615	2
C₂×C₄×C₉⋊C₈	2
#71785	3