Subgroups — #11807116

Order 1152.

744 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	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10588909	C₂	1	4	yes	maximal
3	#10588664	C₂	1	4	yes	maximal
4	C₄⋊C₄×Dic₉	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₉	C₂	1	2	yes	maximal
6	#83230	—	3	1	no	maximal
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₉×C₄⋊Q₈	C₄	1	8	yes	complemented
9	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
14	C₂×C₄×Dic₉	C₂²	1	4	yes	—
15	C₂×C₄×Dic₉	C₂²	1	4	yes	—
16	C₉⋊#95	C₂²	1	4	yes	—
17	C₉⋊#95	C₂²	1	8	yes	—
18	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	#5229	—	3	4	no	—
20	#4984	—	3	4	no	—
21	Dic₃×C₄⋊C₄	—	3	4	no	—
22	C₄×C₄⋊Dic₃	—	3	2	no	—
23	C₂×Q₈×C₉	—	4	2	no	—
24	C₂×Q₈×C₉	—	4	2	no	—
25	C₂×Q₈×C₉	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	4	yes	—
28	C₂²×C₄×C₉	C₂³	1	2	yes	—
29	C₂²×Dic₉	C₂³	1	4	yes	—
30	C₂²×Dic₉	C₂³	1	4	yes	—
31	C₉×C₄⋊C₄	C₂×C₄	1	16	yes	—
32	C₄²×C₉	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₉	—	2	4	no	—
34	C₄⋊Dic₉	—	4	4	no	—
35	C₄⋊Dic₉	Q₈	1	8	yes	complemented
36	C₄×Dic₉	—	2	8	no	—
37	C₄×Dic₉	—	4	4	no	—
38	#2369	—	9	1	no	Sylow
39	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
40	C₃×C₄⋊Q₈	Dic₃	1	8	yes	—
41	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₃	—	3	2	no	—
44	C₂×C₄⋊Dic₃	—	3	2	no	—
45	C₂×C₄⋊Dic₃	—	3	4	no	—
46	C₂×C₄×Dic₃	—	3	4	no	—
47	C₂×C₄×Dic₃	—	3	4	no	—
48	C₃⋊#95	—	3	4	no	—
49	C₃⋊#95	—	3	8	no	—
50	C₂³×C₉	C₂⁴	1	1	yes	characteristic
51	Q₈×C₉	—	4	8	no	—
52	C₂×C₄×C₉	—	2	4	no	—
53	C₂×C₄×C₉	—	2	4	no	—
54	C₂×C₄×C₉	—	2	1	no	—
55	C₂×C₄×C₉	—	2	1	no	—
56	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
58	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₉	#41	1	4	yes	—
61	C₂×Dic₉	—	2	4	no	—
62	C₂×Dic₉	—	2	4	no	—
63	C₂×Dic₉	—	2	4	no	—
64	C₂×Dic₉	—	2	4	no	—
65	C₂×Dic₉	—	2	4	no	—
66	C₂×Dic₉	C₂×Q₈	1	8	yes	—
67	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
68	#391	—	9	4	no	—
69	#389	—	9	4	no	—
70	C₄×C₄⋊C₄	—	9	4	no	—
71	C₄×C₄⋊C₄	—	9	2	no	—
72	C₂×C₃×Q₈	—	4	2	no	—
73	C₂×C₃×Q₈	—	4	2	no	—
74	C₂×C₃×Q₈	C₂×Dic₃	1	8	yes	—
75	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
77	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
78	C₂²×Dic₃	—	3	4	no	—
79	C₂²×Dic₃	—	3	4	no	—
80	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
81	C₃×C₄²	C₂×Dic₃	1	4	yes	—
82	C₄⋊Dic₃	—	3	8	no	—
83	C₄⋊Dic₃	—	6	4	no	—
84	C₄⋊Dic₃	—	12	4	no	—
85	C₄×Dic₃	—	6	8	no	—
86	C₄×Dic₃	—	12	4	no	—
87	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
88	C₂²×C₉	C₂²×Q₈	1	2	yes	—
89	C₂²×C₉	C₂×#41	1	2	yes	—
90	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₉	—	2	8	no	—
93	C₄×C₉	—	2	2	no	—
94	C₄×C₉	C₄×Q₈	1	8	yes	—
95	Dic₉	—	2	8	no	—
96	Dic₉	—	4	4	no	—
97	C₂²×Q₈	C₂×D₉	1	2	yes	—
98	C₄⋊Q₈	Dic₉	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	9	2	no	—
100	C₂×C₄⋊C₄	—	9	2	no	—
101	C₂×C₄⋊C₄	—	9	4	no	—
102	C₂×C₄⋊C₄	C₂×D₉	1	4	yes	—
103	C₂×C₄²	—	9	4	no	—
104	C₂×C₄²	—	9	4	no	—
105	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
106	#95	—	9	8	no	—
107	#95	—	9	4	no	—
108	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
109	C₃×Q₈	—	4	8	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
115	C₂×C₃×C₄	#239	1	4	yes	—
116	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
117	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
118	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
119	C₂×Dic₃	—	3	8	no	—
120	C₂×Dic₃	—	6	4	no	—
121	C₂×Dic₃	—	6	4	no	—
122	C₂×Dic₃	—	6	4	no	—
123	C₂×Dic₃	—	6	4	no	—
124	C₂×Dic₃	—	6	4	no	—
125	C₂×C₉	C₂×C₄×Q₈	1	2	yes	—
126	C₂×C₉	#518	1	1	yes	characteristic
127	C₂×C₉	#554	1	2	yes	—
128	C₂×C₉	#555	1	1	yes	characteristic
129	C₂×C₉	Q₈²	1	1	yes	characteristic
130	C₂×Q₈	—	4	2	no	—
131	C₂×Q₈	—	4	2	no	—
132	C₂×Q₈	C₂×Dic₉	1	8	yes	—
133	C₂²×C₄	—	9	4	no	—
134	C₂²×C₄	—	9	4	no	—
135	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₉	1	4	yes	—
137	C₂²×C₄	C₂²×D₉	1	2	yes	—
138	C₄⋊C₄	—	9	8	no	—
139	C₄⋊C₄	—	18	4	no	—
140	C₄⋊C₄	—	36	4	no	—
141	C₄⋊C₄	C₂×Dic₉	1	16	yes	—
142	C₄²	—	18	8	no	—
143	C₄²	—	36	4	no	—
144	C₄²	C₂×Dic₉	1	4	yes	—
145	C₂²×C₃	#1004	1	1	yes	characteristic
146	C₂²×C₃	#1004	1	1	yes	characteristic
147	C₂²×C₃	C₂×D₃×Q₈	1	2	yes	—
148	C₂²×C₃	C₂×#239	1	2	yes	—
149	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
150	C₃×C₄	—	2	2	no	—
151	C₃×C₄	—	2	8	no	—
152	C₃×C₄	Q₈×Dic₃	1	8	yes	—
153	Dic₃	—	6	8	no	—
154	Dic₃	—	12	4	no	—
155	C₉	#2369	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂³×D₉	1	1	yes	characteristic
157	Q₈	—	4	8	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	9	8	no	—
163	C₂×C₄	—	18	4	no	—
164	C₂×C₄	—	18	4	no	—
165	C₂×C₄	—	18	4	no	—
166	C₂×C₄	—	18	4	no	—
167	C₂×C₄	—	18	4	no	—
168	C₂×C₄	Q₈×D₉	1	4	yes	—
169	C₂×C₄	#3701	1	4	yes	—
170	C₂×C₄	C₂²×Dic₉	1	2	yes	—
171	C₂×C₄	C₂²×Dic₉	1	8	yes	—
172	C₂×C₄	C₂²×Dic₉	1	4	yes	—
173	C₂×C₃	#5722	1	1	yes	characteristic
174	C₂×C₃	#5730	1	2	yes	—
175	C₂×C₃	#5734	1	1	yes	characteristic
176	C₂×C₃	#5798	1	1	yes	characteristic
177	C₂×C₃	C₂×Q₈×Dic₃	1	2	yes	—
178	C₂²	#63710	1	1	yes	characteristic
179	C₂²	#63710	1	1	yes	characteristic
180	C₂²	C₂×Q₈×D₉	1	2	yes	—
181	C₂²	C₂×#3701	1	2	yes	—
182	C₂²	C₂³×Dic₉	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	8	no	—
185	C₄	—	18	8	no	—
186	C₄	—	36	4	no	—
187	C₄	Q₈×Dic₉	1	8	yes	—
188	C₃	#83230	1	1	yes	characteristic
189	C₂	#10589824	1	1	yes	characteristic
190	C₂	#10589832	1	2	yes	—
191	C₂	#10589836	1	1	yes	characteristic
192	C₂	#10589900	1	1	yes	characteristic
193	C₂	C₂×Q₈×Dic₉	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle, prime radical	C₂³×C₃	C₂³×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	#2369
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ 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
35. C₄⋊Dic₉	Q₈	semidirect
98. C₄⋊Q₈	Dic₉	semidirect
155. C₉	#2369	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
#10588909	2
#10588664	2
C₄⋊C₄×Dic₉	2
C₄×C₄⋊Dic₉	2
#83230	3