Subgroups — #11791426

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#71778	—	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	1	no	—
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂×C₄×Dic₉	—	2	1	no	—
14	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
15	#63395	—	2	8	no	—
16	C₉⋊#95	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	—	2	2	no	—
18	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	C₂×#835	—	3	4	no	—
20	C₄×C₄⋊Dic₃	—	3	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₂×Q₈×C₉	—	2	2	no	—
23	C₂×Q₈×C₉	—	2	4	no	—
24	C₂×Q₈×C₉	—	2	4	no	—
25	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₉	—	2	2	no	—
31	C₉×C₄⋊C₄	—	2	4	no	—
32	C₉×C₄⋊C₄	—	2	4	no	—
33	C₄²×C₉	C₂×C₄	1	2	yes	—
34	C₄²×C₉	D₄	1	2	yes	—
35	C₄⋊Dic₉	—	2	2	no	—
36	C₄⋊Dic₉	D₄	1	4	yes	—
37	C₄×Dic₉	—	4	2	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	4	no	—
40	#1981	—	9	1	no	Sylow
41	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
42	C₃×C₄⋊Q₈	—	2	2	no	—
43	C₃×C₄⋊Q₈	Dic₃	1	4	yes	—
44	C₂×C₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₂×C₄⋊Dic₃	—	3	2	no	—
48	C₂×C₄×Dic₃	—	6	1	no	—
49	C₂²×C₃⋊C₈	—	3	2	no	—
50	#835	—	6	8	no	—
51	C₃⋊#95	—	6	1	no	—
52	C₄⋊(C₃⋊C₈)	—	6	2	no	—
53	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₉	—	2	8	no	—
55	Q₈×C₉	—	4	2	no	—
56	Q₈×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	4	no	—
59	C₂×C₄×C₉	—	4	4	no	—
60	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₉	SD₁₆	1	2	yes	—
63	C₂×C₄×C₉	Q₁₆	1	2	yes	—
64	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₉	#41	1	2	yes	—
68	C₂×Dic₉	—	2	4	no	—
69	C₂×Dic₉	—	4	2	no	—
70	C₂×Dic₉	—	4	2	no	—
71	C₉⋊C₈	—	4	2	no	—
72	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	9	1	no	—
74	C₂×#103	—	9	4	no	—
75	C₄×C₄⋊C₄	—	9	1	no	—
76	C₂×C₃×Q₈	—	2	2	no	—
77	C₂×C₃×Q₈	—	2	4	no	—
78	C₂×C₃×Q₈	—	2	4	no	—
79	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	#67	1	1	yes	characteristic
82	C₂²×C₃×C₄	#67	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×Dic₃	—	6	2	no	—
85	C₃×C₄⋊C₄	—	2	4	no	—
86	C₃×C₄⋊C₄	—	2	4	no	—
87	C₃×C₄²	C₂×Dic₃	1	2	yes	—
88	C₃×C₄²	#67	1	2	yes	—
89	C₄⋊Dic₃	—	3	4	no	—
90	C₄⋊Dic₃	—	6	2	no	—
91	C₄×Dic₃	—	12	2	no	—
92	C₂×C₃⋊C₈	—	6	4	no	—
93	C₂×C₃⋊C₈	—	6	2	no	—
94	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₉	#121	1	2	yes	—
96	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂²×C₉	#137	1	2	yes	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	4	4	no	—
101	C₄×C₉	#103	1	4	yes	complemented
102	C₄×C₉	C₄×D₄	1	2	yes	—
103	C₄×C₉	#124	1	1	yes	characteristic
104	C₄×C₉	#127	1	1	yes	characteristic
105	Dic₉	—	4	4	no	—
106	C₂²×Q₈	C₂×D₉	1	2	yes	—
107	C₂²×C₈	—	9	2	no	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	Dic₉	1	4	yes	complemented
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	9	2	no	—
113	C₂×C₄²	—	18	1	no	—
114	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
115	C₄⋊C₈	—	18	2	no	—
116	#103	—	18	8	no	—
117	#95	—	18	1	no	—
118	C₂³×C₃	C₂×#67	1	1	yes	characteristic
119	C₃×Q₈	—	2	8	no	—
120	C₃×Q₈	—	4	2	no	—
121	C₃×Q₈	—	4	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	4	4	no	—
125	C₂×C₃×C₄	#215	1	2	yes	—
126	C₂×C₃×C₄	#216	1	2	yes	—
127	C₂×C₃×C₄	#217	1	2	yes	—
128	C₂×C₃×C₄	#217	1	2	yes	—
129	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
130	C₂×C₃×C₄	#237	1	2	yes	—
131	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
133	C₂×Dic₃	—	6	4	no	—
134	C₂×Dic₃	—	12	2	no	—
135	C₂×Dic₃	—	12	2	no	—
136	C₃⋊C₈	—	12	2	no	—
137	C₂×C₉	#390	1	1	yes	characteristic
138	C₂×C₉	C₂×#103	1	1	yes	characteristic
139	C₂×C₉	#419	1	1	yes	characteristic
140	C₂×C₉	#461	1	2	yes	—
141	C₂×C₉	#462	1	2	yes	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	C₂×Dic₉	1	4	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	18	2	no	—
148	C₂²×C₄	#611	1	1	yes	characteristic
149	C₂²×C₄	#611	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
151	C₂×C₈	—	18	4	no	—
152	C₂×C₈	—	18	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	9	4	no	—
156	C₄⋊C₄	—	18	2	no	—
157	C₄²	—	36	2	no	—
158	C₄²	C₂×Dic₉	1	2	yes	—
159	C₄²	#611	1	2	yes	—
160	C₂²×C₃	#938	1	2	yes	—
161	C₂²×C₃	C₂×#215	1	1	yes	characteristic
162	C₂²×C₃	#942	1	2	yes	—
163	C₂²×C₃	C₂×#216	1	1	yes	characteristic
164	C₂²×C₃	C₂×#217	1	1	yes	characteristic
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	4	4	no	—
167	C₃×C₄	#835	1	4	yes	—
168	C₃×C₄	D₄×Dic₃	1	2	yes	—
169	C₃×C₄	#936	1	1	yes	characteristic
170	C₃×C₄	#940	1	1	yes	characteristic
171	Dic₃	—	12	4	no	—
172	C₉	#1981	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₂×#611	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	Q₈	—	4	2	no	—
176	Q₈	—	4	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	18	4	no	—
181	C₂×C₄	—	36	2	no	—
182	C₂×C₄	—	36	2	no	—
183	C₂×C₄	#3674	1	2	yes	—
184	C₂×C₄	#3675	1	2	yes	—
185	C₂×C₄	#3676	1	2	yes	—
186	C₂×C₄	#3676	1	2	yes	—
187	C₂×C₄	D₄×D₉	1	2	yes	—
188	C₂×C₄	#3699	1	2	yes	—
189	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
190	C₂×C₄	C₂×#611	1	1	yes	characteristic
191	C₈	—	36	2	no	—
192	C₂×C₃	#5083	1	2	yes	—
193	C₂×C₃	#5090	1	2	yes	—
194	C₂×C₃	#5221	1	1	yes	characteristic
195	C₂×C₃	C₂×#835	1	1	yes	characteristic
196	C₂×C₃	#5225	1	1	yes	characteristic
197	C₂²	#63500	1	2	yes	—
198	C₂²	C₂×#3674	1	1	yes	characteristic
199	C₂²	C₂×#3675	1	1	yes	characteristic
200	C₂²	#63505	1	2	yes	—
201	C₂²	C₂×#3676	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	36	4	no	—
205	C₄	#63395	1	4	yes	complemented
206	C₄	D₄×Dic₉	1	2	yes	—
207	C₄	#63498	1	1	yes	characteristic
208	C₄	#63502	1	1	yes	characteristic
209	C₃	#71778	1	1	yes	characteristic
210	C₂	#10588763	1	2	yes	—
211	C₂	#10588768	1	2	yes	—
212	C₂	#10588901	1	1	yes	characteristic
213	C₂	C₂×#63395	1	1	yes	characteristic
214	C₂	#10588906	1	1	yes	characteristic

	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	#1981
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
101. C₄×C₉	#103	semidirect
109. C₄⋊Q₈	Dic₉	semidirect
172. C₉	#1981	semidirect
205. 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
C₄×C₄⋊Dic₉	2
C₂×C₄⋊(C₉⋊C₈)	2
#71778	3