Subgroups — #11790335

Order 1152.

384 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63377	C₂	1	2	yes	maximal
3	#10588615	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
5	#10588229	C₂	1	2	yes	maximal
6	#70454	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₉	—	2	1	no	—
10	C₂×C₄⋊Dic₉	—	2	1	no	—
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₉	—	2	1	no	—
15	C₉×C₄⋊C₈	C₄	1	4	yes	complemented
16	C₉⋊#95	—	2	1	no	—
17	#63377	—	2	4	no	—
18	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
19	C₂×#817	—	3	2	no	—
20	#4935	—	3	1	no	—
21	C₄×C₄⋊Dic₃	—	3	1	no	—
22	#4551	—	3	2	no	—
23	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₉	—	2	1	no	—
27	C₂²×Dic₉	—	2	1	no	—
28	C₂²×Dic₉	—	2	2	no	—
29	C₂×C₈×C₉	—	2	2	no	—
30	C₂×C₈×C₉	—	2	2	no	—
31	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
32	C₄²×C₉	C₂×C₄	1	2	yes	—
33	C₄²×C₉	Q₈	1	2	yes	—
34	C₄⋊Dic₉	—	2	1	no	—
35	C₄⋊Dic₉	—	2	1	no	—
36	C₄⋊Dic₉	—	2	2	no	—
37	C₄⋊Dic₉	—	2	2	no	—
38	C₄⋊Dic₉	—	4	2	no	—
39	C₄⋊Dic₉	—	4	2	no	—
40	C₄⋊Dic₉	D₄	1	2	yes	—
41	C₄⋊Dic₉	Q₈	1	2	yes	—
42	C₄×Dic₉	—	4	2	no	—
43	#1948	—	9	1	no	Sylow
44	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₃	—	3	1	no	—
47	C₂×C₄⋊Dic₃	—	3	1	no	—
48	C₂×C₄⋊Dic₃	—	3	2	no	—
49	C₂×C₄⋊Dic₃	—	6	1	no	—
50	C₂×C₄⋊Dic₃	—	6	1	no	—
51	C₂×C₄×Dic₃	—	6	1	no	—
52	C₃×C₄⋊C₈	Dic₃	1	4	yes	—
53	C₃⋊#95	—	6	1	no	—
54	#817	—	6	4	no	—
55	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₉	—	2	1	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₉	SD₁₆	1	2	yes	—
61	C₂×C₄×C₉	SD₁₆	1	2	yes	—
62	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₉	#41	1	1	yes	characteristic
67	C₂×C₄×C₉	#41	1	1	yes	characteristic
68	C₂×Dic₉	—	2	2	no	—
69	C₂×Dic₉	—	2	2	no	—
70	C₂×Dic₉	—	4	1	no	—
71	C₂×Dic₉	—	4	1	no	—
72	C₂×Dic₉	—	4	1	no	—
73	C₂×Dic₉	—	4	1	no	—
74	C₂×Dic₉	—	4	2	no	—
75	C₂×Dic₉	—	4	2	no	—
76	C₂×Dic₉	—	4	2	no	—
77	C₈×C₉	—	2	4	no	—
78	C₂×#106	—	9	2	no	—
79	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
80	#384	—	9	1	no	—
81	C₄×C₄⋊C₄	—	9	1	no	—
82	#340	—	9	2	no	—
83	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
84	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₃	—	6	1	no	—
87	C₂²×Dic₃	—	6	1	no	—
88	C₂²×Dic₃	—	6	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
92	C₃×C₄²	Dic₆	1	2	yes	—
93	C₃×C₄²	C₂×Dic₃	1	2	yes	—
94	C₄⋊Dic₃	—	3	2	no	—
95	C₄⋊Dic₃	—	3	2	no	—
96	C₄⋊Dic₃	—	6	1	no	—
97	C₄⋊Dic₃	—	6	1	no	—
98	C₄⋊Dic₃	—	6	2	no	—
99	C₄⋊Dic₃	—	6	2	no	—
100	C₄⋊Dic₃	—	12	2	no	—
101	C₄⋊Dic₃	—	12	2	no	—
102	C₄×Dic₃	—	12	2	no	—
103	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₉	#121	1	1	yes	characteristic
105	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂²×C₉	#137	1	1	yes	characteristic
110	C₄×C₉	—	2	2	no	—
111	C₄×C₉	#106	1	4	yes	complemented
112	C₄×C₉	C₄×D₄	1	1	yes	characteristic
113	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
114	C₄×C₉	#125	1	1	yes	characteristic
115	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
116	Dic₉	—	4	2	no	—
117	Dic₉	—	4	2	no	—
118	Dic₉	—	8	2	no	—
119	C₂²×C₈	C₂×D₉	1	2	yes	—
120	C₂×C₄⋊C₄	—	9	1	no	—
121	C₂×C₄⋊C₄	—	9	1	no	—
122	C₂×C₄⋊C₄	—	9	2	no	—
123	C₂×C₄⋊C₄	—	18	1	no	—
124	C₂×C₄⋊C₄	—	18	1	no	—
125	C₂×C₄²	—	18	1	no	—
126	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
127	#106	—	18	4	no	—
128	C₄⋊C₈	Dic₉	1	4	yes	complemented
129	#95	—	18	1	no	—
130	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	#204	1	2	yes	—
134	C₂×C₃×C₄	#204	1	2	yes	—
135	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
136	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
137	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
138	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	#237	1	1	yes	characteristic
140	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
141	C₂×C₃×C₄	#239	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
143	C₂×Dic₃	—	6	2	no	—
144	C₂×Dic₃	—	6	2	no	—
145	C₂×Dic₃	—	12	1	no	—
146	C₂×Dic₃	—	12	1	no	—
147	C₂×Dic₃	—	12	1	no	—
148	C₂×Dic₃	—	12	1	no	—
149	C₂×Dic₃	—	12	2	no	—
150	C₂×Dic₃	—	12	2	no	—
151	C₂×Dic₃	—	12	2	no	—
152	C₃×C₈	—	2	4	no	—
153	C₂×C₉	#389	1	1	yes	characteristic
154	C₂×C₉	C₂×#106	1	1	yes	characteristic
155	C₂×C₉	#428	1	1	yes	characteristic
156	C₂×C₉	#460	1	1	yes	characteristic
157	C₂×C₉	#461	1	1	yes	characteristic
158	C₂×C₉	#475	1	1	yes	characteristic
159	C₂×C₉	#476	1	1	yes	characteristic
160	C₂²×C₄	—	18	1	no	—
161	C₂²×C₄	—	18	1	no	—
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	D₃₆	1	1	yes	characteristic
164	C₂²×C₄	D₃₆	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂×Dic₉	1	4	yes	—
169	C₄⋊C₄	—	9	2	no	—
170	C₄⋊C₄	—	9	2	no	—
171	C₄⋊C₄	—	18	1	no	—
172	C₄⋊C₄	—	18	1	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₄⋊C₄	—	18	2	no	—
175	C₄⋊C₄	—	36	2	no	—
176	C₄⋊C₄	—	36	2	no	—
177	C₄²	—	36	2	no	—
178	C₄²	Dic₁₈	1	2	yes	—
179	C₄²	C₂×Dic₉	1	2	yes	—
180	C₂²×C₃	#895	1	1	yes	characteristic
181	C₂²×C₃	#897	1	1	yes	characteristic
182	C₂²×C₃	C₂×#204	1	1	yes	characteristic
183	C₂²×C₃	C₂×#204	1	1	yes	characteristic
184	C₂²×C₃	#908	1	1	yes	characteristic
185	C₂²×C₃	#909	1	1	yes	characteristic
186	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	#817	1	4	yes	—
189	C₃×C₄	#888	1	1	yes	characteristic
190	C₃×C₄	#890	1	1	yes	characteristic
191	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
192	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
193	Dic₃	—	12	2	no	—
194	Dic₃	—	12	2	no	—
195	Dic₃	—	24	2	no	—
196	C₉	#1948	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×D₃₆	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	18	2	no	—
201	C₂×C₄	—	18	2	no	—
202	C₂×C₄	—	36	1	no	—
203	C₂×C₄	—	36	1	no	—
204	C₂×C₄	—	36	1	no	—
205	C₂×C₄	—	36	1	no	—
206	C₂×C₄	—	36	2	no	—
207	C₂×C₄	—	36	2	no	—
208	C₂×C₄	—	36	2	no	—
209	C₂×C₄	#3664	1	2	yes	—
210	C₂×C₄	#3664	1	2	yes	—
211	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
212	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
213	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
214	C₂×C₄	D₄×D₉	1	1	yes	characteristic
215	C₂×C₄	#3699	1	1	yes	characteristic
216	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
217	C₂×C₄	#3701	1	1	yes	characteristic
218	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₂×C₃	#4841	1	1	yes	characteristic
221	C₂×C₃	#4842	1	1	yes	characteristic
222	C₂×C₃	#4848	1	1	yes	characteristic
223	C₂×C₃	#4851	1	1	yes	characteristic
224	C₂×C₃	#4984	1	1	yes	characteristic
225	C₂×C₃	C₂×#817	1	1	yes	characteristic
226	C₂×C₃	#5121	1	1	yes	characteristic
227	C₂²	#63457	1	1	yes	characteristic
228	C₂²	#63459	1	1	yes	characteristic
229	C₂²	C₂×#3664	1	1	yes	characteristic
230	C₂²	C₂×#3664	1	1	yes	characteristic
231	C₂²	#63470	1	1	yes	characteristic
232	C₂²	#63471	1	1	yes	characteristic
233	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	36	2	no	—
236	C₄	—	36	2	no	—
237	C₄	—	72	2	no	—
238	C₄	#63377	1	4	yes	complemented
239	C₄	#63450	1	1	yes	characteristic
240	C₄	#63452	1	1	yes	characteristic
241	C₄	D₄×Dic₉	1	1	yes	characteristic
242	C₄	Q₈×Dic₉	1	1	yes	characteristic
243	C₃	#70454	1	1	yes	characteristic
244	C₂	#10588512	1	1	yes	characteristic
245	C₂	#10588515	1	1	yes	characteristic
246	C₂	#10588525	1	1	yes	characteristic
247	C₂	#10588526	1	1	yes	characteristic
248	C₂	#10588664	1	1	yes	characteristic
249	C₂	C₂×#63377	1	1	yes	characteristic
250	C₂	#10588805	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	#1948
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₉×C₄⋊C₈	C₄	semidirect
111. C₄×C₉	#106	semidirect
128. C₄⋊C₈	Dic₉	semidirect
196. C₉	#1948	semidirect
238. C₄	#63377	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⋊C₈	2
C₂×#63377	2
#10588615	2
C₄×C₄⋊Dic₉	2
#10588229	2
#70454	3