Subgroups — #11790027

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#10588200	C₂	1	2	yes	maximal
6	#69484	—	3	1	no	maximal
7	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₉×#95	C₄	1	2	yes	—
15	C₄⋊(C₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₉⋊C₈)	—	2	4	no	—
17	C₄×C₉⋊C₈	—	2	2	no	—
18	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊C₈)	—	3	2	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₄×C₃⋊C₈	—	3	1	no	—
22	#4524	—	3	2	no	—
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₉×C₄⋊C₄	—	2	2	no	—
29	C₉×C₄⋊C₄	C₈	1	4	yes	complemented
30	C₄²×C₉	—	2	2	no	—
31	C₄²×C₉	—	2	2	no	—
32	C₄²×C₉	D₄	1	2	yes	—
33	C₄²×C₉	Q₈	1	2	yes	—
34	C₂×C₉⋊C₈	—	2	1	no	—
35	C₂×C₉⋊C₈	—	2	1	no	—
36	C₂×C₉⋊C₈	—	2	1	no	—
37	C₂×C₉⋊C₈	—	2	1	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	2	no	—
41	C₂×C₉⋊C₈	D₄	1	2	yes	—
42	C₂×C₉⋊C₈	Q₈	1	2	yes	—
43	#1939	—	9	1	no	Sylow
44	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
45	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
47	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃⋊C₈	—	3	1	no	—
49	C₂²×C₃⋊C₈	—	3	1	no	—
50	C₂²×C₃⋊C₈	—	3	2	no	—
51	C₃×#95	Dic₃	1	2	yes	—
52	C₄⋊(C₃⋊C₈)	—	6	4	no	—
53	C₄⋊(C₃⋊C₈)	—	6	2	no	—
54	C₄×C₃⋊C₈	—	6	2	no	—
55	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₉	—	2	2	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
66	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
67	C₂×C₄×C₉	M₁₆	1	2	yes	—
68	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	1	yes	characteristic
74	C₉⋊C₈	—	2	2	no	—
75	C₉⋊C₈	—	2	2	no	—
76	C₉⋊C₈	—	4	2	no	—
77	C₂×C₄⋊C₈	—	9	2	no	—
78	C₂×C₄⋊C₈	—	9	1	no	—
79	C₂×C₄×C₈	—	9	1	no	—
80	C₄×C₄⋊C₄	D₉	1	1	yes	characteristic
81	#336	—	9	2	no	—
82	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
83	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	C₃⋊C₈	1	4	yes	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	Dic₆	1	2	yes	—
92	C₃×C₄²	D₁₂	1	2	yes	—
93	C₂×C₃⋊C₈	—	3	2	no	—
94	C₂×C₃⋊C₈	—	3	2	no	—
95	C₂×C₃⋊C₈	—	6	1	no	—
96	C₂×C₃⋊C₈	—	6	1	no	—
97	C₂×C₃⋊C₈	—	6	1	no	—
98	C₂×C₃⋊C₈	—	6	1	no	—
99	C₂×C₃⋊C₈	—	6	2	no	—
100	C₂×C₃⋊C₈	—	6	2	no	—
101	C₂×C₃⋊C₈	—	6	2	no	—
102	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
105	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
106	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂²×C₉	#131	1	1	yes	characteristic
108	C₂²×C₉	#131	1	1	yes	characteristic
109	C₄×C₉	—	2	2	no	—
110	C₄×C₉	—	2	4	no	—
111	C₄×C₉	—	4	2	no	—
112	C₄×C₉	C₄⋊C₈	1	4	yes	complemented
113	C₄×C₉	#121	1	1	yes	characteristic
114	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₄×C₉	#125	1	1	yes	characteristic
116	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₈	—	9	1	no	—
118	C₂²×C₈	—	9	1	no	—
119	C₂²×C₈	—	9	2	no	—
120	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₉	1	2	yes	—
123	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
124	C₄⋊C₈	—	18	4	no	—
125	C₄⋊C₈	—	18	2	no	—
126	C₄×C₈	—	18	2	no	—
127	#95	Dic₉	1	2	yes	—
128	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
137	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
138	C₂×C₃×C₄	#208	1	2	yes	—
139	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
140	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
141	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
143	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄	#237	1	1	yes	characteristic
145	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
146	C₂×C₃×C₄	#239	1	1	yes	characteristic
147	C₃⋊C₈	—	6	2	no	—
148	C₃⋊C₈	—	6	2	no	—
149	C₃⋊C₈	—	12	2	no	—
150	C₂×C₉	#389	1	1	yes	characteristic
151	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₉	#424	1	1	yes	characteristic
153	C₂×C₉	C₈×D₄	1	1	yes	characteristic
154	C₂×C₉	#436	1	1	yes	characteristic
155	C₂×C₉	C₈×Q₈	1	1	yes	characteristic
156	C₂×C₉	#446	1	1	yes	characteristic
157	C₂²×C₄	C₂×Dic₉	1	2	yes	—
158	C₂²×C₄	C₂×Dic₉	1	2	yes	—
159	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
160	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
162	C₂×C₈	—	9	2	no	—
163	C₂×C₈	—	9	2	no	—
164	C₂×C₈	—	18	1	no	—
165	C₂×C₈	—	18	1	no	—
166	C₂×C₈	—	18	1	no	—
167	C₂×C₈	—	18	1	no	—
168	C₂×C₈	—	18	2	no	—
169	C₂×C₈	—	18	2	no	—
170	C₂×C₈	—	18	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	C₉⋊C₈	1	4	yes	complemented
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	Dic₁₈	1	2	yes	—
176	C₄²	D₃₆	1	2	yes	—
177	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
178	C₂²×C₃	C₂×#208	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
180	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
181	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
182	C₂²×C₃	#948	1	1	yes	characteristic
183	C₂²×C₃	#948	1	1	yes	characteristic
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
188	C₃×C₄	#888	1	1	yes	characteristic
189	C₃×C₄	#890	1	1	yes	characteristic
190	C₃×C₄	#895	1	1	yes	characteristic
191	C₃×C₄	#897	1	1	yes	characteristic
192	C₉	#1939	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂²×Dic₉	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	C₂×C₉⋊C₈	1	4	yes	—
202	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
203	C₂×C₄	#3667	1	2	yes	—
204	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
205	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
206	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
207	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
208	C₂×C₄	D₄×D₉	1	1	yes	characteristic
209	C₂×C₄	#3699	1	1	yes	characteristic
210	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
211	C₂×C₄	#3701	1	1	yes	characteristic
212	C₈	—	18	2	no	—
213	C₈	—	18	2	no	—
214	C₈	—	36	2	no	—
215	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
216	C₂×C₃	#4984	1	1	yes	characteristic
217	C₂×C₃	#4999	1	1	yes	characteristic
218	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
219	C₂×C₃	#5012	1	1	yes	characteristic
220	C₂×C₃	Q₈×C₃⋊C₈	1	1	yes	characteristic
221	C₂×C₃	#5024	1	1	yes	characteristic
222	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
223	C₂²	C₂×#3667	1	1	yes	characteristic
224	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
225	C₂²	D₄×Dic₉	1	1	yes	characteristic
226	C₂²	Q₈×Dic₉	1	1	yes	characteristic
227	C₂²	#63510	1	1	yes	characteristic
228	C₂²	#63510	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	C₄⋊(C₉⋊C₈)	1	4	yes	complemented
233	C₄	#63450	1	1	yes	characteristic
234	C₄	#63452	1	1	yes	characteristic
235	C₄	#63457	1	1	yes	characteristic
236	C₄	#63459	1	1	yes	characteristic
237	C₃	#69484	1	1	yes	characteristic
238	C₂	C₂×C₄⋊(C₉⋊C₈)	1	1	yes	characteristic
239	C₂	#10588664	1	1	yes	characteristic
240	C₂	#10588677	1	1	yes	characteristic
241	C₂	D₄×C₉⋊C₈	1	1	yes	characteristic
242	C₂	#10588689	1	1	yes	characteristic
243	C₂	Q₈×C₉⋊C₈	1	1	yes	characteristic
244	C₂	#10588701	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂²×Dic₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₉
hypocenter	C₉	#1939
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
29. C₉×C₄⋊C₄	C₈	semidirect
112. C₄×C₉	C₄⋊C₈	semidirect
172. C₄⋊C₄	C₉⋊C₈	semidirect
192. C₉	#1939	semidirect
232. C₄	C₄⋊(C₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×C₄⋊C₄	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×C₄×C₉⋊C₈	2
#10588200	2
#69484	3