Subgroups — #11785276

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#63376	C₂	1	2	yes	maximal
4	#10588616	C₂	1	1	yes	maximal, characteristic
5	#10588229	C₂	1	2	yes	maximal
6	#70486	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₉⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₉	—	2	1	no	—
14	C₄×Dic₁₈	—	2	2	no	—
15	C₄×Dic₁₈	C₄	1	4	yes	complemented
16	C₉×C₄⋊C₈	—	2	2	no	—
17	C₉⋊#95	—	2	1	no	—
18	C₉⋊#95	—	2	1	no	—
19	#63376	—	2	4	no	—
20	C₂×C₄×Dic₆	—	3	1	no	—
21	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
22	C₂×#816	—	3	2	no	—
23	#4936	—	3	1	no	—
24	#4551	—	3	2	no	—
25	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₉	—	2	1	no	—
29	C₂²×Dic₉	—	2	1	no	—
30	C₂²×Dic₉	—	2	2	no	—
31	C₂×Dic₁₈	—	2	1	no	—
32	C₂×Dic₁₈	—	2	1	no	—
33	C₂×Dic₁₈	—	2	1	no	—
34	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₈	D₄	1	2	yes	—
36	C₂×Dic₁₈	Q₈	1	2	yes	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₄²×C₉	C₂×C₄	1	2	yes	—
41	C₄²×C₉	D₄	1	2	yes	—
42	C₄⋊Dic₉	—	2	1	no	—
43	C₄⋊Dic₉	—	2	2	no	—
44	C₄⋊Dic₉	—	2	2	no	—
45	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
46	Dic₉⋊C₄	—	2	4	no	—
47	C₄×Dic₉	—	2	4	no	—
48	#1805	—	9	1	no	Sylow
49	C₂²×Dic₆	—	3	1	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₃	—	3	1	no	—
53	C₂×C₄⋊Dic₃	—	3	2	no	—
54	C₂×Dic₃⋊C₄	—	6	1	no	—
55	C₂×C₄×Dic₃	—	6	1	no	—
56	C₄×Dic₆	—	3	4	no	—
57	C₄×Dic₆	—	6	2	no	—
58	C₃×C₄⋊C₈	—	2	2	no	—
59	C₃⋊#95	—	6	1	no	—
60	C₃⋊#95	—	6	1	no	—
61	#816	—	6	4	no	—
62	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	1	yes	characteristic
73	C₂×Dic₉	—	2	2	no	—
74	C₂×Dic₉	—	2	2	no	—
75	C₂×Dic₉	—	2	2	no	—
76	C₂×Dic₉	—	2	2	no	—
77	C₂×Dic₉	—	4	2	no	—
78	C₂×Dic₉	—	4	2	no	—
79	C₂×Dic₉	—	4	2	no	—
80	C₂×Dic₉	—	4	2	no	—
81	Dic₁₈	—	2	2	no	—
82	Dic₁₈	—	2	2	no	—
83	Dic₁₈	—	2	2	no	—
84	Dic₁₈	C₄⋊C₄	1	4	yes	—
85	C₈×C₉	—	4	2	no	—
86	C₂×C₄×Q₈	—	9	1	no	—
87	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
88	C₂×#103	—	9	2	no	—
89	#382	—	9	1	no	—
90	#340	—	9	2	no	—
91	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
92	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₃	—	6	1	no	—
95	C₂²×Dic₃	—	6	1	no	—
96	C₂²×Dic₃	—	6	2	no	—
97	C₂×Dic₆	—	3	2	no	—
98	C₂×Dic₆	—	3	2	no	—
99	C₂×Dic₆	—	3	2	no	—
100	C₂×Dic₆	—	6	1	no	—
101	C₂×Dic₆	—	6	1	no	—
102	C₂×Dic₆	—	6	1	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₃×C₄²	C₄×D₃	1	2	yes	—
107	C₃×C₄²	#67	1	2	yes	—
108	C₄⋊Dic₃	—	3	2	no	—
109	C₄⋊Dic₃	—	6	1	no	—
110	C₄⋊Dic₃	—	6	2	no	—
111	C₄⋊Dic₃	—	6	2	no	—
112	Dic₃⋊C₄	—	6	4	no	—
113	C₄×Dic₃	—	6	4	no	—
114	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₉	#121	1	1	yes	characteristic
116	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂²×C₉	#135	1	2	yes	—
118	C₂²×C₉	#137	1	2	yes	—
119	C₄×C₉	—	2	2	no	—
120	C₄×C₉	—	2	2	no	—
121	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₉	#117	1	1	yes	characteristic
123	C₄×C₉	#121	1	1	yes	characteristic
124	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
125	Dic₉	—	2	4	no	—
126	Dic₉	—	4	2	no	—
127	Dic₉	—	8	2	no	—
128	C₂²×Q₈	—	9	1	no	—
129	C₂²×C₈	C₂×D₉	1	2	yes	—
130	C₄×Q₈	—	9	4	no	—
131	C₄×Q₈	—	18	2	no	—
132	C₂×C₄⋊C₄	—	9	1	no	—
133	C₂×C₄⋊C₄	—	9	2	no	—
134	C₂×C₄⋊C₄	—	18	1	no	—
135	C₂×C₄²	—	18	1	no	—
136	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	#103	—	18	4	no	—
139	#95	—	18	1	no	—
140	#95	—	18	1	no	—
141	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
145	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	#237	1	1	yes	characteristic
149	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
150	C₂×C₃×C₄	#239	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
152	C₂×Dic₃	—	6	2	no	—
153	C₂×Dic₃	—	6	2	no	—
154	C₂×Dic₃	—	6	2	no	—
155	C₂×Dic₃	—	6	2	no	—
156	C₂×Dic₃	—	12	2	no	—
157	C₂×Dic₃	—	12	2	no	—
158	C₂×Dic₃	—	12	2	no	—
159	C₂×Dic₃	—	12	2	no	—
160	Dic₆	—	3	4	no	—
161	Dic₆	—	6	2	no	—
162	Dic₆	—	6	2	no	—
163	Dic₆	—	6	2	no	—
164	C₃×C₈	—	4	2	no	—
165	C₂×C₉	#380	1	1	yes	characteristic
166	C₂×C₉	#416	1	1	yes	characteristic
167	C₂×C₉	#419	1	1	yes	characteristic
168	C₂×C₉	#464	1	2	yes	—
169	C₂×C₉	#479	1	2	yes	—
170	C₂×Q₈	—	9	2	no	—
171	C₂×Q₈	—	9	2	no	—
172	C₂×Q₈	—	9	2	no	—
173	C₂×Q₈	—	18	1	no	—
174	C₂×Q₈	—	18	1	no	—
175	C₂×Q₈	—	18	1	no	—
176	C₂²×C₄	—	18	1	no	—
177	C₂²×C₄	—	18	1	no	—
178	C₂²×C₄	—	18	2	no	—
179	C₂²×C₄	D₃₆	1	1	yes	characteristic
180	C₂²×C₄	D₃₆	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₄⋊C₄	—	9	2	no	—
186	C₄⋊C₄	—	18	4	no	—
187	C₄⋊C₄	—	18	1	no	—
188	C₄⋊C₄	—	18	2	no	—
189	C₄⋊C₄	—	18	2	no	—
190	C₄²	—	18	4	no	—
191	C₄²	C₄×D₉	1	2	yes	—
192	C₄²	#611	1	2	yes	—
193	C₂²×C₃	#895	1	1	yes	characteristic
194	C₂²×C₃	#897	1	1	yes	characteristic
195	C₂²×C₃	#904	1	2	yes	—
196	C₂²×C₃	#909	1	2	yes	—
197	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
201	C₃×C₄	#892	1	1	yes	characteristic
202	C₃×C₄	#938	1	1	yes	characteristic
203	C₃×C₄	#946	1	1	yes	characteristic
204	Dic₃	—	6	4	no	—
205	Dic₃	—	12	2	no	—
206	Dic₃	—	24	2	no	—
207	C₉	#1805	1	1	yes	characteristic, complemented, Sylow
208	C₂³	C₂×D₃₆	1	1	yes	characteristic
209	Q₈	—	9	4	no	—
210	Q₈	—	18	2	no	—
211	Q₈	—	18	2	no	—
212	Q₈	—	18	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	18	2	no	—
216	C₂×C₄	—	18	2	no	—
217	C₂×C₄	—	18	2	no	—
218	C₂×C₄	—	18	2	no	—
219	C₂×C₄	—	36	2	no	—
220	C₂×C₄	—	36	2	no	—
221	C₂×C₄	—	36	2	no	—
222	C₂×C₄	—	36	2	no	—
223	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
224	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
225	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
226	C₂×C₄	D₄×D₉	1	1	yes	characteristic
227	C₂×C₄	#3699	1	1	yes	characteristic
228	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
229	C₂×C₄	#3701	1	1	yes	characteristic
230	C₂×C₄	C₂×#611	1	1	yes	characteristic
231	C₈	—	4	2	no	—
232	C₂×C₃	#4829	1	2	yes	—
233	C₂×C₃	#4845	1	2	yes	—
234	C₂×C₃	#4987	1	1	yes	characteristic
235	C₂×C₃	#5113	1	1	yes	characteristic
236	C₂×C₃	#5120	1	1	yes	characteristic
237	C₂²	#63457	1	1	yes	characteristic
238	C₂²	#63459	1	1	yes	characteristic
239	C₂²	#63467	1	2	yes	—
240	C₂²	#63471	1	2	yes	—
241	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	18	4	no	—
245	C₄	—	36	2	no	—
246	C₄	—	72	2	no	—
247	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
248	C₄	#63454	1	1	yes	characteristic
249	C₄	#63500	1	1	yes	characteristic
250	C₄	#63508	1	1	yes	characteristic
251	C₃	#70486	1	1	yes	characteristic
252	C₂	#10588511	1	2	yes	—
253	C₂	#10588529	1	2	yes	—
254	C₂	#10588667	1	1	yes	characteristic
255	C₂	#10588794	1	1	yes	characteristic
256	C₂	#10588804	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	#1805
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. C₄×Dic₁₈	C₄	semidirect
207. C₉	#1805	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈	2
C₂×C₉×C₄⋊C₈	2
C₂×#63376	2
#10588616	2
#10588229	2
#70486	3