Subgroups — #11785336

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63377	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588667	C₂	1	2	yes	maximal
5	#10588206	C₂	1	2	yes	maximal, complemented
6	#70577	—	3	1	no	maximal
7	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×(C₂×D₉)⋊C₄	—	2	2	no	—
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₉	—	2	2	no	—
14	C₂×C₈×D₉	—	2	2	no	—
15	C₂×C₈×D₉	C₄	1	4	yes	complemented
16	C₉×#106	—	2	2	no	—
17	C₉⋊#95	—	2	2	no	—
18	#63377	—	2	2	no	—
19	C₂²×D₃×C₈	—	3	1	no	—
20	C₂×C₃×#106	D₃	1	1	yes	characteristic
21	C₂×#817	—	3	1	no	—
22	#4987	—	3	2	no	—
23	#4529	—	3	2	no	—
24	C₂³×D₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₉	—	2	2	no	—
28	C₂²×Dic₉	D₄	1	1	yes	characteristic
29	C₂×C₄×D₉	—	2	1	no	—
30	C₂×C₄×D₉	—	2	1	no	—
31	C₂×C₄×D₉	—	2	1	no	—
32	C₂×C₄×D₉	—	2	1	no	—
33	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₉	Q₈	1	2	yes	—
35	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₉	D₄	1	2	yes	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	(C₂×D₉)⋊C₄	—	4	4	no	—
41	C₄⋊Dic₉	—	2	2	no	—
42	C₄⋊Dic₉	—	2	2	no	—
43	C₂×C₉⋊C₈	—	2	1	no	—
44	C₂×C₉⋊C₈	—	2	1	no	—
45	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₈×D₉	—	2	8	no	—
47	#1807	—	9	1	no	Sylow
48	C₂²×C₄×D₃	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₂×(C₂×D₃)⋊C₄	—	6	2	no	—
52	C₂×C₄⋊Dic₃	—	3	2	no	—
53	C₂²×C₃⋊C₈	—	3	1	no	—
54	C₂×D₃×C₈	—	3	4	no	—
55	C₂×D₃×C₈	—	6	2	no	—
56	C₂×D₃×C₈	—	6	2	no	—
57	C₃×#106	—	2	2	no	—
58	C₃⋊#95	—	6	2	no	—
59	#817	—	6	2	no	—
60	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×D₉	—	2	1	no	—
62	C₂²×D₉	—	2	1	no	—
63	C₂²×D₉	—	2	2	no	—
64	C₂²×D₉	C₄⋊C₄	1	2	yes	—
65	C₂²×D₉	SD₁₆	1	4	yes	—
66	C₂×C₄×C₉	—	4	2	no	—
67	C₂×C₄×C₉	—	4	2	no	—
68	C₂×C₄×C₉	—	4	2	no	—
69	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	1	yes	characteristic
74	C₂×C₄×C₉	#41	1	1	yes	characteristic
75	C₂×Dic₉	—	2	1	no	—
76	C₂×Dic₉	—	2	1	no	—
77	C₂×Dic₉	—	4	2	no	—
78	C₂×Dic₉	—	4	2	no	—
79	C₂×Dic₉	—	4	2	no	—
80	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
81	C₄×D₉	—	2	2	no	—
82	C₄×D₉	—	2	2	no	—
83	C₄×D₉	—	2	2	no	—
84	C₄×D₉	—	2	2	no	—
85	C₈×C₉	C₂²⋊C₄	1	4	yes	complemented
86	C₉⋊C₈	—	2	2	no	—
87	C₂³×C₈	—	9	1	no	—
88	C₂×#106	—	9	1	no	—
89	C₂×#106	D₉	1	1	yes	characteristic, complemented
90	#380	—	9	2	no	—
91	#340	—	9	2	no	—
92	C₂³×D₃	—	3	1	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×Dic₃	—	3	1	no	—
96	C₂²×Dic₃	—	6	2	no	—
97	C₂×C₄×D₃	—	3	2	no	—
98	C₂×C₄×D₃	—	3	2	no	—
99	C₂×C₄×D₃	—	6	1	no	—
100	C₂×C₄×D₃	—	6	1	no	—
101	C₂×C₄×D₃	—	6	1	no	—
102	C₂×C₄×D₃	—	6	1	no	—
103	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
104	C₂×C₃×C₈	D₁₂	1	2	yes	—
105	C₂×C₃×C₈	#67	1	2	yes	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	(C₂×D₃)⋊C₄	—	12	4	no	—
109	C₄⋊Dic₃	—	6	2	no	—
110	C₄⋊Dic₃	—	6	2	no	—
111	C₂×C₃⋊C₈	—	3	2	no	—
112	C₂×C₃⋊C₈	—	6	1	no	—
113	C₂×C₃⋊C₈	—	6	1	no	—
114	D₃×C₈	—	6	8	no	—
115	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₉	#121	1	1	yes	characteristic
117	C₂²×C₉	#121	1	1	yes	characteristic
118	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
119	C₂²×C₉	#135	1	2	yes	—
120	C₂×D₉	—	2	2	no	—
121	C₂×D₉	—	2	2	no	—
122	C₂×D₉	—	2	4	no	—
123	C₂×D₉	—	2	4	no	—
124	C₂×D₉	#106	1	4	yes	complemented
125	C₄×C₉	—	8	2	no	—
126	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₉	#117	1	1	yes	characteristic
128	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
129	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
130	Dic₉	—	2	2	no	—
131	Dic₉	—	8	2	no	—
132	C₂³×C₄	—	9	1	no	—
133	C₂²×C₈	—	9	1	no	—
134	C₂²×C₈	—	9	4	no	—
135	C₂²×C₈	—	18	2	no	—
136	C₂²×C₈	—	18	2	no	—
137	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	9	2	no	—
139	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
140	C₂×C₂²⋊C₄	—	18	2	no	—
141	#106	—	2	2	no	—
142	#106	—	18	2	no	—
143	#95	—	18	2	no	—
144	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
145	C₂²×D₃	—	3	2	no	—
146	C₂²×D₃	—	3	4	no	—
147	C₂²×D₃	—	6	1	no	—
148	C₂²×D₃	—	6	2	no	—
149	C₂²×D₃	—	6	1	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
155	C₂×C₃×C₄	#237	1	1	yes	characteristic
156	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	#239	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
159	C₂×Dic₃	—	3	2	no	—
160	C₂×Dic₃	—	6	1	no	—
161	C₂×Dic₃	—	6	1	no	—
162	C₂×Dic₃	—	12	2	no	—
163	C₂×Dic₃	—	12	2	no	—
164	C₂×Dic₃	—	12	2	no	—
165	C₄×D₃	—	6	2	no	—
166	C₄×D₃	—	6	2	no	—
167	C₄×D₃	—	6	2	no	—
168	C₄×D₃	—	6	2	no	—
169	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
170	C₃⋊C₈	—	6	2	no	—
171	C₂×C₉	#380	1	1	yes	characteristic
172	C₂×C₉	C₂×#106	1	1	yes	characteristic
173	C₂×C₉	#427	1	1	yes	characteristic
174	C₂×C₉	#465	1	2	yes	—
175	C₂×C₉	#465	1	2	yes	—
176	D₉	—	2	4	no	—
177	C₂⁴	—	9	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	9	1	no	—
180	C₂²×C₄	—	9	2	no	—
181	C₂²×C₄	—	9	2	no	—
182	C₂²×C₄	—	18	1	no	—
183	C₂²×C₄	—	18	1	no	—
184	C₂²×C₄	—	18	1	no	—
185	C₂²×C₄	—	18	1	no	—
186	C₂²×C₄	—	18	2	no	—
187	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
188	C₂×C₈	—	9	2	no	—
189	C₂×C₈	—	18	1	no	—
190	C₂×C₈	—	18	1	no	—
191	C₂×C₈	—	18	8	no	—
192	C₂×C₈	C₄×D₉	1	2	yes	—
193	C₂×C₈	D₃₆	1	2	yes	—
194	C₂×C₈	#611	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	18	2	no	—
198	C₄⋊C₄	—	18	2	no	—
199	C₂²⋊C₄	—	36	4	no	—
200	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₃	#895	1	1	yes	characteristic
202	C₂²×C₃	D₃×SD₁₆	1	2	yes	—
203	C₂²×C₃	#916	1	2	yes	—
204	C₂²×C₃	#938	1	1	yes	characteristic
205	C₂×D₃	—	3	4	no	—
206	C₂×D₃	—	6	4	no	—
207	C₂×D₃	—	6	4	no	—
208	C₂×D₃	—	6	2	no	—
209	C₂×D₃	—	6	2	no	—
210	C₃×C₄	—	8	2	no	—
211	C₃×C₄	#892	1	1	yes	characteristic
212	C₃×C₄	#897	1	1	yes	characteristic
213	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₃×C₄	#946	1	1	yes	characteristic
215	Dic₃	—	6	2	no	—
216	Dic₃	—	24	2	no	—
217	C₉	#1807	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	9	2	no	—
219	C₂³	—	9	4	no	—
220	C₂³	—	18	1	no	—
221	C₂³	—	18	1	no	—
222	C₂³	—	18	2	no	—
223	C₂³	D₄×D₉	1	1	yes	characteristic
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	9	2	no	—
228	C₂×C₄	—	18	1	no	—
229	C₂×C₄	—	18	1	no	—
230	C₂×C₄	—	18	2	no	—
231	C₂×C₄	—	18	2	no	—
232	C₂×C₄	—	18	2	no	—
233	C₂×C₄	—	18	2	no	—
234	C₂×C₄	—	36	2	no	—
235	C₂×C₄	—	36	2	no	—
236	C₂×C₄	—	36	2	no	—
237	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
239	C₂×C₄	#3699	1	1	yes	characteristic
240	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
241	C₂×C₄	#3701	1	1	yes	characteristic
242	C₂×C₄	C₂×#611	1	1	yes	characteristic
243	C₈	—	18	2	no	—
244	C₈	(C₂×D₉)⋊C₄	1	4	yes	complemented
245	C₂×C₃	D₃×#106	1	1	yes	characteristic
246	C₂×C₃	#4860	1	1	yes	characteristic
247	C₂×C₃	#4864	1	2	yes	—
248	C₂×C₃	#4987	1	1	yes	characteristic
249	C₂×C₃	#5171	1	2	yes	—
250	D₃	—	6	4	no	—
251	C₂²	—	9	4	no	—
252	C₂²	—	18	2	no	—
253	C₂²	—	18	4	no	—
254	C₂²	—	18	4	no	—
255	C₂²	—	18	2	no	—
256	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
257	C₂²	#63457	1	1	yes	characteristic
258	C₂²	SD₁₆×D₉	1	2	yes	—
259	C₂²	#63478	1	2	yes	—
260	C₂²	#63500	1	1	yes	characteristic
261	C₄	—	8	2	no	—
262	C₄	—	18	2	no	—
263	C₄	—	72	2	no	—
264	C₄	#63454	1	1	yes	characteristic
265	C₄	#63459	1	1	yes	characteristic
266	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
267	C₄	#63508	1	1	yes	characteristic
268	C₃	#70577	1	1	yes	characteristic
269	C₂	—	18	4	no	—
270	C₂	D₉×#106	1	1	yes	characteristic
271	C₂	#10588540	1	1	yes	characteristic
272	C₂	#10588544	1	2	yes	—
273	C₂	#10588667	1	1	yes	characteristic
274	C₂	#10588851	1	2	yes	—

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#106	C₂
socle	C₂³×C₃	D₃×D₄
prime radical	C₂³×D₉	D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#106	D₉
hypocenter	C₉	#1807
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#106	C₂	semidirect
3. C₂×#63377	C₂	semidirect
5. #10588206	C₂	semidirect
15. C₂×C₈×D₉	C₄	semidirect
85. C₈×C₉	C₂²⋊C₄	semidirect
89. C₂×#106	D₉	semidirect
124. C₂×D₉	#106	semidirect
217. C₉	#1807	semidirect
244. C₈	(C₂×D₉)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₉	2
C₂×C₉×#106	2
C₂×#63377	2
#10588667	2
#10588206	2
#70577	3