Subgroups — #11785613

Order 1152.

396 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	#10588916	C₂	1	2	yes	maximal
3	C₂×#63378	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588234	C₂	1	2	yes	maximal, complemented
6	#72016	—	3	1	no	maximal
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₉×#131	—	2	2	no	—
9	C₉×#131	C₄	1	4	yes	complemented
10	C₂×C₉×M₁₆	—	2	1	no	—
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	—	2	1	no	—
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₂×#3667	C₂²	1	2	yes	—
16	#63395	—	2	2	no	—
17	#63392	—	2	2	no	—
18	#63378	—	2	1	no	—
19	#63378	—	2	1	no	—
20	#63372	—	2	2	no	—
21	C₂×C₃×#131	D₃	1	1	yes	characteristic
22	#5232	—	3	2	no	—
23	C₂×#819	—	3	1	no	—
24	C₂×#818	—	3	1	no	—
25	#4554	—	3	2	no	—
26	C₉×#41	—	2	1	no	—
27	C₉×#41	—	2	1	no	—
28	C₉×#41	C₂×C₄	1	2	yes	—
29	C₉×#41	Q₈	1	2	yes	—
30	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₉	—	2	2	no	—
36	C₉×M₁₆	—	2	1	no	—
37	C₉×M₁₆	—	2	4	no	—
38	C₉×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₉	—	2	1	no	—
40	C₂×C₈×C₉	—	2	4	no	—
41	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₉	D₄	1	1	yes	characteristic
43	C₂×C₈×C₉	D₄	1	1	yes	characteristic
44	C₂×C₈×C₉	D₄	1	2	yes	—
45	C₄⋊Dic₉	—	2	2	no	—
46	C₄⋊Dic₉	—	2	2	no	—
47	#3667	—	2	2	no	—
48	#3667	—	2	2	no	—
49	C₂×C₉⋊C₈	—	2	2	no	—
50	#1816	—	9	1	no	Sylow
51	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
52	C₃×#131	—	2	2	no	—
53	C₃×#131	Dic₃	1	4	yes	—
54	C₂×C₃×M₁₆	—	2	1	no	—
55	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₈	—	2	1	no	—
57	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₄⋊Dic₃	—	3	2	no	—
59	C₂×#208	—	3	2	no	—
60	#835	—	6	2	no	—
61	#832	—	6	2	no	—
62	#819	—	6	2	no	—
63	#818	—	6	1	no	—
64	#818	—	6	1	no	—
65	C₂³×C₉	—	2	1	no	—
66	C₂³×C₉	#41	1	1	yes	characteristic
67	Q₈×C₉	—	2	1	no	—
68	Q₈×C₉	C₄⋊C₄	1	2	yes	—
69	D₄×C₉	—	2	2	no	—
70	D₄×C₉	—	2	2	no	—
71	D₄×C₉	—	2	1	no	—
72	D₄×C₉	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₉	—	2	1	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₉	#41	1	1	yes	characteristic
82	C₂×Dic₉	—	4	2	no	—
83	C₂×Dic₉	—	8	2	no	—
84	C₈×C₉	—	2	2	no	—
85	C₈×C₉	C₂²⋊C₄	1	2	yes	—
86	C₈×C₉	C₂²⋊C₄	1	2	yes	—
87	C₉⋊C₈	—	4	2	no	—
88	C₂×#131	D₉	1	1	yes	characteristic
89	C₂×#108	—	9	1	no	—
90	C₂×#107	—	9	1	no	—
91	#417	—	9	2	no	—
92	#343	—	9	2	no	—
93	C₃×#41	—	2	1	no	—
94	C₃×#41	—	2	1	no	—
95	C₃×#41	Dic₆	1	2	yes	—
96	C₃×#41	C₂×Dic₃	1	2	yes	—
97	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
98	C₂×C₃×D₄	—	2	1	no	—
99	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×Dic₃	—	6	2	no	—
103	C₃×M₁₆	—	2	1	no	—
104	C₃×M₁₆	—	2	4	no	—
105	C₃×M₁₆	C₂×Dic₃	1	2	yes	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
109	C₂×C₃×C₈	#67	1	1	yes	characteristic
110	C₂×C₃×C₈	#67	1	1	yes	characteristic
111	C₂×C₃×C₈	#67	1	2	yes	—
112	C₄⋊Dic₃	—	6	2	no	—
113	C₄⋊Dic₃	—	6	2	no	—
114	#208	—	6	2	no	—
115	#208	—	6	2	no	—
116	C₂×C₃⋊C₈	—	6	2	no	—
117	C₂²×C₉	—	2	1	no	—
118	C₂²×C₉	—	2	1	no	—
119	C₂²×C₉	—	2	2	no	—
120	C₂²×C₉	—	4	1	no	—
121	C₂²×C₉	—	4	1	no	—
122	C₂²×C₉	#117	1	1	yes	characteristic
123	C₂²×C₉	#121	1	1	yes	characteristic
124	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₉	#121	1	1	yes	characteristic
129	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
130	Dic₉	—	8	2	no	—
131	C₂×#41	C₂×D₉	1	1	yes	characteristic
132	#131	—	2	2	no	—
133	#131	Dic₉	1	4	yes	complemented
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	9	2	no	—
136	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	9	2	no	—
140	#108	—	18	2	no	—
141	#107	—	18	1	no	—
142	#107	—	18	1	no	—
143	#103	—	18	2	no	—
144	#102	—	18	2	no	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	#235	1	1	yes	characteristic
147	C₃×Q₈	—	2	1	no	—
148	C₃×Q₈	C₄⋊Dic₃	1	2	yes	—
149	C₃×D₄	—	2	2	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	2	1	no	—
152	C₃×D₄	C₄⋊Dic₃	1	2	yes	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
158	C₂×C₃×C₄	#235	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
162	C₂×Dic₃	—	12	2	no	—
163	C₂×Dic₃	—	24	2	no	—
164	C₃×C₈	—	2	2	no	—
165	C₃×C₈	#217	1	2	yes	—
166	C₃×C₈	#217	1	2	yes	—
167	C₃⋊C₈	—	12	2	no	—
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	—	4	2	no	—
171	C₂×C₉	#380	1	1	yes	characteristic
172	C₂×C₉	#472	1	1	yes	characteristic
173	C₂×C₉	#473	1	1	yes	characteristic
174	#41	—	2	1	no	—
175	#41	—	2	1	no	—
176	#41	Dic₁₈	1	2	yes	—
177	#41	C₂×Dic₉	1	2	yes	—
178	C₂×Q₈	D₃₆	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	D₃₆	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	18	2	no	—
183	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
184	M₁₆	—	2	1	no	—
185	M₁₆	—	2	4	no	—
186	M₁₆	—	18	2	no	—
187	M₁₆	—	18	2	no	—
188	M₁₆	C₂×Dic₉	1	2	yes	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	18	2	no	—
192	C₂×C₈	C₂×Dic₉	1	2	yes	—
193	C₂×C₈	#611	1	1	yes	characteristic
194	C₂×C₈	#611	1	1	yes	characteristic
195	C₂×C₈	#611	1	2	yes	—
196	C₄⋊C₄	—	18	2	no	—
197	C₄⋊C₄	—	18	2	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	#924	1	1	yes	characteristic
204	C₂²×C₃	#926	1	1	yes	characteristic
205	C₂²×C₃	#930	1	1	yes	characteristic
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	#924	1	1	yes	characteristic
208	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
209	C₃×C₄	#930	1	1	yes	characteristic
210	C₃×C₄	C₂×#217	1	1	yes	characteristic
211	Dic₃	—	24	2	no	—
212	C₉	#1816	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	2	1	no	—
214	C₂³	#3697	1	1	yes	characteristic
215	Q₈	—	2	1	no	—
216	Q₈	C₄⋊Dic₉	1	2	yes	—
217	D₄	—	2	2	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	C₄⋊Dic₉	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	36	2	no	—
225	C₂×C₄	—	72	2	no	—
226	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
227	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
228	C₂×C₄	#3697	1	1	yes	characteristic
229	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
230	C₂×C₄	C₂×#611	1	1	yes	characteristic
231	C₂×C₄	C₂×#611	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	36	2	no	—
234	C₈	#3676	1	2	yes	—
235	C₈	#3676	1	2	yes	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	4	2	no	—
239	C₂×C₃	#5142	1	1	yes	characteristic
240	C₂×C₃	#5143	1	1	yes	characteristic
241	C₂×C₃	#5212	1	1	yes	characteristic
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	#63486	1	1	yes	characteristic
248	C₂²	#63488	1	1	yes	characteristic
249	C₂²	#63492	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	72	2	no	—
252	C₄	#63486	1	1	yes	characteristic
253	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
254	C₄	#63492	1	1	yes	characteristic
255	C₄	C₂×#3676	1	1	yes	characteristic
256	C₃	#72016	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	2	no	—
260	C₂	#10588821	1	1	yes	characteristic
261	C₂	#10588823	1	1	yes	characteristic
262	C₂	#10588892	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63492
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂²×C₃	#930
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#131	D₉
hypocenter	C₉	#1816
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
3. C₂×#63378	C₂	semidirect
4. C₂×#63372	C₂	semidirect
5. #10588234	C₂	semidirect
9. C₉×#131	C₄	semidirect
133. #131	Dic₉	semidirect
212. C₉	#1816	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
#10588916	2
C₂×#63378	2
C₂×#63372	2
#10588234	2
#72016	3