Subgroups — #11774732

Order 1152.

414 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	C₉×#532	C₂	1	1	yes	maximal, characteristic
2	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₉⋊C₈)	C₂	1	4	yes	maximal, complemented
5	#71802	—	3	1	no	maximal
6	C₂×C₉×#41	—	2	1	no	—
7	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₉×#127	C₄	1	1	yes	characteristic
9	C₉×#124	—	2	1	no	—
10	C₉×#121	—	2	1	no	—
11	C₉×#121	C₄	1	2	yes	—
12	C₄×D₄×C₉	—	2	1	no	—
13	C₄×D₄×C₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×#3667	—	2	1	no	—
16	C₂²×C₉⋊C₈	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
19	C₄×C₉⋊C₈	—	2	1	no	—
20	C₄×C₉⋊C₈	C₂²	1	2	yes	—
21	C₃×#532	D₃	1	1	yes	characteristic
22	#4924	—	3	1	no	—
23	C₂×C₄×C₃⋊C₈	—	3	1	no	—
24	D₄⋊(C₃⋊C₈)	—	3	4	no	—
25	C₉×#41	—	4	2	no	—
26	C₉×#41	—	4	2	no	—
27	C₂×Q₈×C₉	—	2	1	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	—	2	1	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₉×C₄⋊C₄	—	2	1	no	—
37	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₉×C₂²⋊C₄	—	2	2	no	—
39	C₉×C₂²⋊C₄	—	2	2	no	—
40	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₉	D₄	1	1	yes	characteristic
42	C₄²×C₉	D₄	1	1	yes	characteristic
43	C₄²×C₉	C₂³	1	1	yes	characteristic
44	#3667	—	4	2	no	—
45	C₂×C₉⋊C₈	—	2	4	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	#1492	—	9	1	no	Sylow
49	C₂×C₃×#41	—	2	1	no	—
50	C₃×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
51	C₃×#127	Dic₃	1	1	yes	characteristic
52	C₃×#124	—	2	1	no	—
53	C₃×#121	—	2	1	no	—
54	C₃×#121	Dic₃	1	2	yes	—
55	C₃×C₄×D₄	—	2	1	no	—
56	C₃×C₄×D₄	C₂×D₃	1	2	yes	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×#208	—	6	1	no	—
59	C₂²×C₃⋊C₈	—	6	1	no	—
60	C₄⋊(C₃⋊C₈)	—	3	2	no	—
61	C₄⋊(C₃⋊C₈)	—	6	1	no	—
62	C₄×C₃⋊C₈	—	3	2	no	—
63	C₄×C₃⋊C₈	—	6	1	no	—
64	C₂³×C₉	—	2	2	no	—
65	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
66	Q₈×C₉	—	4	2	no	—
67	D₄×C₉	—	2	4	no	—
68	D₄×C₉	—	4	4	no	—
69	D₄×C₉	—	4	2	no	—
70	D₄×C₉	—	4	2	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	4	2	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	D₈	1	2	yes	—
80	C₂×C₄×C₉	SD₁₆	1	2	yes	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₉⋊C₈	—	2	4	no	—
85	C₉⋊C₈	—	4	2	no	—
86	#532	D₉	1	1	yes	characteristic
87	#423	—	9	1	no	—
88	C₂×C₄×C₈	—	9	1	no	—
89	D₄⋊C₈	—	9	4	no	—
90	C₃×#41	—	4	2	no	—
91	C₃×#41	—	4	2	no	—
92	C₂×C₃×Q₈	—	2	1	no	—
93	C₂×C₃×D₄	—	2	2	no	—
94	C₂×C₃×D₄	—	2	1	no	—
95	C₂×C₃×D₄	—	2	1	no	—
96	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	#67	1	1	yes	characteristic
100	C₂²×C₃×C₄	#67	1	1	yes	characteristic
101	C₃×C₄⋊C₄	—	2	1	no	—
102	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
103	C₃×C₂²⋊C₄	—	2	2	no	—
104	C₃×C₂²⋊C₄	—	2	2	no	—
105	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
106	C₃×C₄²	#67	1	1	yes	characteristic
107	C₃×C₄²	#67	1	1	yes	characteristic
108	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
109	#208	—	12	2	no	—
110	C₂×C₃⋊C₈	—	6	4	no	—
111	C₂×C₃⋊C₈	—	6	2	no	—
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	—	2	1	no	—
115	C₂²×C₉	—	4	2	no	—
116	C₂²×C₉	—	4	2	no	—
117	C₂²×C₉	—	4	2	no	—
118	C₂²×C₉	#102	1	2	yes	—
119	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₉	—	2	1	no	—
121	C₄×C₉	—	2	1	no	—
122	C₄×C₉	—	4	2	no	—
123	C₄×C₉	#102	1	2	yes	—
124	C₄×C₉	#104	1	2	yes	—
125	C₄×C₉	#131	1	2	yes	—
126	C₄×C₉	C₂×D₈	1	1	yes	characteristic
127	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×#41	—	2	1	no	—
129	C₂×M₁₆	—	18	1	no	—
130	C₂²×C₈	—	18	1	no	—
131	C₄⋊Q₈	Dic₉	1	1	yes	characteristic
132	#127	Dic₉	1	1	yes	characteristic
133	#124	—	2	1	no	—
134	#121	—	2	1	no	—
135	#121	Dic₉	1	2	yes	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	C₂×D₉	1	2	yes	—
138	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
139	C₄⋊C₈	—	9	2	no	—
140	C₄⋊C₈	—	18	1	no	—
141	C₄×C₈	—	9	2	no	—
142	C₄×C₈	—	18	1	no	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	#217	1	1	yes	characteristic
145	C₃×Q₈	—	4	2	no	—
146	C₃×D₄	—	2	4	no	—
147	C₃×D₄	—	4	4	no	—
148	C₃×D₄	—	4	2	no	—
149	C₃×D₄	—	4	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	#213	1	2	yes	—
156	C₂×C₃×C₄	#214	1	2	yes	—
157	C₂×C₃×C₄	#217	1	1	yes	characteristic
158	C₂×C₃×C₄	#217	1	1	yes	characteristic
159	C₂×C₃×C₄	#217	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
163	C₃⋊C₈	—	6	4	no	—
164	C₃⋊C₈	—	12	2	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	—	8	2	no	—
167	C₂×C₉	#408	1	1	yes	characteristic
168	C₂×C₉	C₂×#102	1	1	yes	characteristic
169	C₂×C₉	C₂×#104	1	1	yes	characteristic
170	#41	—	4	2	no	—
171	#41	—	4	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	C₂×Dic₉	1	2	yes	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
179	C₂²×C₄	#611	1	1	yes	characteristic
180	C₂²×C₄	#611	1	1	yes	characteristic
181	M₁₆	—	36	2	no	—
182	C₂×C₈	—	18	4	no	—
183	C₂×C₈	—	18	2	no	—
184	C₂×C₈	—	18	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₄²	C₂×Dic₉	1	1	yes	characteristic
190	C₄²	#611	1	1	yes	characteristic
191	C₄²	#611	1	1	yes	characteristic
192	C₄²	C₂²×D₉	1	1	yes	characteristic
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	4	2	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	#832	1	2	yes	—
199	C₂²×C₃	C₂×#217	1	1	yes	characteristic
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	#832	1	2	yes	—
204	C₃×C₄	#837	1	2	yes	—
205	C₃×C₄	C₂×#213	1	1	yes	characteristic
206	C₃×C₄	C₂×#214	1	1	yes	characteristic
207	C₃×C₄	#948	1	2	yes	—
208	C₉	#1492	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	2	no	—
210	C₂³	#3676	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	D₄	—	2	4	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	#3672	1	2	yes	—
222	C₂×C₄	#3673	1	2	yes	—
223	C₂×C₄	#3676	1	1	yes	characteristic
224	C₂×C₄	#3676	1	1	yes	characteristic
225	C₂×C₄	#3676	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
227	C₂×C₄	C₂×#611	1	1	yes	characteristic
228	C₂×C₄	C₂×#611	1	1	yes	characteristic
229	C₈	—	18	4	no	—
230	C₈	—	36	2	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	8	2	no	—
233	C₂×C₃	C₂×#832	1	1	yes	characteristic
234	C₂×C₃	#5234	1	1	yes	characteristic
235	C₂×C₃	C₂×#837	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	#63392	1	2	yes	—
242	C₂²	C₂×#3676	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	2	no	—
246	C₄	#63392	1	2	yes	—
247	C₄	#63396	1	2	yes	—
248	C₄	C₂×#3672	1	1	yes	characteristic
249	C₄	C₂×#3673	1	1	yes	characteristic
250	C₄	#63510	1	2	yes	—
251	C₃	#71802	1	1	yes	characteristic
252	C₂	—	2	2	no	—
253	C₂	—	8	2	no	—
254	C₂	C₂×#63392	1	1	yes	characteristic
255	C₂	#10588912	1	1	yes	characteristic
256	C₂	C₂×#63396	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#532	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₃×#532	D₃
cosocle	C₄²×C₉	C₂³
hypercenter	#532	D₉
hypocenter	C₉	#1492
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
2. #10588604	C₂	semidirect
3. C₂×C₄×C₉⋊C₈	C₂	semidirect
4. D₄⋊(C₉⋊C₈)	C₂	semidirect
208. C₉	#1492	semidirect

Maximal subgroups

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

subgroup	index
C₉×#532	2
#10588604	2
C₂×C₄×C₉⋊C₈	2
D₄⋊(C₉⋊C₈)	2
#71802	3