Subgroups — #11774302

Order 1152.

468 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×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	#10588605	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588605	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₉⋊C₈)	C₂	1	4	yes	maximal, complemented
5	#71597	—	3	1	no	maximal
6	C₂²×D₄×C₉	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₉	—	2	1	no	—
8	C₄×D₄×C₉	—	2	1	no	—
9	C₄×D₄×C₉	—	2	1	no	—
10	C₄×D₄×C₉	C₄	1	2	yes	—
11	C₄×D₄×C₉	C₂²	1	2	yes	—
12	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	#63389	—	2	1	no	—
16	#63389	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
19	C₄×C₉⋊C₈	—	2	1	no	—
20	C₄×C₉⋊C₈	C₂²	1	2	yes	complemented
21	C₂×C₃×C₄×D₄	D₃	1	1	yes	characteristic
22	#4925	—	3	1	no	—
23	#4925	—	3	1	no	—
24	D₄⋊(C₃⋊C₈)	—	3	4	no	—
25	C₂⁴×C₉	—	2	1	no	—
26	C₂×D₄×C₉	—	2	1	no	—
27	C₂×D₄×C₉	—	2	2	no	—
28	C₂×D₄×C₉	C₈	1	4	yes	complemented
29	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	4	4	no	—
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₉×C₂²⋊C₄	—	2	2	no	—
40	C₉×C₂²⋊C₄	—	2	2	no	—
41	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₉	D₄	1	1	yes	characteristic
43	C₄²×C₉	D₄	1	1	yes	characteristic
44	C₄²×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₉⋊C₈	—	2	2	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	#1478	—	9	1	no	Sylow
48	C₂²×C₃×D₄	Dic₃	1	1	yes	characteristic
49	C₂³×C₃×C₄	—	2	1	no	—
50	C₃×C₄×D₄	—	2	1	no	—
51	C₃×C₄×D₄	—	2	1	no	—
52	C₃×C₄×D₄	Dic₃	1	2	yes	—
53	C₃×C₄×D₄	C₂×D₃	1	2	yes	—
54	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
55	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	#830	—	6	1	no	—
58	#830	—	6	1	no	—
59	C₄⋊(C₃⋊C₈)	—	3	2	no	—
60	C₄⋊(C₃⋊C₈)	—	6	1	no	—
61	C₄×C₃⋊C₈	—	3	2	no	—
62	C₄×C₃⋊C₈	—	6	1	no	—
63	C₂³×C₉	—	2	4	no	—
64	C₂³×C₉	—	2	2	no	—
65	C₂³×C₉	—	4	2	no	—
66	C₂³×C₉	—	4	2	no	—
67	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	D₄×C₉	—	2	4	no	—
69	D₄×C₉	—	2	2	no	—
70	D₄×C₉	C₂×C₈	1	4	yes	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	4	1	no	—
74	C₂×C₄×C₉	—	4	4	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	—	4	4	no	—
77	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
81	C₂×C₄×C₉	M₁₆	1	2	yes	—
82	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₉⋊C₈	—	4	2	no	—
86	C₉⋊C₈	—	4	2	no	—
87	C₂×C₄×D₄	D₉	1	1	yes	characteristic
88	#431	—	9	1	no	—
89	#431	—	9	1	no	—
90	D₄⋊C₈	—	9	4	no	—
91	C₂⁴×C₃	—	2	1	no	—
92	C₂×C₃×D₄	—	2	2	no	—
93	C₂×C₃×D₄	—	2	1	no	—
94	C₂×C₃×D₄	C₃⋊C₈	1	4	yes	—
95	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	4	4	no	—
100	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	#67	1	1	yes	characteristic
102	C₂²×C₃×C₄	#67	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	1	no	—
104	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
105	C₃×C₂²⋊C₄	—	2	2	no	—
106	C₃×C₂²⋊C₄	—	2	2	no	—
107	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
108	C₃×C₄²	#67	1	1	yes	characteristic
109	C₃×C₄²	#67	1	1	yes	characteristic
110	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
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	—	2	4	no	—
116	C₂²×C₉	—	4	8	no	—
117	C₂²×C₉	—	4	2	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	C₂²⋊C₈	1	2	yes	—
120	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₉	—	2	1	no	—
122	C₄×C₉	—	2	1	no	—
123	C₄×C₉	—	2	1	no	—
124	C₄×C₉	—	4	2	no	—
125	C₄×C₉	C₂²⋊C₈	1	2	yes	—
126	C₄×C₉	C₂²×C₈	1	1	yes	characteristic
127	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
128	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
129	C₂²×D₄	Dic₉	1	1	yes	characteristic
130	C₂³×C₄	—	2	1	no	—
131	C₄×D₄	—	2	1	no	—
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	Dic₉	1	2	yes	—
134	C₄×D₄	C₂×D₉	1	2	yes	—
135	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	—	2	1	no	—
137	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
138	C₄⋊C₈	—	9	2	no	—
139	C₄⋊C₈	—	18	1	no	—
140	C₂²⋊C₈	—	18	1	no	—
141	C₂²⋊C₈	—	18	1	no	—
142	C₄×C₈	—	9	2	no	—
143	C₄×C₈	—	18	1	no	—
144	C₂³×C₃	—	2	4	no	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	—	4	2	no	—
147	C₂³×C₃	—	4	2	no	—
148	C₂³×C₃	#217	1	1	yes	characteristic
149	C₃×D₄	—	2	4	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	C₂×C₃⋊C₈	1	4	yes	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	—	4	4	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	4	no	—
158	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
159	C₂×C₃×C₄	#208	1	2	yes	—
160	C₂×C₃×C₄	#217	1	1	yes	characteristic
161	C₂×C₃×C₄	#217	1	1	yes	characteristic
162	C₂×C₃×C₄	#217	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
166	C₃⋊C₈	—	12	2	no	—
167	C₃⋊C₈	—	12	2	no	—
168	C₂×C₉	—	2	2	no	—
169	C₂×C₉	—	4	4	no	—
170	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₉	#418	1	1	yes	characteristic
172	C₂×C₉	#421	1	1	yes	characteristic
173	C₂⁴	—	2	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	C₉⋊C₈	1	4	yes	complemented
177	C₂×D₄	C₂×Dic₉	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
183	C₂²×C₄	#611	1	1	yes	characteristic
184	C₂²×C₄	#611	1	1	yes	characteristic
185	C₂×C₈	—	18	2	no	—
186	C₂×C₈	—	18	2	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	C₂×Dic₉	1	1	yes	characteristic
192	C₄²	#611	1	1	yes	characteristic
193	C₄²	#611	1	1	yes	characteristic
194	C₄²	C₂²×D₉	1	1	yes	characteristic
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	4	8	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	#830	1	2	yes	—
202	C₂²×C₃	C₂×#217	1	1	yes	characteristic
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	#830	1	2	yes	—
208	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
209	C₃×C₄	C₂×#208	1	1	yes	characteristic
210	C₃×C₄	#932	1	2	yes	—
211	C₉	#1478	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	#3676	1	1	yes	characteristic
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	D₄	C₂×C₉⋊C₈	1	4	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
227	C₂×C₄	#3667	1	2	yes	—
228	C₂×C₄	#3676	1	1	yes	characteristic
229	C₂×C₄	#3676	1	1	yes	characteristic
230	C₂×C₄	#3676	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
232	C₂×C₄	C₂×#611	1	1	yes	characteristic
233	C₂×C₄	C₂×#611	1	1	yes	characteristic
234	C₈	—	36	2	no	—
235	C₈	—	36	2	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	4	4	no	—
238	C₂×C₃	C₂×#830	1	1	yes	characteristic
239	C₂×C₃	#5215	1	1	yes	characteristic
240	C₂×C₃	#5236	1	1	yes	characteristic
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	#63389	1	2	yes	—
248	C₂²	C₂×#3676	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	#63389	1	2	yes	—
254	C₄	C₂²×C₉⋊C₈	1	1	yes	characteristic
255	C₄	C₂×#3667	1	1	yes	characteristic
256	C₄	#63495	1	2	yes	—
257	C₃	#71597	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	4	4	no	—
260	C₂	C₂×#63389	1	1	yes	characteristic
261	C₂	#10588896	1	1	yes	characteristic
262	C₂	#10588914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₄×C₉	C₂²×C₈
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₂×C₄×D₄×C₉	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂²×C₃×D₄	Dic₃
cosocle	C₄²×C₉	C₂³
hypercenter	C₂×C₄×D₄	D₉
hypocenter	C₉	#1478
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #10588605	C₂	semidirect
3. #10588605	C₂	semidirect
4. D₄⋊(C₉⋊C₈)	C₂	semidirect
18. C₄⋊(C₉⋊C₈)	C₂²	semidirect
20. C₄×C₉⋊C₈	C₂²	semidirect
28. C₂×D₄×C₉	C₈	semidirect
176. C₂×D₄	C₉⋊C₈	semidirect
211. C₉	#1478	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₉	2
#10588605	2
D₄⋊(C₉⋊C₈)	2
#71597	3