Subgroups — #11774546

Order 1152.

468 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₉⋊C₈)	C₂	1	4	yes	maximal, complemented
5	#71598	—	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	C₂×#3667	—	2	1	no	—
16	#63389	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	—	2	1	no	—
19	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
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	#4924	—	3	1	no	—
24	D₄⋊(C₃⋊C₈)	—	3	4	no	—
25	C₂⁴×C₉	—	2	1	no	—
26	C₂×D₄×C₉	—	2	4	no	—
27	C₂×D₄×C₉	—	2	1	no	—
28	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	—	4	2	no	—
33	C₂²×C₄×C₉	—	4	2	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	#3667	—	4	2	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	#1486	—	9	1	no	Sylow
49	C₂²×C₃×D₄	Dic₃	1	1	yes	characteristic
50	C₂³×C₃×C₄	—	2	1	no	—
51	C₃×C₄×D₄	—	2	1	no	—
52	C₃×C₄×D₄	—	2	1	no	—
53	C₃×C₄×D₄	Dic₃	1	2	yes	—
54	C₃×C₄×D₄	C₂×D₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
56	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×#208	—	6	1	no	—
59	#830	—	6	1	no	—
60	C₄⋊(C₃⋊C₈)	—	3	2	no	—
61	C₄⋊(C₃⋊C₈)	—	6	1	no	—
62	C₄⋊(C₃⋊C₈)	—	6	1	no	—
63	C₄×C₃⋊C₈	—	3	2	no	—
64	C₂³×C₉	—	2	4	no	—
65	C₂³×C₉	—	2	2	no	—
66	C₂³×C₉	—	4	2	no	—
67	C₂³×C₉	—	4	2	no	—
68	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
69	D₄×C₉	—	2	4	no	—
70	D₄×C₉	—	2	2	no	—
71	D₄×C₉	M₁₆	1	4	yes	complemented
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	4	1	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	—	4	4	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	—	4	2	no	—
79	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	D₈	1	2	yes	—
83	C₂×C₄×C₉	SD₁₆	1	2	yes	—
84	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
87	C₉⋊C₈	—	4	2	no	—
88	C₉⋊C₈	—	4	2	no	—
89	C₂×C₄×D₄	D₉	1	1	yes	characteristic
90	#431	—	9	1	no	—
91	#423	—	9	1	no	—
92	D₄⋊C₈	—	9	4	no	—
93	C₂⁴×C₃	—	2	1	no	—
94	C₂×C₃×D₄	—	2	4	no	—
95	C₂×C₃×D₄	—	2	1	no	—
96	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	—	4	2	no	—
101	C₂²×C₃×C₄	—	4	2	no	—
102	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	#67	1	1	yes	characteristic
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	2	1	no	—
106	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	#67	1	1	yes	characteristic
111	C₃×C₄²	#67	1	1	yes	characteristic
112	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
113	#208	—	12	2	no	—
114	C₂×C₃⋊C₈	—	6	2	no	—
115	C₂×C₃⋊C₈	—	6	2	no	—
116	C₂²×C₉	—	2	1	no	—
117	C₂²×C₉	—	2	1	no	—
118	C₂²×C₉	—	2	4	no	—
119	C₂²×C₉	—	4	8	no	—
120	C₂²×C₉	—	4	2	no	—
121	C₂²×C₉	—	4	2	no	—
122	C₂²×C₉	#102	1	2	yes	—
123	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₉	—	2	1	no	—
125	C₄×C₉	—	2	1	no	—
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	4	2	no	—
128	C₄×C₉	#102	1	2	yes	—
129	C₄×C₉	C₂×M₁₆	1	2	yes	—
130	C₄×C₉	C₂×D₈	1	1	yes	characteristic
131	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂²×D₄	Dic₉	1	1	yes	characteristic
133	C₂³×C₄	—	2	1	no	—
134	C₂×M₁₆	—	18	1	no	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	Dic₉	1	2	yes	—
138	C₄×D₄	C₂×D₉	1	2	yes	—
139	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
142	C₄⋊C₈	—	9	2	no	—
143	C₄⋊C₈	—	18	1	no	—
144	C₄⋊C₈	—	18	1	no	—
145	C₂²⋊C₈	—	18	1	no	—
146	C₄×C₈	—	9	2	no	—
147	C₂³×C₃	—	2	4	no	—
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	4	2	no	—
150	C₂³×C₃	—	4	2	no	—
151	C₂³×C₃	#217	1	1	yes	characteristic
152	C₃×D₄	—	2	4	no	—
153	C₃×D₄	—	2	2	no	—
154	C₃×D₄	#208	1	4	yes	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	4	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	#213	1	2	yes	—
163	C₂×C₃×C₄	#214	1	2	yes	—
164	C₂×C₃×C₄	#217	1	1	yes	characteristic
165	C₂×C₃×C₄	#217	1	1	yes	characteristic
166	C₂×C₃×C₄	#217	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
170	C₃⋊C₈	—	12	2	no	—
171	C₃⋊C₈	—	12	2	no	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	—	4	4	no	—
174	C₂×C₉	#407	1	1	yes	characteristic
175	C₂×C₉	C₂×#102	1	1	yes	characteristic
176	C₂×C₉	#421	1	1	yes	characteristic
177	C₂⁴	—	2	1	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂×Dic₉	1	2	yes	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
187	C₂²×C₄	#611	1	1	yes	characteristic
188	C₂²×C₄	#611	1	1	yes	characteristic
189	M₁₆	—	36	2	no	—
190	C₂×C₈	—	18	2	no	—
191	C₂×C₈	—	18	2	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	C₂×Dic₉	1	1	yes	characteristic
197	C₄²	#611	1	1	yes	characteristic
198	C₄²	#611	1	1	yes	characteristic
199	C₄²	C₂²×D₉	1	1	yes	characteristic
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	4	no	—
203	C₂²×C₃	—	4	8	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	#832	1	2	yes	—
207	C₂²×C₃	C₂×#217	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	4	2	no	—
212	C₃×C₄	#832	1	2	yes	—
213	C₃×C₄	C₂×#208	1	2	yes	—
214	C₃×C₄	C₂×#213	1	1	yes	characteristic
215	C₃×C₄	C₂×#214	1	1	yes	characteristic
216	C₉	#1486	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	2	4	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	#3676	1	1	yes	characteristic
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	#3667	1	4	yes	complemented
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	#3672	1	2	yes	—
233	C₂×C₄	#3673	1	2	yes	—
234	C₂×C₄	#3676	1	1	yes	characteristic
235	C₂×C₄	#3676	1	1	yes	characteristic
236	C₂×C₄	#3676	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
238	C₂×C₄	C₂×#611	1	1	yes	characteristic
239	C₂×C₄	C₂×#611	1	1	yes	characteristic
240	C₈	—	36	2	no	—
241	C₈	—	36	2	no	—
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	4	4	no	—
244	C₂×C₃	#5207	1	1	yes	characteristic
245	C₂×C₃	C₂×#832	1	1	yes	characteristic
246	C₂×C₃	#5236	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	#63392	1	2	yes	—
254	C₂²	C₂×#3676	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	2	no	—
259	C₄	#63392	1	2	yes	—
260	C₄	C₂×#3667	1	2	yes	—
261	C₄	C₂×#3672	1	1	yes	characteristic
262	C₄	C₂×#3673	1	1	yes	characteristic
263	C₃	#71598	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	4	no	—
266	C₂	#10588887	1	1	yes	characteristic
267	C₂	C₂×#63392	1	1	yes	characteristic
268	C₂	#10588914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
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	#1486
LCS	G ▷ C₂×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. #10588604	C₂	semidirect
4. D₄⋊(C₉⋊C₈)	C₂	semidirect
19. C₄⋊(C₉⋊C₈)	C₂²	semidirect
20. C₄×C₉⋊C₈	C₂²	semidirect
71. D₄×C₉	M₁₆	semidirect
216. C₉	#1486	semidirect
224. D₄	#3667	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
#10588604	2
D₄⋊(C₉⋊C₈)	2
#71598	3