Subgroups — #11774758

Order 1152.

348 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589572	C₂	1	1	yes	maximal, characteristic
2	C₉×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10588160	C₂	1	4	yes	maximal, complemented
5	#70501	—	3	1	no	maximal
6	C₂×C₉×M₁₆	—	2	1	no	—
7	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
8	#63488	—	2	1	no	—
9	#63486	—	2	1	no	—
10	#63486	C₄	1	2	yes	—
11	C₂²×C₉⋊C₈	—	2	1	no	—
12	#63432	—	2	1	no	—
13	#63431	C₄	1	1	yes	characteristic
14	#63431	C₄	1	1	yes	characteristic
15	C₄×Dic₁₈	—	2	1	no	—
16	C₄×Dic₁₈	C₂²	1	2	yes	—
17	C₉×C₄⋊C₈	—	2	1	no	—
18	C₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₄×C₉⋊C₈	—	2	1	no	—
20	C₄×C₉⋊C₈	C₂²	1	2	yes	—
21	#5470	—	3	1	no	—
22	C₃×#423	D₃	1	1	yes	characteristic
23	C₂×C₄×C₃⋊C₈	—	3	1	no	—
24	#4484	—	3	4	no	—
25	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×Dic₁₈	—	2	1	no	—
29	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
30	C₉×M₁₆	—	4	2	no	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₉	D₄	1	1	yes	characteristic
34	C₄²×C₉	D₄	1	1	yes	characteristic
35	C₄²×C₉	C₂³	1	1	yes	characteristic
36	#3676	—	2	2	no	—
37	C₄⋊Dic₉	—	2	1	no	—
38	C₄⋊Dic₉	—	2	1	no	—
39	C₄⋊Dic₉	—	2	1	no	—
40	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
41	Dic₉⋊C₄	—	2	2	no	—
42	Dic₉⋊C₄	—	2	2	no	—
43	C₄×Dic₉	—	2	2	no	—
44	C₂×C₉⋊C₈	—	2	4	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	#1493	—	9	1	no	Sylow
47	C₂×C₃×M₁₆	—	2	1	no	—
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	#926	—	6	1	no	—
50	#924	—	3	2	no	—
51	#924	—	6	1	no	—
52	C₂²×C₃⋊C₈	—	6	1	no	—
53	#870	—	6	1	no	—
54	#869	—	3	1	no	—
55	#869	—	3	1	no	—
56	C₄×Dic₆	—	3	2	no	—
57	C₄×Dic₆	—	6	1	no	—
58	C₃×C₄⋊C₈	—	2	1	no	—
59	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
60	C₄×C₃⋊C₈	—	3	2	no	—
61	C₄×C₃⋊C₈	—	6	1	no	—
62	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	SD₁₆	1	2	yes	—
69	C₂×C₄×C₉	Q₁₆	1	2	yes	—
70	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×Dic₉	—	2	2	no	—
74	C₂×Dic₉	—	2	2	no	—
75	Dic₁₈	—	2	4	no	—
76	Dic₁₈	—	4	2	no	—
77	C₈×C₉	—	4	2	no	—
78	C₉⋊C₈	—	2	4	no	—
79	#533	—	9	1	no	—
80	#423	D₉	1	1	yes	characteristic
81	C₂×C₄×C₈	—	9	1	no	—
82	Q₈⋊C₈	—	9	4	no	—
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
85	C₂²×C₃×C₄	#67	1	1	yes	characteristic
86	C₂×Dic₆	—	3	2	no	—
87	C₂×Dic₆	—	6	1	no	—
88	C₃×M₁₆	—	4	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
91	C₃×C₄²	D₁₂	1	1	yes	characteristic
92	C₃×C₄²	#67	1	1	yes	characteristic
93	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
94	#217	—	6	2	no	—
95	C₄⋊Dic₃	—	3	2	no	—
96	C₄⋊Dic₃	—	6	1	no	—
97	C₄⋊Dic₃	—	6	1	no	—
98	C₄⋊Dic₃	—	6	1	no	—
99	Dic₃⋊C₄	—	6	2	no	—
100	Dic₃⋊C₄	—	6	2	no	—
101	C₄×Dic₃	—	6	2	no	—
102	C₂×C₃⋊C₈	—	6	4	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂²×C₉	—	2	1	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	#103	1	2	yes	—
107	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₉	—	2	1	no	—
109	C₄×C₉	—	2	1	no	—
110	C₄×C₉	#103	1	2	yes	—
111	C₄×C₉	#104	1	2	yes	—
112	C₄×C₉	#131	1	2	yes	—
113	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₄×C₉	C₂×Q₁₆	1	1	yes	characteristic
115	Dic₉	—	4	2	no	—
116	Dic₉	—	4	2	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₂²×C₈	—	18	1	no	—
119	C₄⋊Q₈	—	9	1	no	—
120	C₄⋊Q₈	—	9	1	no	—
121	#125	—	18	1	no	—
122	C₂²⋊Q₈	—	9	2	no	—
123	C₂²⋊Q₈	—	18	1	no	—
124	C₄×Q₈	—	9	2	no	—
125	C₄×Q₈	—	18	1	no	—
126	#117	—	18	1	no	—
127	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
128	C₄⋊C₈	—	2	1	no	—
129	C₄⋊C₈	C₂×D₉	1	2	yes	—
130	C₄×C₈	—	9	2	no	—
131	C₄×C₈	—	18	1	no	—
132	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	#214	1	2	yes	—
139	C₂×C₃×C₄	#216	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
143	C₂×Dic₃	—	6	2	no	—
144	C₂×Dic₃	—	6	2	no	—
145	Dic₆	—	6	4	no	—
146	Dic₆	—	12	2	no	—
147	C₃×C₈	—	4	2	no	—
148	C₃⋊C₈	—	6	4	no	—
149	C₂×C₉	—	2	2	no	—
150	C₂×C₉	#408	1	1	yes	characteristic
151	C₂×C₉	C₂×#103	1	1	yes	characteristic
152	C₂×C₉	C₂×#104	1	1	yes	characteristic
153	C₂×Q₈	—	9	2	no	—
154	C₂×Q₈	—	18	1	no	—
155	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
156	C₂²×C₄	D₃₆	1	1	yes	characteristic
157	C₂²×C₄	#611	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	18	4	no	—
161	C₂×C₈	—	18	2	no	—
162	C₄⋊C₄	—	9	2	no	—
163	C₄⋊C₄	—	18	1	no	—
164	C₄⋊C₄	—	18	1	no	—
165	C₄⋊C₄	—	18	2	no	—
166	C₄⋊C₄	—	18	1	no	—
167	C₄⋊C₄	—	18	2	no	—
168	C₂²⋊C₄	—	18	2	no	—
169	C₄²	—	18	2	no	—
170	C₄²	C₄×D₉	1	1	yes	characteristic
171	C₄²	D₃₆	1	1	yes	characteristic
172	C₄²	#611	1	1	yes	characteristic
173	C₄²	C₂²×D₉	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	#810	1	2	yes	—
177	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	#810	1	2	yes	—
181	C₃×C₄	#825	1	2	yes	—
182	C₃×C₄	#907	1	2	yes	—
183	C₃×C₄	C₂×#214	1	1	yes	characteristic
184	C₃×C₄	C₂×#216	1	1	yes	characteristic
185	Dic₃	—	12	2	no	—
186	Dic₃	—	12	2	no	—
187	C₉	#1493	1	1	yes	characteristic, complemented, Sylow
188	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
189	Q₈	—	18	4	no	—
190	Q₈	—	36	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	18	2	no	—
194	C₂×C₄	—	18	2	no	—
195	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
196	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
197	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#3672	1	2	yes	—
199	C₂×C₄	#3674	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
201	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
202	C₂×C₄	C₂×#611	1	1	yes	characteristic
203	C₈	—	4	2	no	—
204	C₈	—	18	4	no	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	C₂×#810	1	1	yes	characteristic
207	C₂×C₃	#5129	1	1	yes	characteristic
208	C₂×C₃	C₂×#825	1	1	yes	characteristic
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#63368	1	2	yes	—
212	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	36	2	no	—
216	C₄	—	36	2	no	—
217	C₄	#63368	1	2	yes	—
218	C₄	#63384	1	2	yes	—
219	C₄	#63469	1	2	yes	—
220	C₄	C₂×#3672	1	1	yes	characteristic
221	C₄	C₂×#3674	1	1	yes	characteristic
222	C₃	#70501	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₂×#63368	1	1	yes	characteristic
225	C₂	#10588811	1	1	yes	characteristic
226	C₂	C₂×#63384	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×#423	C₂
socle	C₂²×C₃	C₂×(C₂×D₃)⋊C₄
prime radical	C₂³×C₃	(C₂×D₃)⋊C₄
cosocle	C₄²×C₉	C₂³
hypercenter	#423	D₉
hypocenter	C₉	#1493
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
4. #10588160	C₂	semidirect
187. C₉	#1493	semidirect

Maximal subgroups

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

subgroup	index
#10589572	2
C₉×#423	2
C₂×C₄×C₉⋊C₈	2
#10588160	2
#70501	3