Subgroups — #72254907

Order 2304.

762 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
2	#11772234	C₂	1	1	yes	maximal, characteristic, complemented
3	#11771567	C₂	1	2	yes	maximal, complemented
4	#11771294	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₄×D₁₆	C₂	1	1	yes	maximal, characteristic
6	#10706938	—	3	4	no	maximal
7	C₂×C₃²×D₁₆	C₄	1	1	yes	characteristic
8	C₃²×C₄×D₈	C₂²	1	2	yes	—
9	D₄×C₃⋊(C₃⋊C₈)	—	2	2	no	—
10	C₃²×#366	C₄	1	1	yes	characteristic
11	C₃²×#357	C₄	1	2	yes	—
12	C₃²×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	D₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
14	C₄×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	#10589324	C₂²	1	2	yes	—
16	C₈×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₁₆	D₃	1	4	yes	—
18	D₈×C₃⋊C₈	—	3	8	no	—
19	#67763	—	3	8	no	—
20	#67501	—	3	4	no	—
21	#67498	—	3	4	no	—
22	C₂×C₃²×D₈	C₂×C₄	1	2	yes	—
23	C₃²×C₄×D₄	—	2	2	no	—
24	C₂²×C₃⋊(C₃⋊C₈)	—	4	2	no	—
25	C₃²×D₁₆	C₈	1	4	yes	complemented
26	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₃²×#107	C₂×C₄	1	2	yes	—
29	C₃²×#102	—	2	2	no	—
30	C₃²×C₄×C₈	C₂³	1	1	yes	characteristic
31	#63657	—	4	2	no	—
32	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
33	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
34	C₄×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
35	C₄×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
36	#12045	—	9	1	no	Sylow
37	C₂×C₃×D₁₆	Dic₃	1	4	yes	—
38	C₃×C₄×D₈	C₂×D₃	1	8	yes	—
39	D₄×C₃⋊C₈	—	6	8	no	—
40	C₃×#366	Dic₃	1	4	yes	—
41	C₃×#357	Dic₃	1	8	yes	—
42	C₃×C₄×C₁₆	C₂×D₃	1	4	yes	—
43	D₄⋊(C₃⋊C₈)	—	6	8	no	—
44	C₄×C₃⋊C₁₆	—	3	4	no	—
45	#4458	—	3	8	no	—
46	C₈×C₃⋊C₈	—	3	4	no	—
47	C₂×C₃²×D₄	—	2	2	no	—
48	C₂²×C₃²×C₄	—	4	2	no	—
49	C₃²×D₈	—	2	2	no	—
50	C₃²×D₈	C₂×C₈	1	4	yes	complemented
51	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₈	#41	1	1	yes	characteristic
53	C₃²×C₄⋊C₄	—	2	2	no	—
54	C₃²×C₂²⋊C₄	—	4	2	no	—
55	C₃²×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
57	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	8	4	no	—
59	C₂×C₃⋊(C₃⋊C₈)	D₈	1	2	yes	—
60	C₃²×C₁₆	—	2	1	no	—
61	C₃²×C₁₆	C₂×C₈	1	2	yes	complemented
62	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
63	C₄×D₁₆	C₃⋊D₃	1	1	yes	characteristic
64	C₈×D₈	—	9	2	no	—
65	#1372	—	9	1	no	—
66	#1331	—	9	2	no	—
67	#1311	—	9	1	no	—
68	C₂×C₃×D₈	C₂×Dic₃	1	8	yes	—
69	C₃×C₄×D₄	—	2	8	no	—
70	C₂²×C₃⋊C₈	—	12	8	no	—
71	C₃×D₁₆	C₃⋊C₈	1	16	yes	complemented
72	C₂×C₃×C₁₆	C₂×Dic₃	1	4	yes	—
73	C₂×C₃×C₁₆	C₂×Dic₃	1	4	yes	—
74	C₃×#107	C₂×Dic₃	1	8	yes	—
75	C₃×#102	—	2	8	no	—
76	C₃×C₄×C₈	C₂²×D₃	1	4	yes	—
77	#830	—	12	8	no	—
78	C₂×C₃⋊C₁₆	—	6	4	no	—
79	C₄⋊(C₃⋊C₈)	—	6	8	no	—
80	C₄×C₃⋊C₈	—	3	4	no	—
81	C₄×C₃⋊C₈	—	3	4	no	—
82	C₂³×C₃²	—	4	2	no	—
83	C₃²×D₄	—	2	4	no	—
84	C₃²×D₄	—	4	2	no	—
85	C₂×C₃²×C₄	—	4	2	no	—
86	C₂×C₃²×C₄	—	8	2	no	—
87	C₂×C₃²×C₄	—	8	2	no	—
88	C₂×C₃²×C₄	C₄×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂×C₃²×C₄	#135	1	1	yes	characteristic
91	C₃²×C₈	—	2	1	no	—
92	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
93	C₃²×C₈	#131	1	1	yes	characteristic
94	C₃⋊(C₃⋊C₈)	—	2	2	no	—
95	C₃⋊(C₃⋊C₈)	—	4	1	no	—
96	C₃⋊(C₃⋊C₈)	—	8	2	no	—
97	C₂×D₁₆	C₃⋊Dic₃	1	1	yes	characteristic
98	C₄×D₈	C₂×C₃⋊D₃	1	2	yes	—
99	C₈×D₄	—	18	2	no	—
100	#366	C₃⋊Dic₃	1	1	yes	characteristic
101	#357	C₃⋊Dic₃	1	2	yes	—
102	C₄×C₁₆	—	9	1	no	—
103	C₄×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
104	#335	—	9	2	no	—
105	D₄⋊C₈	—	18	2	no	—
106	C₈²	—	9	1	no	—
107	C₂×C₃×D₄	—	2	8	no	—
108	C₂²×C₃×C₄	—	4	8	no	—
109	C₃×D₈	—	2	8	no	—
110	C₃×D₈	C₂×C₃⋊C₈	1	16	yes	complemented
111	C₂×C₃×C₈	#237	1	4	yes	—
112	C₂×C₃×C₈	C₂²×Dic₃	1	4	yes	—
113	C₃×C₄⋊C₄	—	2	8	no	—
114	C₃×C₂²⋊C₄	—	4	8	no	—
115	C₃×C₄²	D₃×D₄	1	4	yes	—
116	C₂×C₃⋊C₈	—	3	8	no	—
117	C₂×C₃⋊C₈	—	6	4	no	—
118	C₂×C₃⋊C₈	—	12	8	no	—
119	C₂×C₃⋊C₈	—	24	16	no	—
120	C₃×C₁₆	—	2	4	no	—
121	C₃×C₁₆	C₂×C₃⋊C₈	1	8	yes	complemented
122	C₃⋊C₁₆	—	6	8	no	—
123	C₂²×C₃²	—	4	4	no	—
124	C₂²×C₃²	—	8	2	no	—
125	C₂²×C₃²	—	8	2	no	—
126	C₂²×C₃²	C₄×D₈	1	1	yes	characteristic
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	8	2	no	—
129	C₃²×C₄	C₈×D₄	1	1	yes	characteristic
130	C₃²×C₄	#443	1	1	yes	characteristic
131	C₃²×C₄	#509	1	2	yes	—
132	C₂×D₈	C₂×C₃⋊Dic₃	1	2	yes	—
133	C₂²×C₈	—	36	2	no	—
134	C₄×D₄	—	2	2	no	—
135	D₁₆	C₃⋊(C₃⋊C₈)	1	4	yes	complemented
136	C₂×C₁₆	—	18	1	no	—
137	C₂×C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
138	C₂×C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
139	#107	C₂×C₃⋊Dic₃	1	2	yes	—
140	C₄⋊C₈	—	18	2	no	—
141	#102	—	2	2	no	—
142	C₂²⋊C₈	—	36	2	no	—
143	C₄×C₈	—	9	1	no	—
144	C₄×C₈	—	9	1	no	—
145	C₄×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
146	C₂³×C₃	—	4	8	no	—
147	C₃×D₄	—	2	16	no	—
148	C₃×D₄	—	4	8	no	—
149	C₂×C₃×C₄	—	4	8	no	—
150	C₂×C₃×C₄	—	8	8	no	—
151	C₂×C₃×C₄	—	8	8	no	—
152	C₂×C₃×C₄	D₃×D₈	1	4	yes	—
153	C₂×C₃×C₄	#912	1	4	yes	—
154	C₂×C₃×C₄	D₄×Dic₃	1	4	yes	—
155	C₃×C₈	—	2	4	no	—
156	C₃×C₈	C₂²×C₃⋊C₈	1	4	yes	—
157	C₃×C₈	#948	1	4	yes	—
158	C₃⋊C₈	—	6	8	no	—
159	C₃⋊C₈	—	12	4	no	—
160	C₃⋊C₈	—	24	8	no	—
161	C₂×C₃²	—	8	4	no	—
162	C₂×C₃²	C₈×D₈	1	1	yes	characteristic
163	C₂×C₃²	#2177	1	1	yes	characteristic
164	C₂×C₃²	#2179	1	1	yes	characteristic
165	C₂×D₄	—	2	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	D₈	—	2	2	no	—
168	D₈	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	complemented
169	C₂×C₈	—	9	2	no	—
170	C₂×C₈	—	18	1	no	—
171	C₂×C₈	—	36	2	no	—
172	C₂×C₈	—	72	4	no	—
173	C₂×C₈	#3830	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₄²	D₄×C₃⋊D₃	1	1	yes	characteristic
178	C₁₆	—	2	1	no	—
179	C₁₆	—	18	2	no	—
180	C₁₆	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	complemented
181	C₂²×C₃	—	4	16	no	—
182	C₂²×C₃	—	8	8	no	—
183	C₂²×C₃	—	8	8	no	—
184	C₂²×C₃	Dic₃×D₈	1	4	yes	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	8	8	no	—
187	C₃×C₄	#4911	1	8	yes	—
188	C₃×C₄	D₄×C₃⋊C₈	1	4	yes	—
189	C₃×C₄	#5196	1	4	yes	—
190	C₃²	#12045	1	1	yes	characteristic, complemented, Sylow
191	C₂³	—	4	2	no	—
192	D₄	—	2	4	no	—
193	D₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	D₈×C₃⋊D₃	1	1	yes	characteristic
198	C₂×C₄	#64121	1	1	yes	characteristic
199	C₂×C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	18	2	no	—
202	C₈	—	36	1	no	—
203	C₈	—	72	2	no	—
204	C₈	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
205	C₈	#64157	1	1	yes	characteristic
206	C₂×C₃	—	8	16	no	—
207	C₂×C₃	D₈×C₃⋊C₈	1	4	yes	—
208	C₂×C₃	#71461	1	4	yes	—
209	C₂×C₃	#71512	1	4	yes	—
210	C₂²	—	4	4	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	8	2	no	—
213	C₂²	D₈×C₃⋊Dic₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	#10592610	1	2	yes	—
217	C₄	D₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
218	C₄	#10592895	1	1	yes	characteristic
219	C₃	#10706938	1	4	yes	complemented
220	C₂	—	8	4	no	—
221	C₂	D₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
222	C₂	#11797810	1	1	yes	characteristic
223	C₂	#11797874	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×C₃⋊D₃
commutator	C₃²×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×C₄×D₁₆	C₂
socle	C₂²×C₃²	C₄×D₈
prime radical	C₂×C₃²×D₁₆	C₄
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	C₄×D₁₆	C₃⋊D₃
hypocenter	C₃²	#12045
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. D₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
2. #11772234	C₂	semidirect
3. #11771567	C₂	semidirect
4. #11771294	C₂	semidirect
25. C₃²×D₁₆	C₈	semidirect
50. C₃²×D₈	C₂×C₈	semidirect
61. C₃²×C₁₆	C₂×C₈	semidirect
71. C₃×D₁₆	C₃⋊C₈	semidirect
110. C₃×D₈	C₂×C₃⋊C₈	semidirect
121. C₃×C₁₆	C₂×C₃⋊C₈	semidirect
135. D₁₆	C₃⋊(C₃⋊C₈)	semidirect
168. D₈	C₂×C₃⋊(C₃⋊C₈)	semidirect
180. C₁₆	C₂×C₃⋊(C₃⋊C₈)	semidirect
190. C₃²	#12045	semidirect
219. C₃	#10706938	semidirect

Maximal subgroups

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

subgroup	index
D₈×C₃⋊(C₃⋊C₈)	2
#11772234	2
#11771567	2
#11771294	2
C₃²×C₄×D₁₆	2
#10706938	3