Subgroups — #11790333

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#63635	C₂	1	2	yes	maximal
3	#10592634	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
5	#10589395	C₂	1	2	yes	maximal
6	#69467	—	3	4	no	maximal
7	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	—	2	1	no	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
13	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
14	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃²×#95	—	2	1	no	—
16	#63635	—	2	4	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
18	C₃×C₄×C₄⋊C₄	D₃	1	4	yes	—
19	C₂×#808	—	3	8	no	—
20	#4935	—	3	4	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
22	#4529	—	3	8	no	—
23	C₂²×C₃²×C₄	—	2	1	no	—
24	C₂²×C₃²×C₄	—	2	1	no	—
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃⋊Dic₃	—	2	2	no	—
29	C₃²×C₄⋊C₄	—	2	1	no	—
30	C₃²×C₄⋊C₄	—	2	1	no	—
31	C₃²×C₄⋊C₄	D₄	1	2	yes	—
32	C₃²×C₄⋊C₄	Q₈	1	2	yes	—
33	C₃²×C₄²	—	4	2	no	—
34	C₃²×C₄²	C₂×C₄	1	2	yes	—
35	C₃²×C₄²	Q₈	1	2	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
39	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
43	#1948	—	9	1	no	Sylow
44	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
45	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
46	C₂×C₃×C₄²	—	2	4	no	—
47	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
48	C₂×C₄⋊Dic₃	—	3	8	no	—
49	C₂×C₄⋊Dic₃	—	6	4	no	—
50	C₂×C₄⋊Dic₃	—	6	4	no	—
51	C₂²×C₃⋊C₈	—	3	8	no	—
52	C₃×#95	—	2	4	no	—
53	#808	—	6	16	no	—
54	C₄⋊(C₃⋊C₈)	—	3	16	no	—
55	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	1	no	—
57	C₂×C₃²×C₄	—	2	1	no	—
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	—	4	1	no	—
61	C₂×C₃²×C₄	—	4	1	no	—
62	C₂×C₃²×C₄	—	4	1	no	—
63	C₂×C₃²×C₄	—	4	1	no	—
64	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
65	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
66	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
67	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
68	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	#41	1	1	yes	characteristic
73	C₂×C₃²×C₄	#41	1	1	yes	characteristic
74	C₂×C₃⋊Dic₃	—	4	2	no	—
75	C₂×C₃⋊Dic₃	—	4	2	no	—
76	C₂×C₃⋊Dic₃	—	4	2	no	—
77	C₃⋊(C₃⋊C₈)	—	2	4	no	—
78	C₂×#106	—	9	2	no	—
79	C₂×C₄⋊C₈	—	9	1	no	—
80	#384	—	9	1	no	—
81	C₄×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
82	#340	—	9	2	no	—
83	C₂²×C₃×C₄	—	2	4	no	—
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	#67	1	4	yes	—
86	C₂²×C₃×C₄	#67	1	4	yes	—
87	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
88	C₂²×Dic₃	—	6	8	no	—
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₄⋊C₄	—	2	4	no	—
91	C₃×C₄⋊C₄	Dic₆	1	8	yes	—
92	C₃×C₄⋊C₄	D₁₂	1	8	yes	—
93	C₃×C₄²	—	4	8	no	—
94	C₃×C₄²	Dic₆	1	8	yes	—
95	C₃×C₄²	C₄×D₃	1	8	yes	—
96	C₄⋊Dic₃	—	6	8	no	—
97	C₄⋊Dic₃	—	6	8	no	—
98	C₄⋊Dic₃	—	12	8	no	—
99	C₄⋊Dic₃	—	12	8	no	—
100	C₂×C₃⋊C₈	—	3	16	no	—
101	C₂×C₃⋊C₈	—	6	8	no	—
102	C₂×C₃⋊C₈	—	6	8	no	—
103	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃²	#121	1	1	yes	characteristic
105	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
108	C₂²×C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂²×C₃²	#137	1	1	yes	characteristic
110	C₃²×C₄	—	2	2	no	—
111	C₃²×C₄	—	4	2	no	—
112	C₃²×C₄	—	4	2	no	—
113	C₃²×C₄	#106	1	4	yes	complemented
114	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
115	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
116	C₃²×C₄	#125	1	1	yes	characteristic
117	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
118	C₃⋊Dic₃	—	8	2	no	—
119	C₂²×C₈	—	9	2	no	—
120	C₂×C₄⋊C₄	—	9	2	no	—
121	C₂×C₄⋊C₄	—	18	1	no	—
122	C₂×C₄⋊C₄	—	18	1	no	—
123	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
127	#106	—	18	4	no	—
128	C₄⋊C₈	—	9	4	no	—
129	#95	—	2	1	no	—
130	C₂³×C₃	C₂×#67	1	4	yes	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	—	2	8	no	—
135	C₂×C₃×C₄	—	4	4	no	—
136	C₂×C₃×C₄	—	4	4	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
140	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
141	C₂×C₃×C₄	#214	1	8	yes	—
142	C₂×C₃×C₄	#215	1	8	yes	—
143	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
144	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
145	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
146	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
147	C₂×C₃×C₄	#235	1	4	yes	—
148	C₂×C₃×C₄	#235	1	4	yes	—
149	C₂×Dic₃	—	12	8	no	—
150	C₂×Dic₃	—	12	8	no	—
151	C₂×Dic₃	—	12	8	no	—
152	C₃⋊C₈	—	6	16	no	—
153	C₂×C₃²	#389	1	1	yes	characteristic
154	C₂×C₃²	C₂×#106	1	1	yes	characteristic
155	C₂×C₃²	#428	1	1	yes	characteristic
156	C₂×C₃²	#460	1	1	yes	characteristic
157	C₂×C₃²	#461	1	1	yes	characteristic
158	C₂×C₃²	#475	1	1	yes	characteristic
159	C₂×C₃²	#476	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	#638	1	1	yes	characteristic
164	C₂²×C₄	#638	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
166	C₂×C₈	—	9	4	no	—
167	C₂×C₈	—	18	2	no	—
168	C₂×C₈	—	18	2	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	18	2	no	—
172	C₄⋊C₄	—	18	2	no	—
173	C₄⋊C₄	—	36	2	no	—
174	C₄⋊C₄	—	36	2	no	—
175	C₄⋊C₄	#634	1	2	yes	—
176	C₄⋊C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
177	C₄²	—	4	2	no	—
178	C₄²	#634	1	2	yes	—
179	C₄²	C₄×C₃⋊D₃	1	2	yes	—
180	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
181	C₂²×C₃	#924	1	4	yes	—
182	C₂²×C₃	#930	1	4	yes	—
183	C₂²×C₃	C₂×#214	1	4	yes	—
184	C₂²×C₃	C₂×#215	1	4	yes	—
185	C₂²×C₃	#949	1	4	yes	—
186	C₂²×C₃	#951	1	4	yes	—
187	C₃×C₄	—	2	8	no	—
188	C₃×C₄	—	4	8	no	—
189	C₃×C₄	—	4	8	no	—
190	C₃×C₄	#808	1	16	yes	complemented
191	C₃×C₄	C₄×Dic₆	1	4	yes	—
192	C₃×C₄	#869	1	4	yes	—
193	C₃×C₄	#870	1	4	yes	—
194	C₃×C₄	C₄×D₁₂	1	4	yes	—
195	Dic₃	—	24	8	no	—
196	C₃²	#1948	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×#638	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	36	2	no	—
207	C₂×C₄	—	36	2	no	—
208	C₂×C₄	—	36	2	no	—
209	C₂×C₄	#3750	1	2	yes	—
210	C₂×C₄	#3750	1	2	yes	—
211	C₂×C₄	#3754	1	2	yes	—
212	C₂×C₄	#3755	1	2	yes	—
213	C₂×C₄	C₂×#634	1	1	yes	characteristic
214	C₂×C₄	C₂×#634	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
217	C₂×C₄	#3828	1	1	yes	characteristic
218	C₂×C₄	#3828	1	1	yes	characteristic
219	C₈	—	18	4	no	—
220	C₂×C₃	#4928	1	4	yes	—
221	C₂×C₃	C₂×#808	1	4	yes	—
222	C₂×C₃	#4995	1	4	yes	—
223	C₂×C₃	#5008	1	4	yes	—
224	C₂×C₃	#5019	1	4	yes	—
225	C₂×C₃	#5020	1	4	yes	—
226	C₂×C₃	#5027	1	4	yes	—
227	C₂²	C₂×#3750	1	1	yes	characteristic
228	C₂²	#64133	1	1	yes	characteristic
229	C₂²	#64139	1	1	yes	characteristic
230	C₂²	C₂×#3754	1	1	yes	characteristic
231	C₂²	C₂×#3755	1	1	yes	characteristic
232	C₂²	#64158	1	1	yes	characteristic
233	C₂²	#64160	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	72	2	no	—
238	C₄	#63635	1	4	yes	complemented
239	C₄	C₄×#634	1	1	yes	characteristic
240	C₄	#64078	1	1	yes	characteristic
241	C₄	#64079	1	1	yes	characteristic
242	C₄	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
243	C₃	#69467	1	4	yes	complemented
244	C₂	#10592627	1	1	yes	characteristic
245	C₂	C₂×#63635	1	1	yes	characteristic
246	C₂	#10592694	1	1	yes	characteristic
247	C₂	#10592707	1	1	yes	characteristic
248	C₂	#10592718	1	1	yes	characteristic
249	C₂	#10592719	1	1	yes	characteristic
250	C₂	#10592726	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#638
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#1948
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
113. C₃²×C₄	#106	semidirect
190. C₃×C₄	#808	semidirect
196. C₃²	#1948	semidirect
238. C₄	#63635	semidirect
243. C₃	#69467	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₄⋊C₄	2
C₂×#63635	2
#10592634	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10589395	2
#69467	3