Subgroups — #11790064

Order 1152.

744 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	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	2	yes	maximal
4	C₂×#63630	C₂	1	1	yes	maximal, characteristic
5	#10589390	C₂	1	2	yes	maximal
6	#69485	—	3	4	no	maximal
7	C₂×C₃²×C₄⋊C₄	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃²×#95	C₄	1	1	yes	characteristic
14	C₃²×#95	C₄	1	1	yes	characteristic
15	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
17	#63630	—	2	2	no	—
18	C₃×C₄×C₄⋊C₄	D₃	1	4	yes	—
19	C₂×C₄⋊(C₃⋊C₈)	—	3	8	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
21	C₂×#803	—	3	4	no	—
22	#4524	—	3	8	no	—
23	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	—	2	4	no	—
29	C₃²×C₄²	—	2	2	no	—
30	C₃²×C₄²	—	2	2	no	—
31	C₃²×C₄²	D₄	1	2	yes	—
32	C₃²×C₄²	Q₈	1	2	yes	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
41	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
42	#1940	—	9	1	no	Sylow
43	C₂×C₃×C₄⋊C₄	Dic₃	1	8	yes	—
44	C₂×C₃×C₄²	C₂×D₃	1	8	yes	—
45	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
46	C₂²×C₃⋊C₈	—	3	4	no	—
47	C₂²×C₃⋊C₈	—	3	4	no	—
48	C₂²×C₃⋊C₈	—	3	8	no	—
49	C₃×#95	Dic₃	1	4	yes	—
50	C₃×#95	Dic₃	1	4	yes	—
51	C₄⋊(C₃⋊C₈)	—	6	16	no	—
52	C₄⋊(C₃⋊C₈)	—	6	8	no	—
53	#803	—	6	8	no	—
54	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	—	2	2	no	—
56	C₂×C₃²×C₄	—	2	2	no	—
57	C₂×C₃²×C₄	—	2	2	no	—
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	—	2	2	no	—
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
64	C₂×C₃²×C₄	M₁₆	1	4	yes	—
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	#41	1	1	yes	characteristic
70	C₂×C₃²×C₄	#41	1	1	yes	characteristic
71	C₃⋊(C₃⋊C₈)	—	2	2	no	—
72	C₃⋊(C₃⋊C₈)	—	2	2	no	—
73	C₃⋊(C₃⋊C₈)	—	4	2	no	—
74	C₂×C₄⋊C₈	—	9	2	no	—
75	C₂×C₄⋊C₈	—	9	1	no	—
76	C₂×#97	—	9	1	no	—
77	C₄×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
78	#336	—	9	2	no	—
79	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
80	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
81	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
82	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
83	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
84	C₃×C₄⋊C₄	—	2	16	no	—
85	C₃×C₄²	—	2	8	no	—
86	C₃×C₄²	—	2	8	no	—
87	C₃×C₄²	Dic₆	1	8	yes	—
88	C₃×C₄²	D₁₂	1	8	yes	—
89	C₂×C₃⋊C₈	—	3	8	no	—
90	C₂×C₃⋊C₈	—	3	8	no	—
91	C₂×C₃⋊C₈	—	6	4	no	—
92	C₂×C₃⋊C₈	—	6	4	no	—
93	C₂×C₃⋊C₈	—	6	4	no	—
94	C₂×C₃⋊C₈	—	6	4	no	—
95	C₂×C₃⋊C₈	—	6	8	no	—
96	C₂×C₃⋊C₈	—	6	8	no	—
97	C₂×C₃⋊C₈	—	6	8	no	—
98	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
101	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
102	C₂²×C₃²	#131	1	2	yes	—
103	C₃²×C₄	—	2	2	no	—
104	C₃²×C₄	—	2	2	no	—
105	C₃²×C₄	—	2	4	no	—
106	C₃²×C₄	—	4	2	no	—
107	C₃²×C₄	#121	1	1	yes	characteristic
108	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
109	C₃²×C₄	#125	1	1	yes	characteristic
110	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
111	C₂²×C₈	—	9	1	no	—
112	C₂²×C₈	—	9	1	no	—
113	C₂²×C₈	—	9	2	no	—
114	C₂×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
115	C₂×C₄²	C₂×C₃⋊D₃	1	2	yes	—
116	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₄⋊C₈	—	18	4	no	—
118	C₄⋊C₈	—	18	2	no	—
119	#97	—	18	2	no	—
120	#95	C₃⋊Dic₃	1	1	yes	characteristic
121	#95	C₃⋊Dic₃	1	1	yes	characteristic
122	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
123	C₂×C₃×C₄	—	2	8	no	—
124	C₂×C₃×C₄	—	2	8	no	—
125	C₂×C₃×C₄	—	2	8	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	—	2	8	no	—
130	C₂×C₃×C₄	#208	1	16	yes	—
131	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
132	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
133	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
134	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
135	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
136	C₂×C₃×C₄	#237	1	4	yes	—
137	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
138	C₂×C₃×C₄	#239	1	4	yes	—
139	C₃⋊C₈	—	6	8	no	—
140	C₃⋊C₈	—	6	8	no	—
141	C₃⋊C₈	—	12	8	no	—
142	C₂×C₃²	#389	1	1	yes	characteristic
143	C₂×C₃²	#423	1	1	yes	characteristic
144	C₂×C₃²	#424	1	1	yes	characteristic
145	C₂×C₃²	#435	1	2	yes	—
146	C₂×C₃²	#446	1	2	yes	—
147	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
148	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
149	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
152	C₂×C₈	—	9	2	no	—
153	C₂×C₈	—	9	2	no	—
154	C₂×C₈	—	18	1	no	—
155	C₂×C₈	—	18	1	no	—
156	C₂×C₈	—	18	1	no	—
157	C₂×C₈	—	18	1	no	—
158	C₂×C₈	—	18	2	no	—
159	C₂×C₈	—	18	2	no	—
160	C₂×C₈	—	18	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	#634	1	2	yes	—
165	C₄²	C₄⋊(C₃⋊D₃)	1	2	yes	—
166	C₂²×C₃	C₂×#208	1	8	yes	—
167	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
168	C₂²×C₃	D₄×Dic₃	1	4	yes	—
169	C₂²×C₃	Q₈×Dic₃	1	4	yes	—
170	C₂²×C₃	#948	1	8	yes	—
171	C₃×C₄	—	2	8	no	—
172	C₃×C₄	—	2	8	no	—
173	C₃×C₄	—	2	16	no	—
174	C₃×C₄	—	4	8	no	—
175	C₃×C₄	#888	1	4	yes	—
176	C₃×C₄	#890	1	4	yes	—
177	C₃×C₄	#895	1	4	yes	—
178	C₃×C₄	#897	1	4	yes	—
179	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	#3748	1	4	yes	—
189	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
190	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
191	C₂×C₄	C₂×#634	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
193	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
194	C₂×C₄	#3830	1	1	yes	characteristic
195	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
196	C₂×C₄	#3832	1	1	yes	characteristic
197	C₈	—	18	2	no	—
198	C₈	—	18	2	no	—
199	C₈	—	36	2	no	—
200	C₂×C₃	#4924	1	4	yes	—
201	C₂×C₃	#4984	1	4	yes	—
202	C₂×C₃	#4999	1	4	yes	—
203	C₂×C₃	#5011	1	8	yes	—
204	C₂×C₃	#5024	1	8	yes	—
205	C₂²	C₂×#3748	1	2	yes	—
206	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
207	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
208	C₂²	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
209	C₂²	#64157	1	2	yes	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	4	no	—
213	C₄	—	4	2	no	—
214	C₄	#64097	1	1	yes	characteristic
215	C₄	#64099	1	1	yes	characteristic
216	C₄	#64104	1	1	yes	characteristic
217	C₄	#64106	1	1	yes	characteristic
218	C₃	#69485	1	4	yes	complemented
219	C₂	#10592623	1	1	yes	characteristic
220	C₂	#10592683	1	1	yes	characteristic
221	C₂	#10592698	1	1	yes	characteristic
222	C₂	#10592710	1	2	yes	—
223	C₂	#10592723	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₃²×C₄×C₄⋊C₄	C₂
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#1940
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
179. C₃²	#1940	semidirect
218. C₃	#69485	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×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₂×#63630	2
#10589390	2
#69485	3