Subgroups — #11784193

Order 1152.

876 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63642	C₂	1	2	yes	maximal
3	C₄²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
5	#10589416	C₂	1	2	yes	maximal
6	#70441	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₃⋊Dic₃	C₄	1	2	yes	—
10	C₂×C₄×C₃⋊Dic₃	C₄	1	2	yes	—
11	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃²×C₄⋊C₈	C₄	1	4	yes	complemented
15	#63642	—	2	4	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
17	C₂×C₃×C₄⋊C₈	D₃	1	4	yes	—
18	C₂×#815	—	3	8	no	—
19	C₄²×Dic₃	—	3	4	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
21	#4550	—	3	8	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
26	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	—	2	2	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	C₂×C₄	1	2	yes	—
32	C₄×C₃⋊Dic₃	—	2	4	no	—
33	C₄×C₃⋊Dic₃	—	2	1	no	—
34	C₄×C₃⋊Dic₃	—	2	1	no	—
35	C₄×C₃⋊Dic₃	—	2	4	no	—
36	C₄×C₃⋊Dic₃	D₄	1	2	yes	—
37	C₄×C₃⋊Dic₃	Q₈	1	2	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
41	#1767	—	9	1	no	Sylow
42	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
43	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
44	C₂×C₄×Dic₃	—	3	4	no	—
45	C₂×C₄×Dic₃	—	3	8	no	—
46	C₂×C₄×Dic₃	—	3	8	no	—
47	C₂×C₄×Dic₃	—	3	4	no	—
48	C₂²×C₃⋊C₈	—	3	8	no	—
49	C₃×C₄⋊C₈	Dic₃	1	16	yes	complemented
50	#815	—	6	16	no	—
51	C₄⋊(C₃⋊C₈)	—	3	16	no	—
52	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	—	2	2	no	—
54	C₂×C₃²×C₄	C₄²	1	2	yes	—
55	C₂×C₃²×C₄	C₄²	1	2	yes	—
56	C₂×C₃²×C₄	M₁₆	1	4	yes	—
57	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃²×C₄	#41	1	1	yes	characteristic
62	C₂×C₃²×C₄	#41	1	1	yes	characteristic
63	C₂×C₃⋊Dic₃	—	2	2	no	—
64	C₂×C₃⋊Dic₃	—	2	2	no	—
65	C₂×C₃⋊Dic₃	—	2	2	no	—
66	C₂×C₃⋊Dic₃	—	2	2	no	—
67	C₂×C₃⋊Dic₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₃⋊Dic₃	M₁₆	1	4	yes	—
69	C₃²×C₈	—	2	4	no	—
70	C₃⋊(C₃⋊C₈)	—	2	4	no	—
71	C₂×C₄⋊C₈	—	9	1	no	—
72	C₂×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
73	C₂×#97	—	9	2	no	—
74	C₄³	—	9	1	no	—
75	#336	—	9	2	no	—
76	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
77	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
78	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
79	C₂²×Dic₃	—	3	8	no	—
80	C₂²×Dic₃	—	3	8	no	—
81	C₂×C₃×C₈	—	2	8	no	—
82	C₂×C₃×C₈	—	2	8	no	—
83	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
84	C₃×C₄²	C₄×D₃	1	8	yes	—
85	C₃×C₄²	C₂×Dic₃	1	8	yes	—
86	C₄×Dic₃	—	3	8	no	—
87	C₄×Dic₃	—	3	8	no	—
88	C₄×Dic₃	—	6	16	no	—
89	C₄×Dic₃	—	6	16	no	—
90	C₄×Dic₃	—	6	4	no	—
91	C₄×Dic₃	—	6	4	no	—
92	C₂×C₃⋊C₈	—	3	16	no	—
93	C₂×C₃⋊C₈	—	6	8	no	—
94	C₂×C₃⋊C₈	—	6	8	no	—
95	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
96	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₃²	#117	1	1	yes	characteristic
98	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
99	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
100	C₃²×C₄	—	2	2	no	—
101	C₃²×C₄	#97	1	4	yes	complemented
102	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
103	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
104	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
105	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
106	C₃⋊Dic₃	—	2	4	no	—
107	C₃⋊Dic₃	—	2	4	no	—
108	C₂²×C₈	—	9	2	no	—
109	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
110	C₂×C₄²	—	9	1	no	—
111	C₂×C₄²	—	9	2	no	—
112	C₂×C₄²	—	9	2	no	—
113	C₂×C₄²	—	9	1	no	—
114	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
115	C₄⋊C₈	—	9	4	no	—
116	C₄⋊C₈	C₃⋊Dic₃	1	4	yes	complemented
117	#97	—	18	4	no	—
118	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	16	yes	—
121	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
122	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
123	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
124	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
125	C₂×C₃×C₄	#237	1	4	yes	—
126	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
127	C₂×C₃×C₄	#239	1	4	yes	—
128	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
129	C₂×Dic₃	—	3	16	no	—
130	C₂×Dic₃	—	3	16	no	—
131	C₂×Dic₃	—	6	8	no	—
132	C₂×Dic₃	—	6	8	no	—
133	C₂×Dic₃	—	6	8	no	—
134	C₂×Dic₃	—	6	8	no	—
135	C₃×C₈	—	2	16	no	—
136	C₃⋊C₈	—	6	16	no	—
137	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃²	C₂×#97	1	1	yes	characteristic
139	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
140	C₂×C₃²	#423	1	2	yes	—
141	C₂×C₃²	#432	1	2	yes	—
142	C₂²×C₄	—	9	2	no	—
143	C₂²×C₄	—	9	2	no	—
144	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
145	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	9	4	no	—
150	C₂×C₈	—	18	2	no	—
151	C₂×C₈	—	18	2	no	—
152	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
153	C₄²	—	9	2	no	—
154	C₄²	—	9	2	no	—
155	C₄²	—	18	4	no	—
156	C₄²	—	18	4	no	—
157	C₄²	—	18	1	no	—
158	C₄²	—	18	1	no	—
159	C₄²	C₄×C₃⋊D₃	1	2	yes	—
160	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
161	C₂²×C₃	D₃×C₄⋊C₄	1	4	yes	—
162	C₂²×C₃	#892	1	4	yes	—
163	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	8	yes	—
164	C₂²×C₃	D₃×M₁₆	1	8	yes	—
165	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
166	C₃×C₄	—	2	8	no	—
167	C₃×C₄	#815	1	16	yes	complemented
168	C₃×C₄	#887	1	4	yes	—
169	C₃×C₄	#893	1	4	yes	—
170	C₃×C₄	D₄×Dic₃	1	4	yes	—
171	C₃×C₄	Q₈×Dic₃	1	4	yes	—
172	Dic₃	—	6	16	no	—
173	Dic₃	—	6	16	no	—
174	C₃²	#1767	1	1	yes	characteristic, complemented, Sylow
175	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	9	4	no	—
178	C₂×C₄	—	9	4	no	—
179	C₂×C₄	—	18	2	no	—
180	C₂×C₄	—	18	2	no	—
181	C₂×C₄	—	18	2	no	—
182	C₂×C₄	—	18	2	no	—
183	C₂×C₄	#3743	1	4	yes	—
184	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
185	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
186	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
187	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
188	C₂×C₄	#3830	1	1	yes	characteristic
189	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
190	C₂×C₄	#3832	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	18	4	no	—
194	C₂×C₃	#4835	1	8	yes	—
195	C₂×C₃	#4837	1	8	yes	—
196	C₂×C₃	Dic₃×C₄⋊C₄	1	4	yes	—
197	C₂×C₃	C₂×#815	1	4	yes	—
198	C₂×C₃	Dic₃×M₁₆	1	4	yes	—
199	C₂²	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
200	C₂²	#64101	1	1	yes	characteristic
201	C₂²	C₂×#3743	1	2	yes	—
202	C₂²	M₁₆×C₃⋊D₃	1	2	yes	—
203	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	18	4	no	—
206	C₄	—	18	4	no	—
207	C₄	#63642	1	4	yes	complemented
208	C₄	#64096	1	1	yes	characteristic
209	C₄	#64102	1	1	yes	characteristic
210	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
211	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
212	C₃	#70441	1	4	yes	complemented
213	C₂	#10592534	1	2	yes	—
214	C₂	#10592536	1	2	yes	—
215	C₂	C₄⋊C₄×C₃⋊Dic₃	1	1	yes	characteristic
216	C₂	C₂×#63642	1	1	yes	characteristic
217	C₂	M₁₆×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂×C₄²
Fitting	C₂×C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	#1767
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. C₃²×C₄⋊C₈	C₄	semidirect
16. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
49. C₃×C₄⋊C₈	Dic₃	semidirect
101. C₃²×C₄	#97	semidirect
116. C₄⋊C₈	C₃⋊Dic₃	semidirect
167. C₃×C₄	#815	semidirect
174. C₃²	#1767	semidirect
207. C₄	#63642	semidirect
212. C₃	#70441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊C₈	2
C₂×#63642	2
C₄²×C₃⋊Dic₃	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10589416	2
#70441	3