Subgroups — #11797031

Order 1152.

385 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	#10592800	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#10589133	C₂	1	4	yes	maximal, complemented
5	#71421	—	3	1	no	maximal
6	#71401	—	3	1	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	#64135	C₄	1	1	yes	characteristic
9	C₂×#3748	C₄	1	1	yes	characteristic
10	#63642	C₄	1	2	yes	—
11	C₈×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₃×#812	—	2	2	no	—
13	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₃×M₃₂	—	3	1	no	—
17	C₂²×C₃×C₁₆	—	3	1	no	—
18	#5101	—	3	1	no	—
19	#5101	—	3	1	no	—
20	#5101	—	6	1	no	—
21	C₂×#812	D₃	1	1	yes	characteristic
22	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
23	Dic₃⋊C₁₆	—	3	4	no	—
24	Dic₃⋊C₁₆	—	3	4	no	—
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	2	yes	—
28	C₂×C₃²×C₈	Q₈	1	2	yes	—
29	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
30	#3757	C₈	1	2	yes	—
31	C₄⋊(C₃⋊Dic₃)	C₈	1	2	yes	—
32	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
33	#3748	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
35	C₃×C₃⋊C₁₆	—	2	2	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	#2151	—	9	1	no	Sylow
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	#926	—	3	1	no	—
42	#926	—	3	1	no	—
43	#926	—	6	1	no	—
44	C₂×#208	—	3	1	no	—
45	C₂×#208	—	3	1	no	—
46	C₂×#208	—	6	1	no	—
47	C₃×M₃₂	—	6	2	no	—
48	C₂×C₃×C₁₆	—	3	2	no	—
49	C₂×C₃×C₁₆	—	3	2	no	—
50	C₂×C₃×C₁₆	—	6	2	no	—
51	#815	—	3	2	no	—
52	#815	—	3	2	no	—
53	#815	—	6	2	no	—
54	C₈×Dic₃	—	3	2	no	—
55	C₈×Dic₃	—	3	2	no	—
56	C₈×Dic₃	—	6	2	no	—
57	#812	—	2	2	no	—
58	C₂×C₃⋊C₁₆	—	2	2	no	—
59	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
60	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
61	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
64	C₂×C₃²×C₄	M₁₆	1	2	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃⋊Dic₃	C₂×C₈	1	4	yes	—
67	C₃²×C₈	C₄⋊C₄	1	2	yes	—
68	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₃⋊(C₃⋊C₈)	—	2	2	no	—
71	C₂×M₃₂	—	3	1	no	—
72	C₂²×C₁₆	—	3	1	no	—
73	#405	—	9	1	no	—
74	C₄⋊C₁₆	—	9	4	no	—
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃×C₈	—	2	1	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	1	no	—
82	C₂×C₃×C₈	Dic₆	1	2	yes	—
83	C₂×C₃×C₈	Dic₆	1	2	yes	—
84	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
85	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃×C₈	#67	1	2	yes	—
87	C₂×C₃×C₈	#67	1	2	yes	—
88	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
90	#217	—	3	2	no	—
91	#217	—	3	2	no	—
92	#217	—	6	2	no	—
93	C₄⋊Dic₃	—	3	2	no	—
94	C₄⋊Dic₃	—	3	2	no	—
95	C₄⋊Dic₃	—	6	2	no	—
96	C₄×Dic₃	—	3	2	no	—
97	C₄×Dic₃	—	3	2	no	—
98	C₄×Dic₃	—	6	2	no	—
99	#208	—	6	4	no	—
100	#208	—	6	2	no	—
101	#208	—	6	2	no	—
102	C₂×C₃⋊C₈	—	3	2	no	—
103	C₂×C₃⋊C₈	—	3	2	no	—
104	C₂×C₃⋊C₈	—	6	2	no	—
105	C₃×C₁₆	—	6	2	no	—
106	C₃×C₁₆	—	6	2	no	—
107	C₃⋊C₁₆	—	2	2	no	—
108	C₃⋊C₁₆	—	2	2	no	—
109	C₂²×C₃²	—	2	2	no	—
110	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
111	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
112	C₃²×C₄	C₄⋊C₈	1	2	yes	—
113	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
115	C₃⋊Dic₃	—	2	4	no	—
116	C₂×M₁₆	—	9	1	no	—
117	C₂²×C₈	D₃²	1	1	yes	characteristic
118	#117	—	9	1	no	—
119	M₃₂	—	6	2	no	—
120	C₂×C₁₆	—	3	2	no	—
121	C₂×C₁₆	—	3	2	no	—
122	C₂×C₁₆	—	6	2	no	—
123	#97	—	9	2	no	—
124	C₄×C₈	—	9	2	no	—
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
127	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
134	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
135	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
136	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
137	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
138	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
140	C₂×Dic₃	—	3	4	no	—
141	C₂×Dic₃	—	3	4	no	—
142	C₂×Dic₃	—	6	4	no	—
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
147	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
148	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
149	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
150	C₃×C₈	C₂×#67	1	1	yes	characteristic
151	C₃×C₈	C₂×#67	1	1	yes	characteristic
152	C₃⋊C₈	—	6	4	no	—
153	C₃⋊C₈	—	6	2	no	—
154	C₃⋊C₈	—	6	2	no	—
155	C₂×C₃²	—	2	2	no	—
156	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
157	C₂×C₃²	#504	1	2	yes	—
158	C₂²×C₄	#624	1	1	yes	characteristic
159	M₁₆	—	18	2	no	—
160	C₂×C₈	—	9	2	no	—
161	C₂×C₈	#624	1	1	yes	characteristic
162	C₂×C₈	#625	1	2	yes	—
163	C₂×C₈	#627	1	2	yes	—
164	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
165	C₄⋊C₄	—	9	2	no	—
166	C₂²⋊C₄	—	9	2	no	—
167	C₄²	—	9	2	no	—
168	C₁₆	—	6	2	no	—
169	C₁₆	—	6	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	4	2	no	—
175	C₂²×C₃	Dic₃⋊C₈	1	2	yes	—
176	C₂²×C₃	Dic₃⋊C₈	1	2	yes	—
177	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
178	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
183	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
184	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
185	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
186	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
187	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
188	Dic₃	—	6	4	no	—
189	Dic₃	—	6	4	no	—
190	Dic₃	—	12	4	no	—
191	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
192	C₂³	#3710	1	1	yes	characteristic
193	C₂×C₄	—	9	4	no	—
194	C₂×C₄	#3710	1	1	yes	characteristic
195	C₂×C₄	#3712	1	2	yes	—
196	C₂×C₄	#3724	1	2	yes	—
197	C₂×C₄	C₂×#624	1	1	yes	characteristic
198	C₈	—	18	2	no	—
199	C₈	#3724	1	2	yes	—
200	C₈	C₂×#625	1	1	yes	characteristic
201	C₈	C₂×#627	1	1	yes	characteristic
202	C₂×C₃	—	2	2	no	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	—	4	2	no	—
207	C₂×C₃	#4901	1	2	yes	—
208	C₂×C₃	#4907	1	2	yes	—
209	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
210	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	#63572	1	2	yes	—
213	C₂²	C₂×#3710	1	1	yes	characteristic
214	C₄	—	18	4	no	—
215	C₄	#63572	1	2	yes	—
216	C₄	C₂×#3712	1	1	yes	characteristic
217	C₄	C₂×#3724	1	1	yes	characteristic
218	C₃	—	2	1	no	—
219	C₃	#71401	1	1	yes	characteristic, complemented
220	C₃	#71421	1	1	yes	characteristic, complemented
221	C₂	—	2	2	no	—
222	C₂	#10590624	1	2	yes	—
223	C₂	C₂×#63572	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂²×C₃²×C₈	C₂²
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2151
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #10589133	C₂	semidirect
191. C₃²	#2151	semidirect
219. C₃	#71401	semidirect
220. C₃	#71421	semidirect

Maximal subgroups

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

subgroup	index
#10592800	2
C₂×C₃×#812	2
C₂²×C₃×C₃⋊C₁₆	2
#10589133	2
#71421	3
#71401	3