Subgroups — #11790133

Order 1152.

744 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10592683	C₂	1	2	yes	maximal
5	#10589395	C₂	1	2	yes	maximal
6	#70612	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃²×#107	C₄	1	4	yes	complemented
14	#63658	—	2	2	no	—
15	#63645	—	2	2	no	—
16	C₈×C₃⋊Dic₃	—	2	2	no	—
17	C₂×C₃×#107	D₃	1	4	yes	—
18	C₂×#818	—	3	4	no	—
19	C₂×C₈×Dic₃	—	3	4	no	—
20	#4984	—	3	8	no	—
21	#4529	—	3	8	no	—
22	C₂²×C₃²×C₄	—	2	2	no	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃⋊Dic₃	—	2	2	no	—
25	C₂²×C₃⋊Dic₃	D₄	1	1	yes	characteristic
26	C₂²×C₃⋊Dic₃	D₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₈	D₄	1	2	yes	—
29	C₂×C₃²×C₈	Q₈	1	2	yes	—
30	C₃²×C₄⋊C₄	—	2	2	no	—
31	C₃²×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
33	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
34	C₄⋊(C₃⋊Dic₃)	—	4	4	no	—
35	C₄×C₃⋊Dic₃	—	2	1	no	—
36	C₄×C₃⋊Dic₃	—	2	1	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
39	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
40	#1942	—	9	1	no	Sylow
41	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
42	C₂×C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
43	C₂×C₄⋊Dic₃	—	3	8	no	—
44	C₂×C₄⋊Dic₃	—	6	8	no	—
45	C₂×C₄×Dic₃	—	3	4	no	—
46	C₂²×C₃⋊C₈	—	3	4	no	—
47	C₃×#107	Dic₃	1	16	yes	complemented
48	C₃⋊#95	—	6	8	no	—
49	#818	—	6	8	no	—
50	C₈×Dic₃	—	6	8	no	—
51	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	—	2	4	no	—
53	C₂×C₃²×C₄	—	4	2	no	—
54	C₂×C₃²×C₄	—	4	2	no	—
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₃²×C₄	#41	1	1	yes	characteristic
60	C₂×C₃²×C₄	#41	1	1	yes	characteristic
61	C₂×C₃⋊Dic₃	—	2	1	no	—
62	C₂×C₃⋊Dic₃	—	2	1	no	—
63	C₂×C₃⋊Dic₃	—	2	1	no	—
64	C₂×C₃⋊Dic₃	—	2	1	no	—
65	C₂×C₃⋊Dic₃	—	4	2	no	—
66	C₂×C₃⋊Dic₃	—	4	2	no	—
67	C₂×C₃⋊Dic₃	—	4	2	no	—
68	C₂×C₃⋊Dic₃	D₈	1	2	yes	—
69	C₂×C₃⋊Dic₃	Q₁₆	1	2	yes	—
70	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
71	C₃⋊(C₃⋊C₈)	—	2	2	no	—
72	C₂×#107	—	9	1	no	—
73	C₂×#107	C₃⋊D₃	1	1	yes	characteristic
74	C₂×C₄×C₈	—	9	1	no	—
75	#389	—	9	2	no	—
76	#340	—	9	2	no	—
77	C₂²×C₃×C₄	—	2	8	no	—
78	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
79	C₂²×Dic₃	—	3	4	no	—
80	C₂²×Dic₃	—	3	4	no	—
81	C₂²×Dic₃	—	6	8	no	—
82	C₂×C₃×C₈	Dic₆	1	8	yes	—
83	C₂×C₃×C₈	D₁₂	1	8	yes	—
84	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
85	C₃×C₄⋊C₄	—	2	8	no	—
86	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
87	C₄⋊Dic₃	—	6	8	no	—
88	C₄⋊Dic₃	—	6	8	no	—
89	C₄⋊Dic₃	—	12	16	no	—
90	C₄×Dic₃	—	6	4	no	—
91	C₄×Dic₃	—	6	4	no	—
92	C₂×C₃⋊C₈	—	3	8	no	—
93	C₂×C₃⋊C₈	—	6	4	no	—
94	C₂×C₃⋊C₈	—	6	4	no	—
95	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
96	C₂²×C₃²	#121	1	1	yes	characteristic
97	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
98	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
99	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
100	C₂²×C₃²	#135	1	1	yes	characteristic
101	C₂²×C₃²	#135	1	1	yes	characteristic
102	C₃²×C₄	—	4	4	no	—
103	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
105	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
106	C₃²×C₄	#125	1	1	yes	characteristic
107	C₃⋊Dic₃	—	2	2	no	—
108	C₃⋊Dic₃	—	2	2	no	—
109	C₃⋊Dic₃	—	8	2	no	—
110	C₂²×C₈	—	9	1	no	—
111	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	9	2	no	—
113	C₂×C₄⋊C₄	—	18	2	no	—
114	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
115	C₂×C₄²	—	9	1	no	—
116	#107	—	18	2	no	—
117	#107	C₃⋊Dic₃	1	4	yes	complemented
118	C₄×C₈	—	18	2	no	—
119	#95	—	18	2	no	—
120	C₂³×C₃	D₃×D₄	1	4	yes	—
121	C₂×C₃×C₄	—	2	16	no	—
122	C₂×C₃×C₄	—	4	8	no	—
123	C₂×C₃×C₄	—	4	8	no	—
124	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
125	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
126	C₂×C₃×C₄	#237	1	4	yes	—
127	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
128	C₂×C₃×C₄	#239	1	4	yes	—
129	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
130	C₂×Dic₃	—	3	8	no	—
131	C₂×Dic₃	—	3	8	no	—
132	C₂×Dic₃	—	6	4	no	—
133	C₂×Dic₃	—	6	4	no	—
134	C₂×Dic₃	—	6	4	no	—
135	C₂×Dic₃	—	6	4	no	—
136	C₂×Dic₃	—	12	8	no	—
137	C₂×Dic₃	—	12	8	no	—
138	C₂×Dic₃	—	12	8	no	—
139	C₃×C₈	C₄⋊Dic₃	1	16	yes	complemented
140	C₃⋊C₈	—	6	8	no	—
141	C₂×C₃²	#389	1	1	yes	characteristic
142	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
143	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
144	C₂×C₃²	#466	1	1	yes	characteristic
145	C₂×C₃²	#467	1	1	yes	characteristic
146	C₂×C₃²	#499	1	1	yes	characteristic
147	C₂×C₃²	#500	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	9	1	no	—
150	C₂²×C₄	—	9	1	no	—
151	C₂²×C₄	—	18	2	no	—
152	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
153	C₂×C₈	—	9	2	no	—
154	C₂×C₈	—	18	1	no	—
155	C₂×C₈	—	18	1	no	—
156	C₂×C₈	#634	1	2	yes	—
157	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
158	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	18	2	no	—
161	C₄⋊C₄	—	18	2	no	—
162	C₄⋊C₄	—	36	4	no	—
163	C₄⋊C₄	C₂×C₃⋊Dic₃	1	4	yes	—
164	C₄²	—	18	1	no	—
165	C₄²	—	18	1	no	—
166	C₂²×C₃	#888	1	4	yes	—
167	C₂²×C₃	#895	1	4	yes	—
168	C₂²×C₃	D₃×D₈	1	4	yes	—
169	C₂²×C₃	#912	1	4	yes	—
170	C₂²×C₃	D₃×Q₁₆	1	4	yes	—
171	C₂²×C₃	#919	1	4	yes	—
172	C₂²×C₃	D₄×Dic₃	1	4	yes	—
173	C₃×C₄	—	4	16	no	—
174	C₃×C₄	#890	1	4	yes	—
175	C₃×C₄	#897	1	4	yes	—
176	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
177	C₃×C₄	Q₈×Dic₃	1	4	yes	—
178	Dic₃	—	6	8	no	—
179	Dic₃	—	6	8	no	—
180	Dic₃	—	24	8	no	—
181	C₃²	#1942	1	1	yes	characteristic, complemented, Sylow
182	C₂³	D₄×C₃⋊D₃	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	9	2	no	—
187	C₂×C₄	—	9	2	no	—
188	C₂×C₄	—	18	1	no	—
189	C₂×C₄	—	18	1	no	—
190	C₂×C₄	—	18	1	no	—
191	C₂×C₄	—	18	1	no	—
192	C₂×C₄	—	36	2	no	—
193	C₂×C₄	—	36	2	no	—
194	C₂×C₄	—	36	2	no	—
195	C₂×C₄	C₂×#634	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
197	C₂×C₄	#3830	1	1	yes	characteristic
198	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
199	C₂×C₄	#3832	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
201	C₈	—	18	2	no	—
202	C₈	C₄⋊(C₃⋊Dic₃)	1	4	yes	complemented
203	C₂×C₃	#4874	1	4	yes	—
204	C₂×C₃	#4878	1	4	yes	—
205	C₂×C₃	#4883	1	4	yes	—
206	C₂×C₃	#4887	1	4	yes	—
207	C₂×C₃	#4984	1	4	yes	—
208	C₂×C₃	Dic₃×D₈	1	4	yes	—
209	C₂×C₃	Dic₃×Q₁₆	1	4	yes	—
210	C₂²	#64097	1	1	yes	characteristic
211	C₂²	#64104	1	1	yes	characteristic
212	C₂²	D₈×C₃⋊D₃	1	1	yes	characteristic
213	C₂²	#64121	1	1	yes	characteristic
214	C₂²	Q₁₆×C₃⋊D₃	1	1	yes	characteristic
215	C₂²	#64128	1	1	yes	characteristic
216	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
217	C₄	—	4	4	no	—
218	C₄	—	18	2	no	—
219	C₄	—	18	2	no	—
220	C₄	—	72	2	no	—
221	C₄	#64099	1	1	yes	characteristic
222	C₄	#64106	1	1	yes	characteristic
223	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
224	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
225	C₃	#70612	1	4	yes	complemented
226	C₂	#10592573	1	1	yes	characteristic
227	C₂	#10592577	1	1	yes	characteristic
228	C₂	#10592582	1	1	yes	characteristic
229	C₂	#10592586	1	1	yes	characteristic
230	C₂	#10592683	1	1	yes	characteristic
231	C₂	D₈×C₃⋊Dic₃	1	1	yes	characteristic
232	C₂	Q₁₆×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×C₃⋊D₃
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#107	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#107	C₃⋊D₃
hypocenter	C₃²	#1942
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
13. C₃²×#107	C₄	semidirect
47. C₃×#107	Dic₃	semidirect
70. C₃²×C₈	C₄⋊C₄	semidirect
117. #107	C₃⋊Dic₃	semidirect
139. C₃×C₈	C₄⋊Dic₃	semidirect
181. C₃²	#1942	semidirect
202. C₈	C₄⋊(C₃⋊Dic₃)	semidirect
225. C₃	#70612	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#107	2
C₂×#63645	2
C₂×C₈×C₃⋊Dic₃	2
#10592683	2
#10589395	2
#70612	3