Subgroups — #11790200

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂×#63642	C₂	1	1	yes	maximal, characteristic
4	#10592683	C₂	1	2	yes	maximal
5	#10589395	C₂	1	2	yes	maximal
6	#70562	—	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×#106	C₄	1	4	yes	complemented
14	#63658	—	2	2	no	—
15	#63645	—	2	2	no	—
16	#63642	—	2	2	no	—
17	C₂×C₃×#106	D₃	1	4	yes	—
18	C₂×#818	—	3	4	no	—
19	C₂×#815	—	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	#1944	—	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×#106	Dic₃	1	16	yes	complemented
48	C₃⋊#95	—	6	8	no	—
49	#818	—	6	8	no	—
50	#815	—	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	2	no	—
64	C₂×C₃⋊Dic₃	—	2	2	no	—
65	C₂×C₃⋊Dic₃	—	4	2	no	—
66	C₂×C₃⋊Dic₃	—	4	2	no	—
67	C₂×C₃⋊Dic₃	—	4	2	no	—
68	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
69	C₃⋊(C₃⋊C₈)	—	2	2	no	—
70	C₂×#107	—	9	1	no	—
71	C₂×#106	C₃⋊D₃	1	1	yes	characteristic
72	C₂×#97	—	9	1	no	—
73	#389	—	9	2	no	—
74	#340	—	9	2	no	—
75	C₂²×C₃×C₄	—	2	8	no	—
76	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
77	C₂²×Dic₃	—	3	4	no	—
78	C₂²×Dic₃	—	3	4	no	—
79	C₂²×Dic₃	—	6	8	no	—
80	C₂×C₃×C₈	Dic₆	1	8	yes	—
81	C₂×C₃×C₈	D₁₂	1	8	yes	—
82	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
83	C₃×C₄⋊C₄	—	2	8	no	—
84	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
85	C₄⋊Dic₃	—	6	8	no	—
86	C₄⋊Dic₃	—	6	8	no	—
87	C₄⋊Dic₃	—	12	16	no	—
88	C₄×Dic₃	—	6	4	no	—
89	C₄×Dic₃	—	6	4	no	—
90	C₂×C₃⋊C₈	—	3	8	no	—
91	C₂×C₃⋊C₈	—	6	4	no	—
92	C₂×C₃⋊C₈	—	6	4	no	—
93	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₃²	#121	1	1	yes	characteristic
95	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
97	C₂²×C₃²	#137	1	2	yes	—
98	C₃²×C₄	—	4	4	no	—
99	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
101	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₃²×C₄	#125	1	1	yes	characteristic
103	C₃⋊Dic₃	—	4	1	no	—
104	C₃⋊Dic₃	—	4	1	no	—
105	C₃⋊Dic₃	—	8	2	no	—
106	C₂²×C₈	—	9	1	no	—
107	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	9	2	no	—
109	C₂×C₄⋊C₄	—	18	2	no	—
110	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
111	C₂×C₄²	—	9	1	no	—
112	#107	—	18	2	no	—
113	#106	C₃⋊Dic₃	1	4	yes	complemented
114	#97	—	18	2	no	—
115	#95	—	18	2	no	—
116	C₂³×C₃	D₃×D₄	1	4	yes	—
117	C₂×C₃×C₄	—	2	16	no	—
118	C₂×C₃×C₄	—	4	8	no	—
119	C₂×C₃×C₄	—	4	8	no	—
120	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
121	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
122	C₂×C₃×C₄	#237	1	4	yes	—
123	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
124	C₂×C₃×C₄	#239	1	4	yes	—
125	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
126	C₂×Dic₃	—	6	4	no	—
127	C₂×Dic₃	—	6	4	no	—
128	C₂×Dic₃	—	6	8	no	—
129	C₂×Dic₃	—	6	8	no	—
130	C₂×Dic₃	—	12	8	no	—
131	C₂×Dic₃	—	12	8	no	—
132	C₂×Dic₃	—	12	8	no	—
133	C₃×C₈	C₄⋊Dic₃	1	16	yes	complemented
134	C₃⋊C₈	—	6	8	no	—
135	C₂×C₃²	#389	1	1	yes	characteristic
136	C₂×C₃²	#440	1	2	yes	—
137	C₂×C₃²	#469	1	1	yes	characteristic
138	C₂×C₃²	#470	1	1	yes	characteristic
139	C₂×C₃²	#501	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	9	1	no	—
142	C₂²×C₄	—	9	1	no	—
143	C₂²×C₄	—	18	2	no	—
144	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
145	C₂×C₈	—	9	2	no	—
146	C₂×C₈	—	18	1	no	—
147	C₂×C₈	—	18	1	no	—
148	C₂×C₈	#634	1	2	yes	—
149	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
150	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	18	2	no	—
153	C₄⋊C₄	—	18	2	no	—
154	C₄⋊C₄	—	36	4	no	—
155	C₄⋊C₄	C₂×C₃⋊Dic₃	1	4	yes	—
156	C₄²	—	18	1	no	—
157	C₄²	—	18	1	no	—
158	C₂²×C₃	#888	1	4	yes	—
159	C₂²×C₃	#895	1	4	yes	—
160	C₂²×C₃	#914	1	8	yes	—
161	C₂²×C₃	#915	1	8	yes	—
162	C₂²×C₃	D₄×Dic₃	1	4	yes	—
163	C₃×C₄	—	4	16	no	—
164	C₃×C₄	#890	1	4	yes	—
165	C₃×C₄	#897	1	4	yes	—
166	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
167	C₃×C₄	Q₈×Dic₃	1	4	yes	—
168	Dic₃	—	12	4	no	—
169	Dic₃	—	12	4	no	—
170	Dic₃	—	24	8	no	—
171	C₃²	#1944	1	1	yes	characteristic, complemented, Sylow
172	C₂³	D₄×C₃⋊D₃	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	18	1	no	—
177	C₂×C₄	—	18	1	no	—
178	C₂×C₄	—	18	2	no	—
179	C₂×C₄	—	18	2	no	—
180	C₂×C₄	—	36	2	no	—
181	C₂×C₄	—	36	2	no	—
182	C₂×C₄	—	36	2	no	—
183	C₂×C₄	C₂×#634	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
185	C₂×C₄	#3830	1	1	yes	characteristic
186	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
187	C₂×C₄	#3832	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
189	C₈	—	18	2	no	—
190	C₈	C₄⋊(C₃⋊Dic₃)	1	4	yes	complemented
191	C₂×C₃	#4856	1	8	yes	—
192	C₂×C₃	#4865	1	4	yes	—
193	C₂×C₃	#4867	1	4	yes	—
194	C₂×C₃	#4984	1	4	yes	—
195	C₂×C₃	#5164	1	8	yes	—
196	C₂²	#64097	1	1	yes	characteristic
197	C₂²	#64104	1	1	yes	characteristic
198	C₂²	#64123	1	2	yes	—
199	C₂²	#64124	1	2	yes	—
200	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
201	C₄	—	4	4	no	—
202	C₄	—	36	1	no	—
203	C₄	—	36	1	no	—
204	C₄	—	72	2	no	—
205	C₄	#64099	1	1	yes	characteristic
206	C₄	#64106	1	1	yes	characteristic
207	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
208	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
209	C₃	#70562	1	4	yes	complemented
210	C₂	#10592555	1	2	yes	—
211	C₂	#10592564	1	1	yes	characteristic
212	C₂	#10592566	1	1	yes	characteristic
213	C₂	#10592683	1	1	yes	characteristic
214	C₂	#10592863	1	2	yes	—

	subgroup	G/H
center	C₂³	D₄×C₃⋊D₃
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#106	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#106	C₃⋊D₃
hypocenter	C₃²	#1944
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
13. C₃²×#106	C₄	semidirect
47. C₃×#106	Dic₃	semidirect
68. C₃²×C₈	C₄⋊C₄	semidirect
113. #106	C₃⋊Dic₃	semidirect
133. C₃×C₈	C₄⋊Dic₃	semidirect
171. C₃²	#1944	semidirect
190. C₈	C₄⋊(C₃⋊Dic₃)	semidirect
209. C₃	#70562	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#106	2
C₂×#63645	2
C₂×#63642	2
#10592683	2
#10589395	2
#70562	3