Subgroups — #11791867

Order 1152.

780 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×#64078	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#10592623	C₂	1	1	yes	maximal, characteristic
4	#10589420	C₂	1	4	yes	maximal
5	#69879	—	3	4	no	maximal
6	C₂²×#634	C₄	1	2	yes	—
7	C₂×C₃²×M₁₆	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)	C₂²	1	2	yes	—
11	C₂×#3748	C₂²	1	2	yes	—
12	#64078	—	2	1	no	—
13	#64078	—	2	1	no	—
14	#64078	—	2	1	no	—
15	#64078	—	2	1	no	—
16	C₃²×#97	—	2	1	no	—
17	C₃²×C₄×C₈	—	2	1	no	—
18	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
19	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
20	C₂×#869	—	3	4	no	—
21	C₃×C₄×M₁₆	D₃	1	4	yes	—
22	#4924	—	3	4	no	—
23	#4554	—	3	16	no	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
28	C₂×#634	—	2	2	no	—
29	C₂×#634	—	2	2	no	—
30	C₂×#634	—	4	4	no	—
31	C₃²×M₁₆	—	2	2	no	—
32	C₃²×M₁₆	Q₈	1	4	yes	complemented
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₃²×C₄²	D₄	1	1	yes	characteristic
35	C₃²×C₄²	D₄	1	1	yes	characteristic
36	C₃²×C₄²	D₄	1	1	yes	characteristic
37	C₃²×C₄²	D₄	1	1	yes	characteristic
38	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
39	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
40	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
41	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
42	#3748	—	2	4	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
44	#1997	—	9	1	no	Sylow
45	C₂²×Dic₆	—	3	8	no	—
46	C₂×C₃×M₁₆	C₂×D₃	1	8	yes	—
47	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
48	C₂×C₄⋊Dic₃	—	3	8	no	—
49	C₂×C₄⋊Dic₃	—	3	8	no	—
50	C₂×#208	—	3	8	no	—
51	#869	—	6	4	no	—
52	#869	—	6	4	no	—
53	#869	—	6	4	no	—
54	#869	—	6	4	no	—
55	C₃×#97	—	2	4	no	—
56	C₃×C₄×C₈	—	2	4	no	—
57	C₄⋊(C₃⋊C₈)	—	6	4	no	—
58	C₄⋊(C₃⋊C₈)	—	6	4	no	—
59	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	—	2	2	no	—
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
69	C₂×C₃²×C₄	#41	1	2	yes	—
70	C₂×C₃⋊Dic₃	—	2	4	no	—
71	C₂×C₃⋊Dic₃	—	4	4	no	—
72	#634	—	4	4	no	—
73	#634	—	4	4	no	—
74	C₃²×C₈	—	2	4	no	—
75	C₃⋊(C₃⋊C₈)	—	4	2	no	—
76	C₂×C₄⋊Q₈	—	9	1	no	—
77	#423	—	9	1	no	—
78	C₄×M₁₆	C₃⋊D₃	1	1	yes	characteristic
79	#343	—	9	4	no	—
80	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
81	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
83	C₂²×Dic₃	—	3	16	no	—
84	C₂×Dic₆	—	6	8	no	—
85	C₂×Dic₆	—	6	8	no	—
86	C₂×Dic₆	—	12	16	no	—
87	C₃×M₁₆	—	2	8	no	—
88	C₃×M₁₆	Dic₆	1	16	yes	complemented
89	C₂×C₃×C₈	—	2	8	no	—
90	C₃×C₄²	D₁₂	1	4	yes	—
91	C₃×C₄²	D₁₂	1	4	yes	—
92	C₃×C₄²	#67	1	4	yes	—
93	C₃×C₄²	#67	1	4	yes	—
94	C₄⋊Dic₃	—	6	8	no	—
95	C₄⋊Dic₃	—	6	8	no	—
96	C₄⋊Dic₃	—	6	8	no	—
97	C₄⋊Dic₃	—	6	8	no	—
98	#208	—	6	16	no	—
99	C₂×C₃⋊C₈	—	6	8	no	—
100	C₂²×C₃²	—	2	1	no	—
101	C₂²×C₃²	—	2	1	no	—
102	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃²	C₄×Q₈	1	2	yes	—
104	C₃²×C₄	—	2	1	no	—
105	C₃²×C₄	—	2	1	no	—
106	C₃²×C₄	#100	1	2	yes	—
107	C₃²×C₄	#101	1	2	yes	—
108	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
109	C₃²×C₄	#124	1	1	yes	characteristic
110	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
111	C₃⋊Dic₃	—	4	4	no	—
112	C₂²×Q₈	—	9	2	no	—
113	C₂×M₁₆	—	9	2	no	—
114	C₂×M₁₆	C₂×C₃⋊D₃	1	2	yes	—
115	C₄⋊Q₈	—	18	1	no	—
116	C₄⋊Q₈	—	18	1	no	—
117	C₄⋊Q₈	—	18	1	no	—
118	C₄⋊Q₈	—	18	1	no	—
119	C₂×C₄⋊C₄	—	9	2	no	—
120	C₂×C₄⋊C₄	—	9	2	no	—
121	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₄⋊C₈	—	18	1	no	—
123	C₄⋊C₈	—	18	1	no	—
124	#97	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
130	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
131	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
132	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
133	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
134	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
135	C₂×C₃×C₄	#235	1	8	yes	—
136	C₂×C₃×C₄	C₂×#67	1	4	yes	—
137	C₂×Dic₃	—	6	16	no	—
138	C₂×Dic₃	—	12	16	no	—
139	Dic₆	—	12	16	no	—
140	Dic₆	—	12	16	no	—
141	C₃×C₈	—	2	16	no	—
142	C₃⋊C₈	—	12	8	no	—
143	C₂×C₃²	—	2	2	no	—
144	C₂×C₃²	#391	1	1	yes	characteristic
145	C₂×C₃²	C₂×#100	1	1	yes	characteristic
146	C₂×C₃²	C₂×#101	1	1	yes	characteristic
147	C₂×Q₈	—	18	2	no	—
148	C₂×Q₈	—	18	2	no	—
149	C₂×Q₈	—	36	4	no	—
150	C₂²×C₄	—	9	4	no	—
151	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
152	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	18	4	no	—
156	M₁₆	#634	1	4	yes	complemented
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	18	2	no	—
159	C₄⋊C₄	—	18	2	no	—
160	C₄⋊C₄	—	18	2	no	—
161	C₄⋊C₄	—	18	2	no	—
162	C₄⋊C₄	—	18	2	no	—
163	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
164	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
165	C₄²	#638	1	1	yes	characteristic
166	C₄²	#638	1	1	yes	characteristic
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	C₄×Dic₆	1	8	yes	—
170	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	#823	1	8	yes	—
174	C₃×C₄	#824	1	8	yes	—
175	C₃×C₄	#869	1	4	yes	—
176	C₃×C₄	#875	1	4	yes	—
177	C₃×C₄	#924	1	8	yes	—
178	Dic₃	—	12	16	no	—
179	C₃²	#1997	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
181	Q₈	—	36	4	no	—
182	Q₈	—	36	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	18	4	no	—
186	C₂×C₄	—	36	4	no	—
187	C₂×C₄	#3752	1	1	yes	characteristic
188	C₂×C₄	#3752	1	1	yes	characteristic
189	C₂×C₄	#3752	1	1	yes	characteristic
190	C₂×C₄	#3752	1	1	yes	characteristic
191	C₂×C₄	C₂×#634	1	2	yes	—
192	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
193	C₂×C₄	#3828	1	2	yes	—
194	C₂×C₄	C₂×#638	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₈	—	36	2	no	—
197	C₂×C₃	—	2	8	no	—
198	C₂×C₃	#4929	1	4	yes	—
199	C₂×C₃	C₂×#823	1	4	yes	—
200	C₂×C₃	C₂×#824	1	4	yes	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₄×#634	1	2	yes	—
204	C₂²	C₂×#3752	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	36	4	no	—
208	C₄	#63650	1	2	yes	—
209	C₄	#63651	1	2	yes	—
210	C₄	#64078	1	1	yes	characteristic
211	C₄	#64084	1	1	yes	characteristic
212	C₄	#64133	1	2	yes	—
213	C₃	#69879	1	4	yes	complemented
214	C₂	—	2	2	no	—
215	C₂	#10592628	1	1	yes	characteristic
216	C₂	C₂×#63650	1	1	yes	characteristic
217	C₂	C₂×#63651	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3752
commutator, prime radical	C₂³×C₃²	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₄×M₁₆	C₃⋊D₃
hypocenter	C₃²	#1997
LCS	G ▷ C₂³×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
32. C₃²×M₁₆	Q₈	semidirect
88. C₃×M₁₆	Dic₆	semidirect
156. M₁₆	#634	semidirect
179. C₃²	#1997	semidirect
213. C₃	#69879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64078	2
C₃²×C₄×M₁₆	2
#10592623	2
#10589420	2
#69879	3