Subgroups — #11861337

Order 1152.

1092 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#533	C₂	1	1	yes	maximal, characteristic
2	#10592723	C₂	1	4	yes	maximal, complemented
3	Q₈×C₃⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#10592698	C₂	1	2	yes	maximal
5	#10592696	C₂	1	2	yes	maximal
6	#10592623	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	#83328	—	3	4	no	maximal
9	C₃²×C₄⋊Q₈	C₄	1	2	yes	—
10	C₃²×#125	C₄	1	2	yes	—
11	C₃²×C₂²⋊Q₈	C₄	1	4	yes	—
12	C₃²×C₄×Q₈	C₂²	1	4	yes	—
13	C₃²×#117	C₂²	1	2	yes	—
14	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
15	C₂×#3748	C₂²	1	2	yes	—
16	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	4	yes	—
19	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	8	yes	—
20	#63630	C₂²	1	4	yes	—
21	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
22	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
23	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	—
24	C₃×#533	D₃	1	4	yes	—
25	#5024	—	3	16	no	—
26	Q₈×C₃⋊C₈	—	3	16	no	—
27	#4999	—	3	8	no	—
28	#4997	—	3	8	no	—
29	#4924	—	3	4	no	—
30	C₂×C₄×C₃⋊C₈	—	3	4	no	—
31	C₂×C₃²×Q₈	C₂×C₄	1	4	yes	—
32	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₃²×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₃²×C₄⋊C₄	C₂×C₄	1	8	yes	—
36	C₃²×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₃²×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₃²×C₄²	C₂³	1	2	yes	—
41	C₃²×C₄²	C₂³	1	4	yes	—
42	#3748	—	2	4	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
44	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	4	yes	complemented
45	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
46	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	—
47	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
48	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
49	#2960	—	9	1	no	Sylow
50	C₃×C₄⋊Q₈	Dic₃	1	8	yes	—
51	C₃×#125	Dic₃	1	8	yes	—
52	C₃×C₂²⋊Q₈	Dic₃	1	16	yes	—
53	C₃×C₄×Q₈	C₂×D₃	1	16	yes	—
54	C₃×#117	C₂×D₃	1	8	yes	—
55	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
56	C₂×#208	—	3	8	no	—
57	C₂²×C₃⋊C₈	—	3	4	no	—
58	C₂²×C₃⋊C₈	—	3	4	no	—
59	C₄⋊(C₃⋊C₈)	—	3	16	no	—
60	C₄⋊(C₃⋊C₈)	—	3	32	no	—
61	#803	—	3	16	no	—
62	C₄×C₃⋊C₈	—	3	8	no	—
63	C₄×C₃⋊C₈	—	3	8	no	—
64	C₄×C₃⋊C₈	—	3	16	no	—
65	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
66	C₃²×Q₈	—	2	8	no	—
67	C₂×C₃²×C₄	—	2	4	no	—
68	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
69	C₂×C₃²×C₄	C₂²×C₄	1	8	yes	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
72	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
73	C₂×C₃²×C₄	#41	1	2	yes	—
74	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
75	C₃⋊(C₃⋊C₈)	—	2	2	no	—
76	C₃⋊(C₃⋊C₈)	—	2	4	no	—
77	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	2	yes	—
78	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	2	yes	—
79	#533	C₃⋊D₃	1	1	yes	characteristic
80	#446	—	9	4	no	—
81	C₈×Q₈	—	9	4	no	—
82	#424	—	9	2	no	—
83	#423	—	9	1	no	—
84	#405	—	9	2	no	—
85	C₂×C₄×C₈	—	9	1	no	—
86	C₂×C₃×Q₈	C₂×Dic₃	1	16	yes	—
87	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
88	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
89	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
90	C₃×C₄⋊C₄	C₂×Dic₃	1	32	yes	—
91	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
92	C₃×C₂²⋊C₄	C₂×Dic₃	1	16	yes	—
93	C₃×C₄²	C₂×Dic₃	1	4	yes	—
94	C₃×C₄²	C₂×Dic₃	1	4	yes	—
95	C₃×C₄²	C₂²×D₃	1	8	yes	—
96	C₃×C₄²	C₂²×D₃	1	16	yes	—
97	#208	—	6	16	no	—
98	C₂×C₃⋊C₈	—	3	16	no	—
99	C₂×C₃⋊C₈	—	3	4	no	—
100	C₂×C₃⋊C₈	—	3	4	no	—
101	C₂×C₃⋊C₈	—	3	8	no	—
102	C₂×C₃⋊C₈	—	3	16	no	—
103	C₂×C₃⋊C₈	—	6	8	no	—
104	C₂²×C₃²	—	2	2	no	—
105	C₂²×C₃²	C₄×Q₈	1	2	yes	—
106	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
107	C₃²×C₄	—	2	2	no	—
108	C₃²×C₄	—	2	8	no	—
109	C₃²×C₄	C₄×Q₈	1	2	yes	—
110	C₃²×C₄	#131	1	4	yes	—
111	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
112	C₃²×C₄	C₂×#41	1	1	yes	characteristic
113	C₂×M₁₆	—	9	2	no	—
114	C₂²×C₈	—	9	1	no	—
115	C₂²×C₈	—	9	1	no	—
116	C₄⋊Q₈	C₃⋊Dic₃	1	2	yes	—
117	#125	C₃⋊Dic₃	1	2	yes	—
118	C₂²⋊Q₈	C₃⋊Dic₃	1	4	yes	—
119	C₄×Q₈	C₂×C₃⋊D₃	1	4	yes	—
120	#117	C₂×C₃⋊D₃	1	2	yes	—
121	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₄⋊C₈	—	9	4	no	—
123	C₄⋊C₈	—	9	8	no	—
124	#97	—	9	4	no	—
125	C₄×C₈	—	9	2	no	—
126	C₄×C₈	—	9	2	no	—
127	C₄×C₈	—	9	4	no	—
128	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
129	C₃×Q₈	—	2	32	no	—
130	C₂×C₃×C₄	—	2	16	no	—
131	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
132	C₂×C₃×C₄	#239	1	8	yes	—
133	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
134	C₂×C₃×C₄	C₂²×Dic₃	1	32	yes	—
135	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
136	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
137	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
138	C₃⋊C₈	—	3	8	no	—
139	C₃⋊C₈	—	3	8	no	—
140	C₃⋊C₈	—	6	8	no	—
141	C₃⋊C₈	—	6	16	no	—
142	C₂×C₃²	—	2	2	no	—
143	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₃²	C₂×#131	1	2	yes	—
145	C₂×Q₈	C₂×C₃⋊Dic₃	1	4	yes	—
146	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
147	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
148	M₁₆	—	18	4	no	—
149	C₂×C₈	—	9	1	no	—
150	C₂×C₈	—	9	1	no	—
151	C₂×C₈	—	9	4	no	—
152	C₂×C₈	—	9	2	no	—
153	C₂×C₈	—	9	4	no	—
154	C₂×C₈	—	18	2	no	—
155	C₄⋊C₄	C₂×C₃⋊Dic₃	1	4	yes	—
156	C₄⋊C₄	C₂×C₃⋊Dic₃	1	8	yes	—
157	C₄⋊C₄	C₂×C₃⋊Dic₃	1	4	yes	—
158	C₂²⋊C₄	C₂×C₃⋊Dic₃	1	4	yes	—
159	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
160	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
161	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
162	C₄²	C₂²×C₃⋊D₃	1	4	yes	—
163	C₂²×C₃	—	2	8	no	—
164	C₂²×C₃	Q₈×Dic₃	1	8	yes	—
165	C₂²×C₃	C₂³×Dic₃	1	4	yes	—
166	C₃×C₄	—	2	8	no	—
167	C₃×C₄	—	2	32	no	—
168	C₃×C₄	Q₈×Dic₃	1	8	yes	—
169	C₃×C₄	#948	1	16	yes	—
170	C₃×C₄	C₂×D₃×Q₈	1	4	yes	—
171	C₃×C₄	C₂×#239	1	4	yes	—
172	C₃²	#2960	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	Q₈×C₃⋊D₃	1	2	yes	—
177	C₂×C₄	#3832	1	2	yes	—
178	C₂×C₄	C₂²×C₃⋊Dic₃	1	2	yes	—
179	C₂×C₄	C₂²×C₃⋊Dic₃	1	8	yes	—
180	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₃⋊Dic₃	1	2	yes	—
182	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
183	C₈	—	9	2	no	—
184	C₈	—	9	2	no	—
185	C₈	—	18	2	no	—
186	C₈	—	18	4	no	—
187	C₂×C₃	—	2	8	no	—
188	C₂×C₃	C₂×Q₈×Dic₃	1	4	yes	—
189	C₂×C₃	C₂×#948	1	8	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	Q₈×C₃⋊Dic₃	1	2	yes	—
192	C₂²	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	8	no	—
195	C₄	Q₈×C₃⋊Dic₃	1	2	yes	—
196	C₄	#64157	1	4	yes	—
197	C₄	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
198	C₄	C₂×#3832	1	1	yes	characteristic
199	C₃	#83328	1	4	yes	complemented
200	C₂	—	2	2	no	—
201	C₂	C₂×Q₈×C₃⋊Dic₃	1	1	yes	characteristic
202	C₂	C₂×#64157	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂³×C₄
Fitting	C₃²×#533	C₂
prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	#533	C₃⋊D₃
hypocenter	C₃²	#2960
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
2. #10592723	C₂	semidirect
3. Q₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
44. C₂×C₃⋊(C₃⋊C₈)	Q₈	semidirect
172. C₃²	#2960	semidirect
199. C₃	#83328	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#533	2
#10592723	2
Q₈×C₃⋊(C₃⋊C₈)	2
#10592698	2
#10592696	2
#10592623	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#83328	3