Subgroups — #11874980

Order 1152.

1068 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10595941	C₂	1	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#10592805	C₂	1	4	yes	maximal
4	#10592382	C₂	1	4	yes	maximal, complemented
5	#10592381	C₂	1	2	yes	maximal, complemented
6	#10592379	C₂	1	2	yes	maximal, complemented
7	#78442	—	3	4	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	#64135	C₂²	1	4	yes	—
11	#64133	—	2	4	no	—
12	#64079	C₂²	1	4	yes	—
13	#64078	C₂²	1	4	yes	—
14	C₄×#634	—	2	4	no	—
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃²×C₄×C₈	C₂²	1	2	yes	—
18	#63645	C₂²	1	8	yes	—
19	#63644	C₂²	1	8	yes	—
20	#5470	—	3	8	no	—
21	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
22	#5106	—	3	16	no	—
23	#4683	—	3	16	no	—
24	#4682	—	3	8	no	—
25	#4680	—	3	8	no	—
26	C₂²×C₃²×C₄	D₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂×#634	—	2	4	no	—
29	C₂×C₃²×C₈	Q₈	1	8	yes	complemented
30	C₂×C₃²×C₈	C₂³	1	2	yes	—
31	C₂×C₃²×C₈	C₂³	1	2	yes	—
32	C₃²×C₄²	D₄	1	2	yes	—
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	C₃²×C₄²	C₂³	1	1	yes	characteristic
35	#3757	—	2	4	no	—
36	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
37	C₄⋊(C₃⋊Dic₃)	C₂³	1	8	yes	—
38	#3750	—	2	8	no	—
39	C₄×C₃⋊Dic₃	—	2	4	no	—
40	#3160	—	9	1	no	Sylow
41	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
42	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
43	#926	—	3	16	no	—
44	#924	—	6	16	no	—
45	#870	—	3	16	no	—
46	#869	—	3	16	no	—
47	C₄×Dic₆	—	6	16	no	—
48	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
49	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
50	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
51	#818	—	3	32	no	—
52	#817	—	3	32	no	—
53	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
56	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
59	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
60	C₂×C₃⋊Dic₃	—	2	8	no	—
61	#634	—	4	8	no	—
62	C₃²×C₈	C₂×Q₈	1	4	yes	—
63	C₃²×C₈	C₂×Q₈	1	4	yes	—
64	#533	—	9	2	no	—
65	#500	—	9	2	no	—
66	#499	—	9	4	no	—
67	#498	—	9	2	no	—
68	#427	—	9	4	no	—
69	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
70	C₂²×C₃×C₄	D₁₂	1	8	yes	—
71	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
72	C₂×Dic₆	—	6	16	no	—
73	C₂×C₃×C₈	Dic₆	1	32	yes	complemented
74	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
75	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
76	C₃×C₄²	D₁₂	1	8	yes	—
77	C₃×C₄²	C₂²×D₃	1	4	yes	—
78	C₃×C₄²	C₂²×D₃	1	4	yes	—
79	#217	—	6	16	no	—
80	C₄⋊Dic₃	—	3	32	no	—
81	C₄⋊Dic₃	—	6	16	no	—
82	Dic₃⋊C₄	—	6	32	no	—
83	C₄×Dic₃	—	6	16	no	—
84	C₂²×C₃²	—	2	2	no	—
85	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
86	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
87	C₃²×C₄	—	2	2	no	—
88	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
89	C₃²×C₄	#135	1	4	yes	—
90	C₃²×C₄	C₂²×Q₈	1	2	yes	—
91	C₃⋊Dic₃	—	4	8	no	—
92	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
93	C₄⋊Q₈	—	9	4	no	—
94	#125	—	9	4	no	—
95	C₂²⋊Q₈	—	18	4	no	—
96	C₄×Q₈	—	18	4	no	—
97	#117	—	9	4	no	—
98	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
99	#107	—	9	8	no	—
100	#106	—	9	8	no	—
101	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
102	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
103	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₂³×C₃	C₂×D₁₂	1	4	yes	—
105	C₂×C₃×C₄	—	2	16	no	—
106	C₂×C₃×C₄	C₂×Dic₆	1	16	yes	—
107	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
108	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
109	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
110	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
111	C₂×Dic₃	—	6	32	no	—
112	Dic₆	—	12	32	no	—
113	C₃×C₈	C₂×Dic₆	1	16	yes	—
114	C₃×C₈	C₂×Dic₆	1	16	yes	—
115	C₂×C₃²	—	2	2	no	—
116	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃²	C₂×#135	1	2	yes	—
118	C₂×Q₈	—	18	4	no	—
119	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
120	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
121	C₂×C₈	#634	1	8	yes	complemented
122	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
123	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
124	C₄⋊C₄	—	9	8	no	—
125	C₄⋊C₄	—	18	8	no	—
126	C₄⋊C₄	—	18	4	no	—
127	C₂²⋊C₄	—	18	4	no	—
128	C₄²	—	18	4	no	—
129	C₄²	C₄⋊(C₃⋊D₃)	1	2	yes	—
130	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
131	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	#869	1	8	yes	—
134	C₂²×C₃	C₂²×D₁₂	1	4	yes	—
135	C₃×C₄	—	2	8	no	—
136	C₃×C₄	#869	1	8	yes	—
137	C₃×C₄	#904	1	16	yes	—
138	C₃×C₄	C₂²×Dic₆	1	8	yes	—
139	Dic₃	—	12	32	no	—
140	C₃²	#3160	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
142	Q₈	—	36	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	18	8	no	—
145	C₂×C₄	C₂×#634	1	4	yes	—
146	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
148	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
149	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
150	C₈	C₂×#634	1	4	yes	—
151	C₈	C₂×#634	1	4	yes	—
152	C₂×C₃	—	2	8	no	—
153	C₂×C₃	C₂×#869	1	4	yes	—
154	C₂×C₃	C₂×#904	1	8	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	#64078	1	2	yes	—
157	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	36	8	no	—
160	C₄	#64078	1	2	yes	—
161	C₄	#64113	1	4	yes	—
162	C₄	C₂²×#634	1	2	yes	—
163	C₃	#78442	1	4	yes	complemented
164	C₂	—	2	2	no	—
165	C₂	C₂×#64078	1	1	yes	characteristic
166	C₂	C₂×#64113	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#3160
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10592382	C₂	semidirect
5. #10592381	C₂	semidirect
6. #10592379	C₂	semidirect
29. C₂×C₃²×C₈	Q₈	semidirect
73. C₂×C₃×C₈	Dic₆	semidirect
121. C₂×C₈	#634	semidirect
140. C₃²	#3160	semidirect
163. C₃	#78442	semidirect

Maximal subgroups

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

subgroup	index
#10595941	2
C₂×C₃²×C₄×C₈	2
#10592805	2
#10592382	2
#10592381	2
#10592379	2
#78442	3