Finite Group Databasebeta

Subgroups — #11782656

Order 1152.

232 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10596408	C₂	1	1	yes	maximal, characteristic
2	AGL(1,𝔽₉)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#10596404	C₂	1	1	yes	maximal, characteristic, complemented
4	#10596402	C₂	1	1	yes	maximal, characteristic
5	#10593536	C₂	1	1	yes	maximal, characteristic, complemented
6	#10593292	C₂	1	1	yes	maximal, characteristic
7	#10593251	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AΓL(1,𝔽₉)	—	2	2	no	—
9	C₄⋊PSU(3,𝔽₂)	C₂²	1	1	yes	characteristic
10	C₄⋊PSU(3,𝔽₂)	C₂²	1	1	yes	characteristic
11	#64231	C₂²	1	1	yes	characteristic
12	#64227	—	2	1	no	—
13	#64222	C₂²	1	1	yes	characteristic
14	C₄×AGL(1,𝔽₉)	C₂²	1	1	yes	characteristic
15	C₄×AGL(1,𝔽₉)	C₂²	1	1	yes	characteristic
16	AGL(1,𝔽₉)⋊C₄	—	2	2	no	—
17	PSU(3,𝔽₂)⋊C₄	—	2	2	no	—
18	GO⁺(4,𝔽₂)⋊C₄	—	2	2	no	—
19	#63796	—	2	1	no	—
20	#63769	—	2	1	no	—
21	C₈×#157	C₂²	1	1	yes	characteristic
22	#63746	—	2	1	no	—
23	C₂×PSU(3,𝔽₂)	—	2	1	no	—
24	C₂×PSU(3,𝔽₂)	—	2	1	no	—
25	C₂×GO⁺(4,𝔽₂)	—	2	1	no	—
26	C₂×GO⁺(4,𝔽₂)	—	2	1	no	—
27	C₂×AGL(1,𝔽₉)	D₄	1	2	yes	—
28	C₂×AGL(1,𝔽₉)	D₄	1	2	yes	—
29	AΓL(1,𝔽₉)	—	4	4	no	—
30	#3802	—	2	1	no	—
31	#3797	—	2	1	no	—
32	C₄⋊#157	—	2	1	no	—
33	C₄⋊#157	—	2	1	no	—
34	C₄×#157	C₂³	1	1	yes	characteristic
35	#3787	D₄	1	1	yes	characteristic
36	#3777	—	2	1	no	—
37	#3777	—	2	1	no	—
38	#3775	—	4	2	no	—
39	#3773	—	2	1	no	—
40	#3773	—	2	1	no	—
41	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
42	C₃×#204	—	4	1	no	—
43	#3715	—	4	1	no	—
44	#1714	—	9	1	no	maximal, Sylow
45	D₃×SD₁₆	—	12	1	no	—
46	C₂×D₃²	—	4	1	no	—
47	C₂×#157	—	4	1	no	—
48	C₂×#157	—	4	1	no	—
49	C₂×#157	C₂×D₄	1	1	yes	characteristic
50	C₂×#157	C₂×D₄	1	1	yes	characteristic
51	PSU(3,𝔽₂)	—	4	2	no	—
52	PSU(3,𝔽₂)	—	4	2	no	—
53	GO⁺(4,𝔽₂)	—	4	2	no	—
54	GO⁺(4,𝔽₂)	—	4	2	no	—
55	AGL(1,𝔽₉)	—	2	2	no	—
56	AGL(1,𝔽₉)	—	2	2	no	—
57	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
58	C₃×D₁₂	—	4	1	no	—
59	C₃×Dic₆	—	4	1	no	—
60	#627	—	4	1	no	—
61	#625	—	4	1	no	—
62	#624	—	4	1	no	—
63	#622	SD₁₆	1	2	yes	complemented
64	C₃²×C₈	SD₁₆	1	1	yes	characteristic, complemented
65	C₃⋊(C₃⋊C₈)	SD₁₆	1	1	yes	characteristic, complemented
66	#500	—	9	1	no	—
67	#498	—	9	1	no	—
68	#492	—	9	1	no	—
69	#492	—	9	1	no	—
70	#487	—	9	1	no	—
71	#486	—	9	1	no	—
72	#322	—	9	1	no	—
73	D₃×Q₈	—	12	1	no	—
74	D₃×D₄	—	12	1	no	—
75	C₃×SD₁₆	—	12	1	no	—
76	#215	—	12	1	no	—
77	#214	—	12	1	no	—
78	#204	—	4	1	no	—
79	D₃×C₈	—	12	1	no	—
80	C₂×C₃⋊D₃	#127	1	1	yes	characteristic
81	C₂×C₃×D₃	—	8	1	no	—
82	D₃²	—	8	1	no	—
83	D₃²	—	8	1	no	—
84	#157	—	8	1	no	—
85	#157	—	8	1	no	—
86	#157	AGL(1,ℤ/8)	1	2	yes	—
87	#157	#137	1	2	yes	—
88	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
89	C₃⋊Dic₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₃×Dic₃	—	8	1	no	—
91	C₂×SD₁₆	—	18	1	no	—
92	C₂×SD₁₆	—	18	2	no	—
93	C₂×SD₁₆	—	18	1	no	—
94	C₄⋊Q₈	—	9	1	no	—
95	C₄⋊Q₈	—	9	1	no	—
96	C₄⋊Q₈	—	9	1	no	—
97	#127	—	9	1	no	—
98	#107	—	18	1	no	—
99	#107	—	18	1	no	—
100	#106	—	18	2	no	—
101	#103	—	18	2	no	—
102	#102	—	18	2	no	—
103	C₄×C₈	—	9	1	no	—
104	C₄×C₈	—	9	1	no	—
105	C₄×C₈	—	9	1	no	—
106	C₂²×D₃	—	24	1	no	—
107	C₃×Q₈	—	12	1	no	—
108	C₃×D₄	—	12	1	no	—
109	#67	—	24	1	no	—
110	D₁₂	—	4	1	no	—
111	C₄×D₃	—	12	1	no	—
112	C₄×D₃	—	24	1	no	—
113	Dic₆	—	4	1	no	—
114	Dic₆	—	24	1	no	—
115	C₃×C₈	—	4	1	no	—
116	C₃⋊C₈	—	12	1	no	—
117	C₂×C₃²	#492	1	1	yes	characteristic
118	C₃⋊D₃	#496	1	1	yes	characteristic
119	C₃⋊D₃	#497	1	1	yes	characteristic
120	C₃×D₃	—	16	1	no	—
121	C₂×Q₈	—	18	1	no	—
122	C₂×Q₈	—	18	1	no	—
123	C₂×Q₈	—	18	1	no	—
124	C₂×D₄	—	18	1	no	—
125	C₂×D₄	—	18	1	no	—
126	C₂×D₄	—	18	1	no	—
127	SD₁₆	—	12	1	no	—
128	SD₁₆	—	36	1	no	—
129	SD₁₆	—	36	2	no	—
130	SD₁₆	—	36	4	no	—
131	C₂×C₈	—	9	1	no	—
132	C₂×C₈	—	9	1	no	—
133	C₂×C₈	—	9	2	no	—
134	C₂×C₈	—	9	2	no	—
135	C₄⋊C₄	—	18	1	no	—
136	C₄⋊C₄	—	18	1	no	—
137	C₄⋊C₄	—	18	1	no	—
138	C₄⋊C₄	—	18	1	no	—
139	C₄⋊C₄	—	18	1	no	—
140	C₄⋊C₄	—	18	1	no	—
141	C₄²	—	9	1	no	—
142	C₂²×C₃	—	24	1	no	—
143	C₂×D₃	—	8	1	no	—
144	C₂×D₃	—	12	1	no	—
145	C₂×D₃	—	48	1	no	—
146	C₂×D₃	—	48	1	no	—
147	C₃×C₄	—	4	1	no	—
148	C₃×C₄	—	24	1	no	—
149	Dic₃	—	8	1	no	—
150	Dic₃	—	12	1	no	—
151	C₃²	#1714	1	1	yes	characteristic, complemented, Sylow
152	C₂³	—	36	1	no	—
153	Q₈	—	12	1	no	—
154	Q₈	—	36	1	no	—
155	Q₈	—	36	2	no	—
156	Q₈	—	36	2	no	—
157	D₄	—	12	1	no	—
158	D₄	—	36	1	no	—
159	D₄	—	36	2	no	—
160	D₄	—	36	2	no	—
161	C₂×C₄	—	9	1	no	—
162	C₂×C₄	—	9	1	no	—
163	C₂×C₄	—	9	1	no	—
164	C₂×C₄	—	36	1	no	—
165	C₂×C₄	—	36	1	no	—
166	C₂×C₄	—	36	1	no	—
167	C₈	—	9	2	no	—
168	C₈	—	9	1	no	—
169	C₈	—	18	2	no	—
170	C₈	—	18	2	no	—
171	C₈	AΓL(1,𝔽₉)	1	1	yes	characteristic, complemented
172	C₂×C₃	—	4	1	no	—
173	C₂×C₃	—	48	1	no	—
174	D₃	—	12	1	no	—
175	D₃	—	12	1	no	—
176	D₃	—	16	1	no	—
177	C₂²	—	9	1	no	—
178	C₂²	—	24	1	no	—
179	C₂²	—	72	1	no	—
180	C₂²	—	72	1	no	—
181	C₄	—	9	2	no	—
182	C₄	—	9	2	no	—
183	C₄	—	9	1	no	—
184	C₄	—	24	1	no	—
185	C₄	—	72	1	no	—
186	C₄	—	72	1	no	—
187	C₄	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic
188	C₃	—	4	1	no	—
189	C₂	—	9	1	no	—
190	C₂	—	9	1	no	—
191	C₂	—	48	1	no	—
192	C₂	#10596408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10596408
commutator, cosocle	C₄×#157	C₂³
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₃²×C₈	SD₁₆
socle	C₂×C₃²	#492
prime radical	#64231	C₂²
hypercenter	C₈	AΓL(1,𝔽₉)
hypocenter	C₃²	#1714
LCS	G ▷ C₄×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. AGL(1,𝔽₉)⋊Q₈	C₂	semidirect
3. #10596404	C₂	semidirect
5. #10593536	C₂	semidirect
7. #10593251	C₂	semidirect
63. #622	SD₁₆	semidirect
64. C₃²×C₈	SD₁₆	semidirect
65. C₃⋊(C₃⋊C₈)	SD₁₆	semidirect
151. C₃²	#1714	semidirect
171. C₈	AΓL(1,𝔽₉)	semidirect

Maximal subgroups

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

subgroup	index
#10596408	2
AGL(1,𝔽₉)⋊Q₈	2
#10596404	2
#10596402	2
#10593536	2
#10593292	2
#10593251	2
#1714	9