Subgroups — Q₈×#3748 #11861505

Order 1152.

1164 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	#10592723	C₂	1	6	yes	maximal, complemented
3	Q₈×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#10592623	C₂	1	3	yes	maximal
5	C₄×#3748	C₂	1	3	yes	maximal
6	Q₈×#208	—	3	4	no	maximal
7	C₂²×C₃²×Q₈	C₄	1	1	yes	characteristic
8	C₃²×C₄×Q₈	C₄	1	3	yes	—
9	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
10	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Q₈	C₂²	1	3	yes	—
12	C₂×C₃²×C₄⋊C₄	C₄	1	3	yes	—
13	C₂×C₃²×C₄²	C₂²	1	3	yes	—
14	C₂×#3748	C₂²	1	3	yes	—
15	C₂×#3748	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	6	yes	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	6	yes	—
18	#63630	C₂²	1	6	yes	—
19	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	6	yes	—
20	C₂×C₃×C₄×Q₈	D₃	1	4	yes	—
21	#5024	—	3	24	no	—
22	Q₈×C₃⋊C₈	—	3	8	no	—
23	#4924	—	3	12	no	—
24	C₄×#208	—	3	12	no	—
25	C₂×C₃²×Q₈	—	2	4	no	—
26	C₂×C₃²×Q₈	C₂×C₄	1	3	yes	—
27	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
29	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₃²×C₄⋊C₄	C₂×C₄	1	6	yes	—
32	C₃²×C₄⋊C₄	C₂×C₄	1	3	yes	—
33	C₃²×C₄⋊C₄	C₂×C₄	1	3	yes	—
34	C₃²×C₄²	C₂×C₄	1	3	yes	—
35	C₃²×C₄²	C₂×C₄	1	3	yes	—
36	C₃²×C₄²	C₂³	1	3	yes	—
37	C₃²×C₄²	C₂³	1	3	yes	—
38	#3748	—	2	6	no	—
39	#3748	Q₈	1	4	yes	direct product factor
40	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	6	yes	—
41	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
42	Q₈×M₁₆	—	9	1	no	Sylow
43	C₂²×C₃×Q₈	Dic₃	1	4	yes	—
44	C₃×C₄×Q₈	Dic₃	1	12	yes	—
45	C₃×C₄×Q₈	Dic₃	1	4	yes	—
46	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
47	C₃×C₄×Q₈	C₂×D₃	1	12	yes	—
48	C₂×C₃×C₄⋊C₄	Dic₃	1	12	yes	—
49	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
50	C₂×#208	—	3	4	no	—
51	C₂×#208	—	3	12	no	—
52	C₄⋊(C₃⋊C₈)	—	3	24	no	—
53	C₄⋊(C₃⋊C₈)	—	3	24	no	—
54	#803	—	3	24	no	—
55	C₄×C₃⋊C₈	—	3	24	no	—
56	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
57	C₃²×Q₈	—	2	6	no	—
58	C₃²×Q₈	M₁₆	1	4	yes	complemented
59	C₂×C₃²×C₄	—	2	6	no	—
60	C₂×C₃²×C₄	—	4	3	no	—
61	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
62	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
63	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
64	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
66	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃²×C₄	#41	1	2	yes	—
68	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
69	C₃⋊(C₃⋊C₈)	—	2	6	no	—
70	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	4	yes	complemented
71	C₂×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
72	#446	—	9	6	no	—
73	C₈×Q₈	—	9	2	no	—
74	#423	—	9	3	no	—
75	C₄×M₁₆	—	9	3	no	—
76	C₂×C₃×Q₈	—	2	16	no	—
77	C₂×C₃×Q₈	C₂×Dic₃	1	12	yes	—
78	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
79	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
80	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
81	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
82	C₃×C₄⋊C₄	C₂×Dic₃	1	24	yes	—
83	C₃×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
84	C₃×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
85	C₃×C₄²	C₂×Dic₃	1	12	yes	—
86	C₃×C₄²	C₂×Dic₃	1	12	yes	—
87	C₃×C₄²	C₂²×D₃	1	12	yes	—
88	C₃×C₄²	C₂²×D₃	1	12	yes	—
89	#208	—	3	16	no	—
90	#208	—	6	24	no	—
91	C₂×C₃⋊C₈	—	3	8	no	—
92	C₂×C₃⋊C₈	—	3	24	no	—
93	C₂²×C₃²	—	2	1	no	—
94	C₂²×C₃²	—	2	1	no	—
95	C₂²×C₃²	C₄×Q₈	1	2	yes	—
96	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
97	C₃²×C₄	—	2	3	no	—
98	C₃²×C₄	—	2	3	no	—
99	C₃²×C₄	—	2	3	no	—
100	C₃²×C₄	C₄×Q₈	1	2	yes	—
101	C₃²×C₄	C₂×M₁₆	1	6	yes	—
102	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
103	C₃²×C₄	C₂×#41	1	1	yes	characteristic
104	C₂²×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
105	C₂×M₁₆	—	9	1	no	—
106	C₂×M₁₆	—	9	3	no	—
107	C₄×Q₈	C₃⋊Dic₃	1	3	yes	—
108	C₄×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
109	C₄×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
110	C₄×Q₈	C₂×C₃⋊D₃	1	3	yes	—
111	C₂×C₄⋊C₄	C₃⋊Dic₃	1	3	yes	—
112	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
113	C₄⋊C₈	—	9	6	no	—
114	C₄⋊C₈	—	9	6	no	—
115	#97	—	9	6	no	—
116	C₄×C₈	—	9	6	no	—
117	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
118	C₃×Q₈	—	2	24	no	—
119	C₃×Q₈	#208	1	16	yes	complemented
120	C₂×C₃×C₄	—	2	24	no	—
121	C₂×C₃×C₄	—	4	12	no	—
122	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
123	C₂×C₃×C₄	#239	1	8	yes	—
124	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
125	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
126	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
127	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
128	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
129	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
130	C₃⋊C₈	—	3	16	no	—
131	C₃⋊C₈	—	6	24	no	—
132	C₂×C₃²	—	2	2	no	—
133	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂×C₃²	C₂²×M₁₆	1	1	yes	characteristic
135	C₂×C₃²	#568	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	C₂×C₃⋊Dic₃	1	3	yes	—
138	C₂×Q₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
140	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
141	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
142	M₁₆	—	9	4	no	—
143	M₁₆	—	18	6	no	—
144	C₂×C₈	—	9	2	no	—
145	C₂×C₈	—	9	6	no	—
146	C₄⋊C₄	C₂×C₃⋊Dic₃	1	6	yes	—
147	C₄⋊C₄	C₂×C₃⋊Dic₃	1	3	yes	—
148	C₄⋊C₄	C₂×C₃⋊Dic₃	1	3	yes	—
149	C₄²	C₂×C₃⋊Dic₃	1	3	yes	—
150	C₄²	C₂×C₃⋊Dic₃	1	3	yes	—
151	C₄²	C₂²×C₃⋊D₃	1	3	yes	—
152	C₄²	C₂²×C₃⋊D₃	1	3	yes	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	Q₈×Dic₃	1	8	yes	—
156	C₂²×C₃	C₂³×Dic₃	1	4	yes	—
157	C₃×C₄	—	2	12	no	—
158	C₃×C₄	—	2	12	no	—
159	C₃×C₄	—	2	12	no	—
160	C₃×C₄	C₂×#208	1	24	yes	—
161	C₃×C₄	Q₈×Dic₃	1	8	yes	—
162	C₃×C₄	C₂×D₃×Q₈	1	4	yes	—
163	C₃×C₄	C₂×#239	1	4	yes	—
164	C₃²	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
166	Q₈	—	2	6	no	—
167	Q₈	#3748	1	4	yes	direct product factor
168	C₂×C₄	—	2	6	no	—
169	C₂×C₄	—	4	3	no	—
170	C₂×C₄	Q₈×C₃⋊D₃	1	2	yes	—
171	C₂×C₄	#3832	1	2	yes	—
172	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
173	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
174	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
175	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
176	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
177	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
178	C₈	—	9	4	no	—
179	C₈	—	18	6	no	—
180	C₂×C₃	—	2	8	no	—
181	C₂×C₃	C₂²×#208	1	4	yes	—
182	C₂×C₃	C₂×Q₈×Dic₃	1	4	yes	—
183	C₂×C₃	#5819	1	4	yes	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	Q₈×C₃⋊Dic₃	1	2	yes	—
187	C₂²	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	3	no	—
191	C₄	C₂×#3748	1	6	yes	—
192	C₄	Q₈×C₃⋊Dic₃	1	2	yes	—
193	C₄	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
194	C₄	C₂×#3832	1	1	yes	characteristic
195	C₃	Q₈×#208	1	4	yes	complemented
196	C₂	—	2	2	no	—
197	C₂	C₂²×#3748	1	1	yes	characteristic
198	C₂	C₂×Q₈×C₃⋊Dic₃	1	1	yes	characteristic
199	C₂	#10596290	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂³×C₄
Fitting	C₂×C₃²×C₄×Q₈	C₂
prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	Q₈×M₁₆
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

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
39. #3748	Q₈	direct (N × H)
58. C₃²×Q₈	M₁₆	semidirect
70. C₃⋊(C₃⋊C₈)	C₂×Q₈	semidirect
119. C₃×Q₈	#208	semidirect
164. C₃²	Q₈×M₁₆	semidirect
167. Q₈	#3748	direct (N × H)
195. C₃	Q₈×#208	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×Q₈	2
#10592723	2
Q₈×C₃⋊(C₃⋊C₈)	2
#10592623	2
C₄×#3748	2
Q₈×#208	3

Subgroups — Q8×#3748 #11861505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#3748 #11861505