Subgroups — C₂×#10592777 #11874940

Order 1152.

1272 conjugacy classes of subgroups.

The subgroup lattice has 207 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	C₂×#64078	C₂	1	1	yes	maximal, characteristic
3	#10592777	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63634	C₂	1	4	yes	maximal
5	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×#5078	—	3	4	no	maximal
7	C₂²×C₃²×Q₈	—	2	1	no	—
8	C₂²×#634	—	2	1	no	—
9	C₃²×C₄⋊Q₈	—	2	2	no	—
10	C₃²×C₄⋊Q₈	C₂²	1	4	yes	—
11	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
15	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
16	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	#64078	—	2	2	no	—
18	#64078	C₂²	1	4	yes	—
19	#63634	C₂²	1	16	yes	—
20	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	—
21	C₂×C₃×C₄⋊Q₈	D₃	1	4	yes	—
22	C₂×#869	—	3	4	no	—
23	#5078	—	3	32	no	—
24	C₂×#807	—	3	16	no	—
25	C₂×C₄×C₃⋊C₈	—	3	4	no	—
26	C₂×C₃²×Q₈	—	2	4	no	—
27	C₂×C₃²×Q₈	—	4	4	no	—
28	C₂²×C₃²×C₄	—	2	2	no	—
29	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊Dic₃	—	2	2	no	—
33	C₂×#634	—	2	4	no	—
34	C₂×#634	—	4	4	no	—
35	C₃²×C₄⋊C₄	—	2	4	no	—
36	C₃²×C₄⋊C₄	—	2	2	no	—
37	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
38	C₃²×C₄²	D₄	1	2	yes	—
39	C₃²×C₄²	C₂³	1	2	yes	—
40	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
41	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
42	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
43	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	8	yes	complemented
44	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	—
45	C₂×#500	—	9	1	no	Sylow
46	C₂²×C₃×Q₈	—	2	4	no	—
47	C₂²×Dic₆	—	6	4	no	—
48	C₃×C₄⋊Q₈	—	2	8	no	—
49	C₃×C₄⋊Q₈	C₂×D₃	1	16	yes	—
50	C₂×C₃×C₄⋊C₄	—	2	4	no	—
51	C₂×C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
52	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
53	C₂×C₄⋊Dic₃	—	3	8	no	—
54	C₂×C₄⋊Dic₃	—	6	4	no	—
55	C₂²×C₃⋊C₈	—	3	8	no	—
56	#869	—	3	16	no	—
57	#869	—	6	8	no	—
58	#807	—	3	64	no	—
59	C₄×C₃⋊C₈	—	3	16	no	—
60	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
61	C₃²×Q₈	—	4	8	no	—
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	—	4	4	no	—
64	C₂×C₃²×C₄	D₈	1	4	yes	—
65	C₂×C₃²×C₄	Q₁₆	1	4	yes	—
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
69	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
70	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
71	C₂×C₃⋊Dic₃	—	2	4	no	—
72	C₂×C₃⋊Dic₃	—	4	4	no	—
73	#634	—	4	8	no	—
74	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	8	yes	complemented
75	C₂×C₄⋊Q₈	—	9	1	no	—
76	C₂×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
77	#500	—	9	8	no	—
78	C₂×#107	—	9	4	no	—
79	C₂×C₄×C₈	—	9	1	no	—
80	C₂×C₃×Q₈	—	2	16	no	—
81	C₂×C₃×Q₈	—	4	16	no	—
82	C₂²×C₃×C₄	—	2	8	no	—
83	C₂²×C₃×C₄	#67	1	4	yes	—
84	C₂²×C₃×C₄	#67	1	4	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
86	C₂²×Dic₃	—	6	8	no	—
87	C₂×Dic₆	—	6	16	no	—
88	C₂×Dic₆	—	12	16	no	—
89	C₃×C₄⋊C₄	—	2	16	no	—
90	C₃×C₄⋊C₄	—	2	8	no	—
91	C₃×C₄⋊C₄	C₂²×D₃	1	16	yes	—
92	C₃×C₄²	#67	1	8	yes	—
93	C₃×C₄²	C₂²×D₃	1	8	yes	—
94	C₄⋊Dic₃	—	3	16	no	—
95	C₄⋊Dic₃	—	6	16	no	—
96	C₄⋊Dic₃	—	6	8	no	—
97	C₂×C₃⋊C₈	—	3	32	no	—
98	C₂×C₃⋊C₈	—	3	16	no	—
99	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
100	C₂²×C₃²	C₂×D₈	1	2	yes	—
101	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
102	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
103	C₃²×C₄	—	4	4	no	—
104	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
105	C₃²×C₄	C₂×D₈	1	4	yes	—
106	C₃²×C₄	C₂×Q₁₆	1	4	yes	—
107	C₃²×C₄	C₂²×Q₈	1	2	yes	—
108	C₃⋊Dic₃	—	4	4	no	—
109	C₂²×Q₈	—	2	1	no	—
110	C₂²×Q₈	—	18	1	no	—
111	C₂²×C₈	—	9	2	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	—	9	4	no	—
114	C₄⋊Q₈	—	18	2	no	—
115	C₄⋊Q₈	C₂×C₃⋊D₃	1	4	yes	—
116	C₂×C₄⋊C₄	—	2	1	no	—
117	C₂×C₄⋊C₄	—	9	2	no	—
118	C₂×C₄⋊C₄	—	18	1	no	—
119	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
120	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
121	#107	—	9	16	no	—
122	C₄×C₈	—	9	4	no	—
123	C₂³×C₃	C₂×#67	1	4	yes	—
124	C₃×Q₈	—	4	32	no	—
125	C₂×C₃×C₄	—	2	16	no	—
126	C₂×C₃×C₄	—	4	16	no	—
127	C₂×C₃×C₄	#213	1	16	yes	—
128	C₂×C₃×C₄	#216	1	16	yes	—
129	C₂×C₃×C₄	D₃×Q₈	1	16	yes	—
130	C₂×C₃×C₄	C₂×#67	1	8	yes	—
131	C₂×C₃×C₄	C₂×#67	1	8	yes	—
132	C₂×C₃×C₄	C₂×#67	1	4	yes	—
133	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
134	C₂×Dic₃	—	6	16	no	—
135	C₂×Dic₃	—	12	16	no	—
136	Dic₆	—	12	32	no	—
137	C₃⋊C₈	—	3	32	no	—
138	C₂×C₃²	#500	1	4	yes	complemented
139	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
141	C₂×C₃²	C₂²×Q₁₆	1	1	yes	characteristic
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	18	4	no	—
145	C₂×Q₈	—	36	4	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	18	2	no	—
148	C₂²×C₄	#638	1	1	yes	characteristic
149	C₂²×C₄	#638	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
151	C₂×C₈	—	9	4	no	—
152	C₂×C₈	—	9	8	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	9	4	no	—
156	C₄⋊C₄	—	18	2	no	—
157	C₄⋊C₄	—	18	4	no	—
158	C₄⋊C₄	C₂²×C₃⋊D₃	1	4	yes	—
159	C₄²	#638	1	2	yes	—
160	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
161	C₂²×C₃	C₂×#213	1	8	yes	—
162	C₂²×C₃	C₂×#216	1	8	yes	—
163	C₂²×C₃	#944	1	8	yes	—
164	C₂²×C₃	C₂²×#67	1	4	yes	—
165	C₃×C₄	—	4	16	no	—
166	C₃×C₄	C₂×#213	1	16	yes	—
167	C₃×C₄	C₂×#216	1	16	yes	—
168	C₃×C₄	#944	1	8	yes	—
169	C₃×C₄	C₂×D₃×Q₈	1	8	yes	—
170	Dic₃	—	12	16	no	—
171	C₃²	C₂×#500	1	1	yes	characteristic, complemented, Sylow
172	C₂³	C₂×#638	1	1	yes	characteristic
173	Q₈	—	4	8	no	—
174	Q₈	—	36	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	18	4	no	—
178	C₂×C₄	—	36	4	no	—
179	C₂×C₄	#3753	1	4	yes	—
180	C₂×C₄	#3756	1	4	yes	—
181	C₂×C₄	Q₈×C₃⋊D₃	1	4	yes	—
182	C₂×C₄	C₂×#638	1	2	yes	—
183	C₂×C₄	C₂×#638	1	2	yes	—
184	C₂×C₄	C₂×#638	1	1	yes	characteristic
185	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
186	C₈	—	9	8	no	—
187	C₂×C₃	#5078	1	16	yes	complemented
188	C₂×C₃	C₂²×#213	1	4	yes	—
189	C₂×C₃	C₂²×#216	1	4	yes	—
190	C₂×C₃	C₂×#944	1	4	yes	—
191	C₂²	C₂×#3753	1	2	yes	—
192	C₂²	C₂×#3756	1	2	yes	—
193	C₂²	#64153	1	2	yes	—
194	C₂²	C₂²×#638	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	36	4	no	—
197	C₄	C₂×#3753	1	4	yes	—
198	C₄	C₂×#3756	1	4	yes	—
199	C₄	#64153	1	2	yes	—
200	C₄	C₂×Q₈×C₃⋊D₃	1	2	yes	—
201	C₃	C₂×#5078	1	4	yes	complemented
202	C₂	#10592777	1	4	yes	direct product factor
203	C₂	C₂²×#3753	1	1	yes	characteristic
204	C₂	C₂²×#3756	1	1	yes	characteristic
205	C₂	C₂×#64153	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×#500
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #10592777	C₂	direct (N × H)
43. C₂×C₃⋊(C₃⋊C₈)	Q₈	semidirect
74. C₃⋊(C₃⋊C₈)	C₂×Q₈	semidirect
138. C₂×C₃²	#500	semidirect
171. C₃²	C₂×#500	semidirect
187. C₂×C₃	#5078	semidirect
201. C₃	C₂×#5078	semidirect
202. C₂	#10592777	direct (N × H)

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
C₂×#64078	2
#10592777	2
C₂×#63634	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
C₂×#5078	3

Subgroups — C2×#10592777 #11874940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10592777 #11874940