Subgroups — C₂×Q₈×C₄⋊(C₃⋊D₃) #11902654

Order 1152.

4704 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₄⋊(C₃⋊D₃)	C₂	1	16	yes	maximal, direct product factor
4	C₂×#64106	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×C₄⋊(C₃⋊D₃)	C₂	1	3	yes	maximal
6	C₂×#64078	C₂	1	3	yes	maximal, complemented
7	C₂×Q₈×D₁₂	—	3	4	no	maximal
8	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₃⋊D₃	—	2	16	no	—
10	C₂×Q₈×C₃⋊D₃	C₂²	1	16	yes	complemented
11	C₂²×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₃⋊D₃	C₂²	1	6	yes	—
13	C₂²×#634	C₂²	1	6	yes	—
14	C₃²×C₄×Q₈	C₂²	1	8	yes	complemented
15	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
16	C₂×C₃²×C₄²	C₂²	1	3	yes	—
17	C₂×#3752	C₂²	1	6	yes	—
18	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	3	yes	—
19	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
20	#64106	C₂²	1	48	yes	complemented
21	C₄×C₄⋊(C₃⋊D₃)	C₂²	1	24	yes	—
22	#64078	C₂²	1	24	yes	complemented
23	C₂²×D₃×Q₈	—	3	8	no	—
24	C₂×C₃×C₄×Q₈	D₃	1	4	yes	complemented
25	Q₈×D₁₂	—	3	64	no	—
26	C₂×#897	—	3	24	no	—
27	C₂×C₄×D₁₂	—	3	12	no	—
28	C₂×#869	—	3	12	no	—
29	C₂³×C₃⋊D₃	C₂³	1	2	yes	—
30	C₂×C₃²×Q₈	D₄	1	8	yes	complemented
31	C₂×C₃²×Q₈	C₂³	1	4	yes	—
32	C₂²×C₃²×C₄	C₂³	1	3	yes	—
33	C₂²×C₃²×C₄	C₂³	1	3	yes	—
34	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₃⋊Dic₃	C₂³	1	6	yes	—
36	Q₈×C₃⋊D₃	—	2	64	no	—
37	C₂×C₄⋊(C₃⋊D₃)	Q₈	1	8	yes	direct product factor
38	C₂×C₄⋊(C₃⋊D₃)	C₂³	1	4	yes	—
39	C₂×C₄×C₃⋊D₃	—	2	24	no	—
40	C₂×C₄×C₃⋊D₃	C₂³	1	24	yes	—
41	C₂×#634	—	2	24	no	—
42	C₂×#634	C₂³	1	24	yes	—
43	C₃²×C₄⋊C₄	C₂³	1	12	yes	—
44	C₃²×C₄²	C₂³	1	12	yes	—
45	#3752	C₂³	1	24	yes	—
46	C₄⋊(C₃⋊Dic₃)	C₂³	1	12	yes	—
47	C₄⋊(C₃⋊Dic₃)	C₂³	1	24	yes	—
48	C₂×D₄×Q₈	—	9	1	no	Sylow
49	C₂²×C₃×Q₈	C₂×D₃	1	4	yes	—
50	C₂×D₃×Q₈	—	3	64	no	—
51	C₂×D₃×Q₈	—	6	64	no	—
52	C₂²×D₁₂	—	3	4	no	—
53	C₂²×C₄×D₃	—	3	24	no	—
54	C₂²×Dic₆	—	3	24	no	—
55	C₃×C₄×Q₈	C₂×D₃	1	32	yes	complemented
56	C₂×C₃×C₄⋊C₄	C₂×D₃	1	12	yes	—
57	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
58	C₂×(C₂×D₃)⋊C₄	—	3	24	no	—
59	C₂×C₄⋊Dic₃	—	3	12	no	—
60	C₂×C₄⋊Dic₃	—	3	24	no	—
61	#897	—	3	192	no	—
62	C₄×D₁₂	—	3	96	no	—
63	#869	—	3	96	no	—
64	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
65	C₂²×C₃⋊D₃	—	2	4	no	—
66	C₂²×C₃⋊D₃	—	2	4	no	—
67	C₂²×C₃⋊D₃	C₂×Q₈	1	8	yes	complemented
68	C₂²×C₃⋊D₃	C₂⁴	1	4	yes	—
69	C₃²×Q₈	C₂×D₄	1	16	yes	complemented
70	C₂×C₃²×C₄	—	2	6	no	—
71	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
72	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	complemented
73	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
74	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
75	C₂×C₃²×C₄	C₂⁴	1	2	yes	—
76	C₂×C₃⋊Dic₃	—	2	12	no	—
77	C₂×C₃⋊Dic₃	C₂⁴	1	12	yes	—
78	C₄⋊(C₃⋊D₃)	C₂×Q₈	1	16	yes	direct product factor
79	C₄×C₃⋊D₃	—	2	48	no	—
80	#634	—	2	48	no	—
81	C₂³×Q₈	—	9	2	no	—
82	D₄×Q₈	—	9	16	no	—
83	C₂×C₄⋊Q₈	—	9	3	no	—
84	C₂×C₂²⋊Q₈	—	9	6	no	—
85	C₂×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
86	C₂×C₄×D₄	—	9	3	no	—
87	C₂³×D₃	—	3	8	no	—
88	C₂×C₃×Q₈	D₁₂	1	32	yes	complemented
89	C₂×C₃×Q₈	C₂²×D₃	1	16	yes	—
90	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
91	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
92	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
93	C₂²×Dic₃	—	3	24	no	—
94	D₃×Q₈	—	6	256	no	—
95	C₂×D₁₂	—	3	32	no	—
96	C₂×D₁₂	—	3	16	no	—
97	C₂×C₄×D₃	—	3	96	no	—
98	C₂×C₄×D₃	—	6	96	no	—
99	C₂×Dic₆	—	3	96	no	—
100	C₂×Dic₆	—	6	96	no	—
101	C₃×C₄⋊C₄	C₂²×D₃	1	48	yes	—
102	C₃×C₄²	C₂²×D₃	1	48	yes	—
103	(C₂×D₃)⋊C₄	—	3	96	no	—
104	C₄⋊Dic₃	—	3	48	no	—
105	C₄⋊Dic₃	—	3	96	no	—
106	C₂²×C₃²	C₂²×D₄	1	2	yes	—
107	C₂²×C₃²	C₂²×Q₈	1	2	yes	—
108	C₂²×C₃²	2₋¹⁺⁴	1	2	yes	—
109	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
110	C₂×C₃⋊D₃	—	2	16	no	—
111	C₂×C₃⋊D₃	—	2	4	no	—
112	C₂×C₃⋊D₃	—	2	4	no	—
113	C₂×C₃⋊D₃	C₂²×Q₈	1	8	yes	complemented
114	C₃²×C₄	—	2	6	no	—
115	C₃²×C₄	C₂²×D₄	1	12	yes	—
116	C₃²×C₄	C₂²×Q₈	1	4	yes	complemented
117	C₃⋊Dic₃	—	2	12	no	—
118	C₂²×Q₈	—	9	16	no	—
119	C₂²×Q₈	—	9	6	no	—
120	C₂²×Q₈	—	18	16	no	—
121	C₂²×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₂²×D₄	—	9	1	no	—
123	C₂³×C₄	—	9	6	no	—
124	C₄⋊Q₈	—	9	24	no	—
125	C₂²⋊Q₈	—	9	48	no	—
126	C₄×Q₈	C₂×C₃⋊D₃	1	8	yes	complemented
127	C₄×D₄	—	9	24	no	—
128	C₂×C₄⋊C₄	—	9	3	no	—
129	C₂×C₄⋊C₄	—	9	6	no	—
130	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	3	yes	—
131	C₂×C₂²⋊C₄	—	9	6	no	—
132	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
133	C₂³×C₃	C₂³×D₃	1	4	yes	—
134	C₂²×D₃	—	3	32	no	—
135	C₂²×D₃	—	3	16	no	—
136	C₂²×D₃	—	6	16	no	—
137	C₂²×D₃	—	6	16	no	—
138	C₃×Q₈	C₂×D₁₂	1	64	yes	complemented
139	C₂×C₃×C₄	—	2	24	no	—
140	C₂×C₃×C₄	C₂×D₁₂	1	48	yes	—
141	C₂×C₃×C₄	D₃×Q₈	1	16	yes	complemented
142	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
143	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
144	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
145	C₂×Dic₃	—	3	48	no	—
146	C₂×Dic₃	—	6	48	no	—
147	D₁₂	—	3	64	no	—
148	C₄×D₃	—	6	192	no	—
149	Dic₆	—	6	192	no	—
150	C₂×C₃²	D₄×Q₈	1	4	yes	complemented
151	C₂×C₃²	C₂³×D₄	1	1	yes	characteristic
152	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
153	C₂×C₃²	C₂×2₋¹⁺⁴	1	1	yes	characteristic
154	C₃⋊D₃	—	2	8	no	—
155	C₂⁴	—	9	2	no	—
156	C₂×Q₈	—	9	24	no	—
157	C₂×Q₈	—	18	64	no	—
158	C₂×Q₈	—	18	24	no	—
159	C₂×Q₈	C₄⋊(C₃⋊D₃)	1	8	yes	direct product factor
160	C₂×Q₈	C₂²×C₃⋊D₃	1	4	yes	—
161	C₂×D₄	—	9	4	no	—
162	C₂×D₄	—	9	8	no	—
163	C₂²×C₄	—	9	6	no	—
164	C₂²×C₄	—	9	24	no	—
165	C₂²×C₄	—	18	24	no	—
166	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
167	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
168	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
169	C₄⋊C₄	—	9	12	no	—
170	C₄⋊C₄	—	9	24	no	—
171	C₄⋊C₄	C₂²×C₃⋊D₃	1	12	yes	—
172	C₂²⋊C₄	—	9	24	no	—
173	C₄²	C₂²×C₃⋊D₃	1	12	yes	—
174	C₂²×C₃	C₂²×D₁₂	1	8	yes	—
175	C₂²×C₃	C₂×D₃×Q₈	1	8	yes	—
176	C₂²×C₃	#1010	1	8	yes	—
177	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
178	C₂×D₃	—	3	32	no	—
179	C₂×D₃	—	6	16	no	—
180	C₂×D₃	—	6	16	no	—
181	C₂×D₃	—	6	64	no	—
182	C₃×C₄	—	2	24	no	—
183	C₃×C₄	C₂²×D₁₂	1	48	yes	—
184	C₃×C₄	C₂×D₃×Q₈	1	16	yes	complemented
185	Dic₃	—	6	48	no	—
186	C₃²	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	9	8	no	—
188	C₂³	—	9	4	no	—
189	C₂³	—	18	4	no	—
190	C₂³	—	18	4	no	—
191	C₂³	C₂³×C₃⋊D₃	1	1	yes	characteristic
192	Q₈	—	18	48	no	—
193	Q₈	C₂×C₄⋊(C₃⋊D₃)	1	16	yes	direct product factor
194	D₄	—	9	16	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	9	12	no	—
197	C₂×C₄	—	18	12	no	—
198	C₂×C₄	—	18	48	no	—
199	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	12	yes	—
200	C₂×C₄	Q₈×C₃⋊D₃	1	4	yes	complemented
201	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
202	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
203	C₂×C₄	C₂³×C₃⋊D₃	1	2	yes	—
204	C₂×C₃	Q₈×D₁₂	1	16	yes	complemented
205	C₂×C₃	C₂³×D₁₂	1	4	yes	—
206	C₂×C₃	C₂²×D₃×Q₈	1	4	yes	—
207	C₂×C₃	C₂×#1010	1	4	yes	—
208	D₃	—	6	32	no	—
209	C₂²	—	9	8	no	—
210	C₂²	—	18	4	no	—
211	C₂²	—	18	4	no	—
212	C₂²	—	18	16	no	—
213	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	2	yes	—
214	C₂²	C₂×Q₈×C₃⋊D₃	1	2	yes	—
215	C₂²	#64367	1	2	yes	—
216	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
217	C₄	—	2	6	no	—
218	C₄	—	18	12	no	—
219	C₄	C₂²×C₄⋊(C₃⋊D₃)	1	12	yes	—
220	C₄	C₂×Q₈×C₃⋊D₃	1	4	yes	complemented
221	C₃	C₂×Q₈×D₁₂	1	4	yes	complemented
222	C₂	—	18	8	no	—
223	C₂	Q₈×C₄⋊(C₃⋊D₃)	1	4	yes	direct product factor
224	C₂	C₂³×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
225	C₂	C₂²×Q₈×C₃⋊D₃	1	1	yes	characteristic
226	C₂	C₂×#64367	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁵
Frattini	C₂²	C₂⁴×C₃⋊D₃
Fitting	C₂×C₃²×C₄×Q₈	C₂
socle	C₂³×C₃²	C₂⁴
prime radical	C₂²×C₄⋊(C₃⋊D₃)	C₂²
hypercenter	C₂×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₃⋊D₃	C₂	semidirect
2. C₂×C₃²×C₄×Q₈	C₂	semidirect
3. Q₈×C₄⋊(C₃⋊D₃)	C₂	direct (N × H)
4. C₂×#64106	C₂	semidirect
6. C₂×#64078	C₂	semidirect
10. C₂×Q₈×C₃⋊D₃	C₂²	semidirect
14. C₃²×C₄×Q₈	C₂²	semidirect
20. #64106	C₂²	semidirect
22. #64078	C₂²	semidirect
24. C₂×C₃×C₄×Q₈	D₃	semidirect
30. C₂×C₃²×Q₈	D₄	semidirect
37. C₂×C₄⋊(C₃⋊D₃)	Q₈	direct (N × H)
55. C₃×C₄×Q₈	C₂×D₃	semidirect
67. C₂²×C₃⋊D₃	C₂×Q₈	semidirect
69. C₃²×Q₈	C₂×D₄	semidirect
72. C₂×C₃²×C₄	C₂×Q₈	semidirect
78. C₄⋊(C₃⋊D₃)	C₂×Q₈	direct (N × H)
85. C₂×C₄×Q₈	C₃⋊D₃	semidirect
88. C₂×C₃×Q₈	D₁₂	semidirect
113. C₂×C₃⋊D₃	C₂²×Q₈	semidirect
116. C₃²×C₄	C₂²×Q₈	semidirect
126. C₄×Q₈	C₂×C₃⋊D₃	semidirect
138. C₃×Q₈	C₂×D₁₂	semidirect
141. C₂×C₃×C₄	D₃×Q₈	semidirect
150. C₂×C₃²	D₄×Q₈	semidirect
159. C₂×Q₈	C₄⋊(C₃⋊D₃)	direct (N × H)
184. C₃×C₄	C₂×D₃×Q₈	semidirect
186. C₃²	C₂×D₄×Q₈	semidirect
193. Q₈	C₂×C₄⋊(C₃⋊D₃)	direct (N × H)
200. C₂×C₄	Q₈×C₃⋊D₃	semidirect
204. C₂×C₃	Q₈×D₁₂	semidirect
220. C₄	C₂×Q₈×C₃⋊D₃	semidirect
221. C₃	C₂×Q₈×D₁₂	semidirect
223. C₂	Q₈×C₄⋊(C₃⋊D₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₃⋊D₃	2
C₂×C₃²×C₄×Q₈	2
Q₈×C₄⋊(C₃⋊D₃)	2
C₂×#64106	2
C₂×C₄×C₄⋊(C₃⋊D₃)	2
C₂×#64078	2
C₂×Q₈×D₁₂	3

Subgroups — C2×Q8×C4⋊(C3⋊D3) #11902654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₄⋊(C₃⋊D₃) #11902654