Subgroups — C₂×C₃×Q₈×C₃⋊C₈ #11861254

Order 1152.

1029 conjugacy classes of subgroups.

The subgroup lattice has 246 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	C₃×Q₈×C₃⋊C₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	3	yes	maximal
4	C₂×C₃×C₄×C₃⋊C₈	C₂	1	3	yes	maximal
5	C₂×C₃×C₈×Q₈	—	3	1	no	maximal
6	C₂×Q₈×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×Q₈	C₄	1	1	yes	characteristic
8	C₃²×C₄×Q₈	C₄	1	4	yes	—
9	C₃²×C₄×Q₈	C₂²	1	4	yes	—
10	C₂×C₃²×C₄⋊C₄	C₄	1	3	yes	—
11	C₂×C₃²×C₄²	C₂²	1	3	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	3	yes	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	12	yes	—
15	C₃×C₄×C₃⋊C₈	C₂²	1	12	yes	—
16	C₂×C₃×C₄×Q₈	—	2	1	no	—
17	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
18	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₃×C₈×Q₈	—	3	8	no	—
20	C₂×C₃×C₄⋊C₈	—	3	3	no	—
21	C₂×C₃×C₄×C₈	—	3	3	no	—
22	Q₈×C₃⋊C₈	C₂×C₃	1	8	yes	direct product factor
23	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	3	yes	—
24	C₂×C₄×C₃⋊C₈	C₂×C₃	1	3	yes	—
25	C₂×C₃²×Q₈	C₈	1	8	yes	complemented
26	C₂×C₃²×Q₈	C₂×C₄	1	4	yes	—
27	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
28	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₃²×C₄⋊C₄	C₂×C₄	1	12	yes	—
31	C₃²×C₄²	C₂×C₄	1	6	yes	—
32	C₃²×C₄²	C₂³	1	6	yes	—
33	C₂×C₃×C₃⋊C₈	—	2	6	no	—
34	C₂×C₃×C₃⋊C₈	Q₈	1	4	yes	direct product factor
35	C₂×C₃×C₃⋊C₈	C₂³	1	6	yes	—
36	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
37	C₂×C₈×Q₈	—	3	1	no	Sylow
38	C₂²×C₃×Q₈	—	2	1	no	—
39	C₂²×C₃×Q₈	Dic₃	1	1	yes	characteristic
40	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	—	3	1	no	—
42	C₂²×C₃×C₈	—	3	3	no	—
43	C₃×C₄×Q₈	—	2	4	no	—
44	C₃×C₄×Q₈	—	2	4	no	—
45	C₃×C₄×Q₈	Dic₃	1	4	yes	—
46	C₃×C₄×Q₈	C₃×C₄	1	4	yes	—
47	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
48	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
49	C₂×C₃×C₄⋊C₄	—	2	3	no	—
50	C₂×C₃×C₄⋊C₄	Dic₃	1	3	yes	—
51	C₂×C₃×C₄⋊C₄	C₃×C₄	1	3	yes	—
52	C₂×C₃×C₄²	—	2	3	no	—
53	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
54	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
55	C₂²×C₃⋊C₈	C₂²×C₃	1	3	yes	—
56	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	C₃×C₄⋊C₈	—	3	12	no	—
58	C₃×C₄×C₈	—	3	12	no	—
59	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	12	yes	—
60	C₄×C₃⋊C₈	C₂²×C₃	1	12	yes	—
61	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
62	C₃²×Q₈	C₂×C₈	1	16	yes	complemented
63	C₂×C₃²×C₄	—	2	6	no	—
64	C₂×C₃²×C₄	C₂×C₈	1	12	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
66	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
68	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
69	C₂×C₃²×C₄	#41	1	2	yes	—
70	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
71	C₃×C₃⋊C₈	—	2	6	no	—
72	C₃×C₃⋊C₈	C₂×Q₈	1	4	yes	direct product factor
73	C₂×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
74	C₈×Q₈	—	3	8	no	—
75	C₂×C₄⋊C₈	—	3	3	no	—
76	C₂×C₄×C₈	—	3	3	no	—
77	C₂×C₃×Q₈	—	2	8	no	—
78	C₂×C₃×Q₈	—	2	4	no	—
79	C₂×C₃×Q₈	C₃⋊C₈	1	8	yes	direct product factor
80	C₂×C₃×Q₈	C₃×C₈	1	8	yes	complemented
81	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
82	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
83	C₂²×C₃×C₄	—	2	3	no	—
84	C₂²×C₃×C₄	—	2	3	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
87	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
88	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
89	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
90	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₃×C₈	—	3	4	no	—
93	C₂×C₃×C₈	—	3	2	no	—
94	C₂×C₃×C₈	—	3	6	no	—
95	C₂×C₃×C₈	—	6	6	no	—
96	C₃×C₄⋊C₄	—	2	12	no	—
97	C₃×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
98	C₃×C₄⋊C₄	C₂×C₃×C₄	1	12	yes	—
99	C₃×C₄²	—	2	6	no	—
100	C₃×C₄²	—	2	6	no	—
101	C₃×C₄²	C₂×Dic₃	1	6	yes	—
102	C₃×C₄²	C₂×C₃×C₄	1	6	yes	—
103	C₃×C₄²	C₂²×D₃	1	6	yes	—
104	C₃×C₄²	C₂³×C₃	1	6	yes	—
105	C₂×C₃⋊C₈	—	2	6	no	—
106	C₂×C₃⋊C₈	C₃×Q₈	1	4	yes	direct product factor
107	C₂×C₃⋊C₈	C₂³×C₃	1	6	yes	—
108	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
109	C₂²×C₃²	C₄×Q₈	1	2	yes	—
110	C₂²×C₃²	C₂²×C₈	1	2	yes	—
111	C₂²×C₃²	#131	1	2	yes	—
112	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
113	C₃²×C₄	—	2	6	no	—
114	C₃²×C₄	C₄×Q₈	1	2	yes	—
115	C₃²×C₄	C₂²×C₈	1	12	yes	—
116	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
117	C₃²×C₄	C₂×#41	1	1	yes	characteristic
118	C₂²×Q₈	C₃×Dic₃	1	1	yes	characteristic
119	C₂²×C₈	—	3	1	no	—
120	C₂²×C₈	—	3	3	no	—
121	C₄×Q₈	C₃×Dic₃	1	4	yes	—
122	C₄×Q₈	C₂×C₃×D₃	1	4	yes	—
123	C₂×C₄⋊C₄	C₃×Dic₃	1	3	yes	—
124	C₂×C₄²	C₂×C₃×D₃	1	3	yes	—
125	C₄⋊C₈	—	3	12	no	—
126	C₄×C₈	—	3	12	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
129	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₃×Q₈	—	2	16	no	—
131	C₃×Q₈	C₂×C₃⋊C₈	1	16	yes	direct product factor
132	C₃×Q₈	C₂×C₃×C₈	1	16	yes	complemented
133	C₂×C₃×C₄	—	2	6	no	—
134	C₂×C₃×C₄	—	2	6	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	12	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	6	no	—
141	C₂×C₃×C₄	—	2	6	no	—
142	C₂×C₃×C₄	—	4	6	no	—
143	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
144	C₂×C₃×C₄	C₂×C₃×C₈	1	12	yes	—
145	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
146	C₂×C₃×C₄	#239	1	2	yes	—
147	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
148	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
149	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
151	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
153	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
154	C₂×C₃×C₄	C₃×#41	1	2	yes	—
155	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
157	C₃×C₈	—	3	4	no	—
158	C₃×C₈	—	6	6	no	—
159	C₃⋊C₈	—	2	6	no	—
160	C₃⋊C₈	C₂×C₃×Q₈	1	4	yes	direct product factor
161	C₂×C₃²	C₈×Q₈	1	4	yes	complemented
162	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
164	C₂×C₃²	C₂×#131	1	1	yes	characteristic
165	C₂×Q₈	C₃×C₃⋊C₈	1	8	yes	direct product factor
166	C₂×Q₈	C₂×C₃×Dic₃	1	4	yes	—
167	C₂²×C₄	C₂×C₃×Dic₃	1	3	yes	—
168	C₂²×C₄	C₂×C₃×Dic₃	1	3	yes	—
169	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂×C₈	—	3	2	no	—
171	C₂×C₈	—	3	4	no	—
172	C₂×C₈	—	3	6	no	—
173	C₂×C₈	—	6	6	no	—
174	C₄⋊C₄	C₂×C₃×Dic₃	1	12	yes	—
175	C₄²	C₂×C₃×Dic₃	1	6	yes	—
176	C₄²	C₂²×C₃×D₃	1	6	yes	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	C₂²×C₃⋊C₈	1	2	yes	—
182	C₂²×C₃	Q₈×Dic₃	1	2	yes	—
183	C₂²×C₃	#948	1	2	yes	—
184	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
185	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
186	C₂²×C₃	C₃×#131	1	2	yes	—
187	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
188	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
189	C₃×C₄	—	2	6	no	—
190	C₃×C₄	—	2	6	no	—
191	C₃×C₄	—	2	12	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	4	6	no	—
196	C₃×C₄	C₂²×C₃⋊C₈	1	12	yes	—
197	C₃×C₄	Q₈×Dic₃	1	2	yes	—
198	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
199	C₃×C₄	C₂²×C₃×C₈	1	12	yes	—
200	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
201	C₃×C₄	C₂×#239	1	1	yes	characteristic
202	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
203	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
204	C₃²	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
205	C₂³	C₂²×C₃×Dic₃	1	1	yes	characteristic
206	Q₈	C₂×C₃×C₃⋊C₈	1	16	yes	direct product factor
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	C₂×C₃×C₃⋊C₈	1	12	yes	—
209	C₂×C₄	C₃×D₃×Q₈	1	2	yes	—
210	C₂×C₄	C₃×#239	1	2	yes	—
211	C₂×C₄	C₂²×C₃×Dic₃	1	6	yes	—
212	C₂×C₄	C₂²×C₃×Dic₃	1	6	yes	—
213	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
215	C₈	—	3	4	no	—
216	C₈	—	6	6	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	Q₈×C₃⋊C₈	1	4	yes	direct product factor
222	C₂×C₃	C₃×C₈×Q₈	1	4	yes	complemented
223	C₂×C₃	C₂³×C₃⋊C₈	1	1	yes	characteristic
224	C₂×C₃	C₂×Q₈×Dic₃	1	1	yes	characteristic
225	C₂×C₃	C₂×#948	1	1	yes	characteristic
226	C₂×C₃	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
227	C₂×C₃	C₂³×C₃×C₈	1	1	yes	characteristic
228	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
229	C₂²	C₂²×C₃×C₃⋊C₈	1	2	yes	—
230	C₂²	C₃×Q₈×Dic₃	1	2	yes	—
231	C₂²	C₃×#948	1	2	yes	—
232	C₂²	C₂³×C₃×Dic₃	1	1	yes	characteristic
233	C₄	—	2	6	no	—
234	C₄	C₂²×C₃×C₃⋊C₈	1	12	yes	—
235	C₄	C₃×Q₈×Dic₃	1	2	yes	—
236	C₄	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
237	C₄	C₂×C₃×#239	1	1	yes	characteristic
238	C₃	—	2	1	no	—
239	C₃	C₂×Q₈×C₃⋊C₈	1	1	yes	characteristic, direct product factor
240	C₃	C₂×C₃×C₈×Q₈	1	1	yes	characteristic, complemented
241	C₂	C₃×Q₈×C₃⋊C₈	1	4	yes	direct product factor
242	C₂	C₂³×C₃×C₃⋊C₈	1	1	yes	characteristic
243	C₂	C₂×C₃×Q₈×Dic₃	1	1	yes	characteristic
244	C₂	C₂×C₃×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂×C₃	C₂³×C₃×C₈
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴×C₃
hypercenter	C₂×C₃×C₄×Q₈	D₃
hypocenter	C₃	C₂×C₃×C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₃⋊C₈	C₂	direct (N × H)
6. C₂×Q₈×C₃⋊C₈	C₃	direct (N × H)
22. Q₈×C₃⋊C₈	C₂×C₃	direct (N × H)
25. C₂×C₃²×Q₈	C₈	semidirect
34. C₂×C₃×C₃⋊C₈	Q₈	direct (N × H)
62. C₃²×Q₈	C₂×C₈	semidirect
72. C₃×C₃⋊C₈	C₂×Q₈	direct (N × H)
79. C₂×C₃×Q₈	C₃⋊C₈	direct (N × H)
80. C₂×C₃×Q₈	C₃×C₈	semidirect
106. C₂×C₃⋊C₈	C₃×Q₈	direct (N × H)
131. C₃×Q₈	C₂×C₃⋊C₈	direct (N × H)
132. C₃×Q₈	C₂×C₃×C₈	semidirect
160. C₃⋊C₈	C₂×C₃×Q₈	direct (N × H)
161. C₂×C₃²	C₈×Q₈	semidirect
165. C₂×Q₈	C₃×C₃⋊C₈	direct (N × H)
204. C₃²	C₂×C₈×Q₈	semidirect
206. Q₈	C₂×C₃×C₃⋊C₈	direct (N × H)
221. C₂×C₃	Q₈×C₃⋊C₈	direct (N × H)
222. C₂×C₃	C₃×C₈×Q₈	semidirect
239. C₃	C₂×Q₈×C₃⋊C₈	direct (N × H)
240. C₃	C₂×C₃×C₈×Q₈	semidirect
241. C₂	C₃×Q₈×C₃⋊C₈	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
C₃×Q₈×C₃⋊C₈	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×C₃×C₈×Q₈	3
C₂×Q₈×C₃⋊C₈	3

Subgroups — C2×C3×Q8×C3⋊C8 #11861254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Q₈×C₃⋊C₈ #11861254