Subgroups — C₃×D₈×C₃⋊C₈ #11777141

Order 1152.

459 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃×D₄×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃×D₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×D₈	—	3	1	no	maximal
7	D₈×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×D₈	C₄	1	1	yes	characteristic
9	C₃²×C₄×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	—	2	2	no	—
11	C₃²×#107	C₄	1	1	yes	characteristic
12	C₃²×#102	C₄	1	2	yes	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#830	—	2	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×D₈	—	2	1	no	—
19	C₃×C₄×D₈	D₃	1	1	yes	characteristic
20	C₃×C₄×D₈	C₂×C₃	1	1	yes	characteristic
21	C₃×C₈×D₄	—	3	2	no	—
22	D₄×C₃⋊C₈	C₂×C₃	1	2	yes	complemented
23	C₃×#335	—	3	1	no	—
24	C₃×D₄⋊C₈	—	3	2	no	—
25	C₃×C₈²	—	3	1	no	—
26	D₄⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	complemented
27	#4458	C₂×C₃	1	1	yes	characteristic, complemented
28	C₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₃²×D₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃²×C₄	—	2	2	no	—
31	C₃²×D₈	C₈	1	4	yes	complemented
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₃²×C₂²⋊C₄	—	2	2	no	—
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₃⋊C₈	—	2	1	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	4	4	no	—
40	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
42	C₈×D₈	—	3	1	no	Sylow
43	C₂×C₃×D₈	—	2	1	no	—
44	C₂×C₃×D₈	Dic₃	1	1	yes	characteristic
45	C₂×C₃×D₈	C₃×C₄	1	1	yes	characteristic
46	C₂²×C₃×C₈	—	6	2	no	—
47	C₃×C₄×D₄	—	2	2	no	—
48	C₃×C₄×D₄	C₂×D₃	1	2	yes	—
49	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
50	C₂²×C₃⋊C₈	—	2	2	no	—
51	C₃×#107	—	2	1	no	—
52	C₃×#107	Dic₃	1	1	yes	characteristic
53	C₃×#107	C₃×C₄	1	1	yes	characteristic
54	C₃×C₄⋊C₈	—	3	2	no	—
55	C₃×#102	—	2	2	no	—
56	C₃×#102	Dic₃	1	2	yes	—
57	C₃×#102	C₃×C₄	1	2	yes	—
58	C₃×C₂²⋊C₈	—	6	2	no	—
59	C₃×C₄×C₈	—	2	1	no	—
60	C₃×C₄×C₈	—	3	1	no	—
61	C₃×C₄×C₈	—	3	1	no	—
62	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
64	#830	—	2	2	no	—
65	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
66	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
67	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂³×C₃²	—	2	2	no	—
69	C₃²×D₄	—	2	2	no	—
70	C₃²×D₄	C₂×C₈	1	4	yes	complemented
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	4	2	no	—
73	C₂×C₃²×C₄	—	4	2	no	—
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	#41	1	1	yes	characteristic
77	C₃²×C₈	—	2	1	no	—
78	C₃²×C₈	C₂×C₈	1	2	yes	complemented
79	C₃×C₃⋊C₈	—	2	2	no	—
80	C₃×C₃⋊C₈	—	2	1	no	—
81	C₃×C₃⋊C₈	—	4	2	no	—
82	C₃×C₃⋊C₈	D₈	1	2	yes	direct product factor
83	C₄×D₈	C₃×D₃	1	1	yes	characteristic
84	C₈×D₄	—	3	2	no	—
85	#335	—	3	1	no	—
86	D₄⋊C₈	—	3	2	no	—
87	C₈²	—	3	1	no	—
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
90	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	4	2	no	—
94	C₃×D₈	—	2	4	no	—
95	C₃×D₈	C₃⋊C₈	1	4	yes	direct product factor
96	C₃×D₈	C₃×C₈	1	4	yes	complemented
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	3	1	no	—
100	C₂×C₃×C₈	—	3	1	no	—
101	C₂×C₃×C₈	—	6	1	no	—
102	C₂×C₃×C₈	—	6	2	no	—
103	C₂×C₃×C₈	—	12	4	no	—
104	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
110	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
111	C₃×C₂²⋊C₄	—	2	2	no	—
112	C₃×C₂²⋊C₄	—	2	2	no	—
113	C₃×C₂²⋊C₄	—	4	2	no	—
114	C₃×C₄²	—	2	1	no	—
115	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
116	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₃⋊C₈	—	2	1	no	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	—	4	4	no	—
120	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
122	C₂²×C₃²	—	2	4	no	—
123	C₂²×C₃²	—	4	2	no	—
124	C₂²×C₃²	—	4	2	no	—
125	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	—	4	2	no	—
128	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₃²×C₄	#131	1	1	yes	characteristic
130	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
131	C₃²×C₄	#135	1	1	yes	characteristic
132	C₂×D₈	C₃×Dic₃	1	1	yes	characteristic
133	C₂²×C₈	—	6	2	no	—
134	C₄×D₄	C₂×C₃×D₃	1	2	yes	—
135	#107	C₃×Dic₃	1	1	yes	characteristic
136	C₄⋊C₈	—	3	2	no	—
137	#102	C₃×Dic₃	1	2	yes	—
138	C₂²⋊C₈	—	6	2	no	—
139	C₄×C₈	—	3	1	no	—
140	C₄×C₈	—	3	1	no	—
141	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
142	C₂³×C₃	—	2	2	no	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	—	4	2	no	—
145	C₃×D₄	—	2	2	no	—
146	C₃×D₄	—	2	2	no	—
147	C₃×D₄	—	2	4	no	—
148	C₃×D₄	—	4	2	no	—
149	C₃×D₄	C₂×C₃⋊C₈	1	4	yes	complemented
150	C₃×D₄	C₂×C₃×C₈	1	4	yes	complemented
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	8	2	no	—
162	C₂×C₃×C₄	—	8	2	no	—
163	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	#237	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	2	1	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	3	2	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	—	6	2	no	—
175	C₃×C₈	—	6	1	no	—
176	C₃×C₈	—	12	2	no	—
177	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	complemented
178	C₃×C₈	C₂×C₃×C₈	1	2	yes	complemented
179	C₃⋊C₈	—	2	2	no	—
180	C₃⋊C₈	—	2	1	no	—
181	C₃⋊C₈	—	4	2	no	—
182	C₃⋊C₈	C₃×D₈	1	2	yes	direct product factor
183	C₂×C₃²	—	4	4	no	—
184	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
185	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
186	C₂×C₃²	#443	1	1	yes	characteristic
187	C₂×D₄	C₂×C₃×Dic₃	1	2	yes	—
188	C₂²×C₄	—	2	2	no	—
189	D₈	C₃×C₃⋊C₈	1	4	yes	direct product factor
190	C₂×C₈	—	3	1	no	—
191	C₂×C₈	—	3	1	no	—
192	C₂×C₈	—	6	1	no	—
193	C₂×C₈	—	6	2	no	—
194	C₂×C₈	—	12	4	no	—
195	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂×C₃×Dic₃	1	2	yes	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	4	4	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	8	2	no	—
209	C₂²×C₃	—	8	2	no	—
210	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
211	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	4	1	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	8	2	no	—
222	C₃×C₄	D₃×D₈	1	1	yes	characteristic
223	C₃×C₄	#912	1	1	yes	characteristic
224	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
225	C₃×C₄	#948	1	1	yes	characteristic
226	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
227	C₃×C₄	C₃×#131	1	1	yes	characteristic
228	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
229	C₃×C₄	C₃×#135	1	1	yes	characteristic
230	C₃²	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	C₂×C₃×C₃⋊C₈	1	4	yes	complemented
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
238	C₂×C₄	C₃×#237	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	3	2	no	—
242	C₈	—	6	2	no	—
243	C₈	—	6	1	no	—
244	C₈	—	12	2	no	—
245	C₈	C₂×C₃×C₃⋊C₈	1	2	yes	complemented
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	4	4	no	—
250	C₂×C₃	—	4	4	no	—
251	C₂×C₃	—	8	4	no	—
252	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
253	C₂×C₃	Dic₃×D₈	1	1	yes	characteristic
254	C₂×C₃	#5196	1	1	yes	characteristic
255	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
256	C₂×C₃	C₃×C₄×D₈	1	1	yes	characteristic
257	C₂×C₃	C₃×#443	1	1	yes	characteristic
258	C₂²	—	2	4	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	C₃×D₃×D₈	1	1	yes	characteristic
265	C₄	C₃×#912	1	1	yes	characteristic
266	C₄	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
267	C₄	C₃×#948	1	1	yes	characteristic
268	C₃	—	2	1	no	—
269	C₃	D₈×C₃⋊C₈	1	1	yes	characteristic, direct product factor
270	C₃	C₃×C₈×D₈	1	1	yes	characteristic, complemented
271	C₂	—	4	4	no	—
272	C₂	C₃×D₄×C₃⋊C₈	1	1	yes	characteristic
273	C₂	C₃×Dic₃×D₈	1	1	yes	characteristic
274	C₂	C₃×#5196	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₄×D₈	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃²×D₈	C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×C₄×D₈	D₃
hypocenter	C₃	C₃×C₈×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₃×D₄×C₃⋊C₈	C₂	semidirect
3. C₃×D₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₃×#4458	C₂	semidirect
5. C₃×C₈×C₃⋊C₈	C₂	semidirect
7. D₈×C₃⋊C₈	C₃	direct (N × H)
22. D₄×C₃⋊C₈	C₂×C₃	semidirect
26. D₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
27. #4458	C₂×C₃	semidirect
28. C₈×C₃⋊C₈	C₂×C₃	semidirect
31. C₃²×D₈	C₈	semidirect
70. C₃²×D₄	C₂×C₈	semidirect
78. C₃²×C₈	C₂×C₈	semidirect
82. C₃×C₃⋊C₈	D₈	direct (N × H)
95. C₃×D₈	C₃⋊C₈	direct (N × H)
96. C₃×D₈	C₃×C₈	semidirect
149. C₃×D₄	C₂×C₃⋊C₈	semidirect
150. C₃×D₄	C₂×C₃×C₈	semidirect
177. C₃×C₈	C₂×C₃⋊C₈	semidirect
178. C₃×C₈	C₂×C₃×C₈	semidirect
182. C₃⋊C₈	C₃×D₈	direct (N × H)
189. D₈	C₃×C₃⋊C₈	direct (N × H)
230. C₃²	C₈×D₈	semidirect
233. D₄	C₂×C₃×C₃⋊C₈	semidirect
245. C₈	C₂×C₃×C₃⋊C₈	semidirect
269. C₃	D₈×C₃⋊C₈	direct (N × H)
270. C₃	C₃×C₈×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₈	2
C₃×D₄×C₃⋊C₈	2
C₃×D₄⋊(C₃⋊C₈)	2
C₃×#4458	2
C₃×C₈×C₃⋊C₈	2
C₃×C₈×D₈	3
D₈×C₃⋊C₈	3

Subgroups — C3×D8×C3⋊C8 #11777141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₈×C₃⋊C₈ #11777141