Subgroups — C₃×#69291 #11799620

Order 1152.

297 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	C₃×C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#4683	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4509	C₂	1	1	yes	maximal, characteristic
5	C₃×#4509	C₂	1	1	yes	maximal, characteristic
6	C₃×#4505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4504	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2228	—	3	1	no	maximal
9	#69291	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₂₄	C₂²	1	1	yes	characteristic
11	C₃×C₄×D₁₂	—	2	1	no	—
12	C₃×#870	—	2	1	no	—
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×#821	—	2	1	no	—
17	C₃×#818	C₂²	1	1	yes	characteristic
18	C₃×#818	C₂²	1	1	yes	characteristic
19	C₃×#818	C₂²	1	1	yes	characteristic
20	C₃×#817	—	2	1	no	—
21	C₃×#499	—	3	1	no	—
22	C₃×C₄×D₈	—	3	1	no	—
23	C₄×D₂₄	C₂×C₃	1	1	yes	characteristic
24	#4683	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#367	—	3	1	no	—
26	C₃×#366	—	3	1	no	—
27	C₃×C₄⋊C₁₆	—	2	1	no	—
28	C₃×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
29	C₃×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×#357	—	3	1	no	—
31	C₃×#357	—	3	1	no	—
32	#4509	C₂×C₃	1	1	yes	characteristic
33	#4509	C₂×C₃	1	1	yes	characteristic
34	#4505	C₂×C₃	1	1	yes	characteristic, complemented
35	#4504	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×C₃×D₁₂	—	2	1	no	—
37	C₂×C₃×C₄×D₃	—	4	1	no	—
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₃²×C₄²	D₄	1	1	yes	characteristic
41	C₃×(C₂×D₃)⋊C₄	—	4	1	no	—
42	C₃×C₄⋊Dic₃	—	2	1	no	—
43	C₃×C₄⋊Dic₃	—	2	1	no	—
44	C₃×C₄⋊Dic₃	—	2	1	no	—
45	C₃×Dic₃⋊C₄	—	4	1	no	—
46	C₃×Dic₃⋊C₄	—	4	1	no	—
47	C₃×D₂₄	—	2	1	no	—
48	C₃×D₂₄	Q₈	1	2	yes	—
49	C₃²×C₁₆	—	2	1	no	—
50	C₃²×C₁₆	—	2	1	no	—
51	#2228	—	3	1	no	Sylow
52	C₂×C₃×D₈	—	3	1	no	—
53	C₃×#125	—	6	1	no	—
54	C₃×C₄×D₄	—	6	1	no	—
55	C₂×D₂₄	C₂²×C₃	1	1	yes	characteristic
56	C₄×D₁₂	—	2	1	no	—
57	#870	—	2	1	no	—
58	C₂×C₃×C₁₆	—	2	1	no	—
59	C₂×C₃×C₁₆	—	2	1	no	—
60	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
64	C₃×#107	—	3	1	no	—
65	C₃×#107	—	3	1	no	—
66	C₃×#107	—	3	1	no	—
67	C₃×#106	—	6	1	no	—
68	C₃×#102	—	6	1	no	—
69	C₃×C₄×C₈	—	2	1	no	—
70	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
72	#821	—	2	1	no	—
73	#818	C₂²×C₃	1	1	yes	characteristic
74	#818	C₂²×C₃	1	1	yes	characteristic
75	#818	C₂²×C₃	1	1	yes	characteristic
76	#817	—	2	1	no	—
77	C₂²×C₃×D₃	—	4	1	no	—
78	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
79	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	4	1	no	—
82	C₂×C₃×Dic₃	—	4	1	no	—
83	C₂×C₃×Dic₃	—	4	1	no	—
84	C₃×D₁₂	—	2	2	no	—
85	C₃×D₁₂	—	4	1	no	—
86	C₃×C₄×D₃	—	8	2	no	—
87	C₃²×C₈	—	2	1	no	—
88	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₃²×C₈	#41	1	1	yes	characteristic
90	#499	—	3	1	no	—
91	C₄×D₈	—	3	1	no	—
92	#367	—	3	1	no	—
93	#366	—	3	1	no	—
94	C₄⋊C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
95	#357	—	3	1	no	—
96	#357	—	3	1	no	—
97	C₂×C₃×D₄	—	6	1	no	—
98	C₂²×C₃×C₄	—	12	1	no	—
99	C₂×D₁₂	—	2	1	no	—
100	C₂×C₄×D₃	—	4	1	no	—
101	C₃×D₈	—	3	2	no	—
102	C₃×D₈	—	6	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
106	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	6	1	no	—
110	C₃×C₄⋊C₄	—	6	1	no	—
111	C₃×C₄⋊C₄	—	6	1	no	—
112	C₃×C₄⋊C₄	—	12	1	no	—
113	C₃×C₄⋊C₄	—	12	1	no	—
114	C₃×C₂²⋊C₄	—	12	1	no	—
115	C₃×C₄²	—	2	1	no	—
116	C₃×C₄²	D₁₂	1	1	yes	characteristic
117	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
118	(C₂×D₃)⋊C₄	—	4	1	no	—
119	C₄⋊Dic₃	—	2	1	no	—
120	C₄⋊Dic₃	—	2	1	no	—
121	C₄⋊Dic₃	—	2	1	no	—
122	Dic₃⋊C₄	—	4	1	no	—
123	Dic₃⋊C₄	—	4	1	no	—
124	D₂₄	—	2	1	no	—
125	D₂₄	C₃×Q₈	1	2	yes	—
126	C₃×C₁₆	—	2	1	no	—
127	C₃×C₁₆	—	2	1	no	—
128	C₃×C₁₆	—	2	1	no	—
129	C₃×C₁₆	—	2	1	no	—
130	C₃×C₁₆	—	4	1	no	—
131	C₃×C₁₆	—	4	1	no	—
132	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
133	C₂×C₃×D₃	—	4	2	no	—
134	C₂×C₃×D₃	—	8	1	no	—
135	C₂×C₃×D₃	—	8	1	no	—
136	C₃²×C₄	—	2	1	no	—
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
139	C₃²×C₄	#137	1	1	yes	characteristic
140	C₃×Dic₃	—	8	1	no	—
141	C₃×Dic₃	—	8	1	no	—
142	C₃×Dic₃	—	8	1	no	—
143	C₂×D₈	—	3	1	no	—
144	#125	—	6	1	no	—
145	C₄×D₄	—	6	1	no	—
146	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
148	#107	—	3	1	no	—
149	#107	—	3	1	no	—
150	#107	—	3	1	no	—
151	#106	—	6	1	no	—
152	#102	—	6	1	no	—
153	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
154	C₂³×C₃	—	12	1	no	—
155	C₂²×D₃	—	4	1	no	—
156	C₃×D₄	—	6	2	no	—
157	C₃×D₄	—	12	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	12	1	no	—
162	C₂×C₃×C₄	—	12	1	no	—
163	C₂×C₃×C₄	—	12	1	no	—
164	C₂×C₃×C₄	—	24	2	no	—
165	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
166	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×Dic₃	—	4	1	no	—
172	C₂×Dic₃	—	4	1	no	—
173	C₂×Dic₃	—	4	1	no	—
174	D₁₂	—	2	2	no	—
175	D₁₂	—	4	1	no	—
176	C₄×D₃	—	8	2	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	1	no	—
179	C₃×C₈	—	2	1	no	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	4	1	no	—
182	C₃×C₈	#237	1	1	yes	characteristic
183	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
184	C₃×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
185	C₃×C₈	C₃×#41	1	1	yes	characteristic
186	C₂×C₃²	#474	1	1	yes	characteristic
187	C₂×C₃²	#508	1	1	yes	characteristic
188	C₂×C₃²	#509	1	1	yes	characteristic
189	C₃×D₃	—	8	2	no	—
190	C₂×D₄	—	6	1	no	—
191	C₂²×C₄	—	12	1	no	—
192	D₈	—	3	2	no	—
193	D₈	—	6	1	no	—
194	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
196	C₄⋊C₄	—	6	1	no	—
197	C₄⋊C₄	—	6	1	no	—
198	C₄⋊C₄	—	6	1	no	—
199	C₄⋊C₄	—	12	1	no	—
200	C₄⋊C₄	—	12	1	no	—
201	C₂²⋊C₄	—	12	1	no	—
202	C₄²	C₃×D₁₂	1	1	yes	characteristic
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	12	2	no	—
207	C₂²×C₃	—	24	1	no	—
208	C₂²×C₃	—	24	1	no	—
209	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
210	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
211	C₂×D₃	—	4	2	no	—
212	C₂×D₃	—	8	1	no	—
213	C₂×D₃	—	8	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	—	24	1	no	—
223	C₃×C₄	—	24	1	no	—
224	C₃×C₄	—	24	1	no	—
225	C₃×C₄	#897	1	1	yes	characteristic
226	C₃×C₄	#909	1	1	yes	characteristic
227	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
228	C₃×C₄	C₃×#137	1	1	yes	characteristic
229	Dic₃	—	8	1	no	—
230	Dic₃	—	8	1	no	—
231	Dic₃	—	8	1	no	—
232	C₃²	#2228	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	12	1	no	—
234	D₄	—	6	2	no	—
235	D₄	—	12	1	no	—
236	C₂×C₄	—	12	1	no	—
237	C₂×C₄	—	12	1	no	—
238	C₂×C₄	—	12	1	no	—
239	C₂×C₄	—	24	2	no	—
240	C₂×C₄	C₃×D₂₄	1	1	yes	characteristic
241	C₂×C₄	C₃×D₂₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	C₃×#237	1	1	yes	characteristic
245	C₈	C₃×D₃×Q₈	1	1	yes	characteristic
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	24	2	no	—
250	C₂×C₃	#4846	1	1	yes	characteristic
251	C₂×C₃	#4904	1	1	yes	characteristic
252	C₂×C₃	#4908	1	1	yes	characteristic
253	C₂×C₃	C₃×#474	1	1	yes	characteristic
254	C₂×C₃	C₃×#508	1	1	yes	characteristic
255	C₂×C₃	C₃×#509	1	1	yes	characteristic
256	D₃	—	8	2	no	—
257	C₂²	—	12	2	no	—
258	C₂²	—	24	1	no	—
259	C₂²	—	24	1	no	—
260	C₂²	C₂×C₃×D₂₄	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	24	1	no	—
264	C₄	—	24	1	no	—
265	C₄	—	24	1	no	—
266	C₄	C₃×#897	1	1	yes	characteristic
267	C₄	C₃×#909	1	1	yes	characteristic
268	C₃	—	2	1	no	—
269	C₃	#69291	1	1	yes	characteristic, direct product factor
270	C₃	C₃×#2228	1	1	yes	characteristic, complemented
271	C₂	—	24	2	no	—
272	C₂	C₃×#4846	1	1	yes	characteristic
273	C₂	C₃×#4904	1	1	yes	characteristic
274	C₂	C₃×#4908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³×C₃
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄⋊C₁₆	C₂
socle	C₂²×C₃²	C₂×D₈
prime radical	C₂×C₃×D₂₄	C₂²
hypercenter	C₃×C₄⋊C₁₆	D₃
hypocenter	C₃	C₃×#2228
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ C₃×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×#4683	C₂	semidirect
3. C₃²×C₄⋊C₁₆	C₂	semidirect
6. C₃×#4505	C₂	semidirect
7. C₃×#4504	C₂	semidirect
9. #69291	C₃	direct (N × H)
24. #4683	C₂×C₃	semidirect
28. C₃×C₄⋊C₁₆	D₃	semidirect
29. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
34. #4505	C₂×C₃	semidirect
35. #4504	C₂×C₃	semidirect
94. C₄⋊C₁₆	C₃×D₃	semidirect
232. C₃²	#2228	semidirect
269. C₃	#69291	direct (N × H)
270. C₃	C₃×#2228	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₂₄	2
C₃×#4683	2
C₃²×C₄⋊C₁₆	2
C₃×#4509	2
C₃×#4505	2
C₃×#4504	2
C₃×#2228	3
#69291	3

Subgroups — C3×#69291 #11799620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69291 #11799620