Subgroups — C₃×#69732 #11776917

Order 1152.

417 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#4926	C₂	1	2	yes	maximal
3	C₃×#4458	C₂	1	2	yes	maximal, complemented
4	C₃×#4457	C₂	1	2	yes	maximal, complemented
5	C₃×#1568	—	3	1	no	maximal
6	#69732	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#830	—	2	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
16	C₃×#803	—	2	2	no	—
17	C₃×#432	—	3	2	no	—
18	C₂×C₃×C₄×C₈	—	2	1	no	—
19	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
20	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
21	#4926	C₂×C₃	1	2	yes	—
22	C₃×#335	—	3	2	no	—
23	C₃×#334	—	3	2	no	—
24	#4458	C₂×C₃	1	2	yes	complemented
25	#4457	C₂×C₃	1	2	yes	complemented
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
29	C₂×C₃²×C₈	—	2	2	no	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄²	D₄	1	1	yes	characteristic
37	C₃²×C₄²	Q₈	1	1	yes	characteristic
38	C₃²×C₄²	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₃⋊C₈	—	2	4	no	—
40	#1568	—	3	1	no	Sylow
41	C₂²×C₃×C₈	—	2	1	no	—
42	C₂²×C₃×C₈	—	2	1	no	—
43	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₃×C₄	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₃×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₃×C₄⋊C₈	—	3	4	no	—
51	C₃×C₂²⋊C₈	—	6	2	no	—
52	C₃×#97	—	6	2	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	2	1	no	—
56	C₃×C₄×C₈	—	2	1	no	—
57	C₃×C₄×C₈	Dic₃	1	1	yes	characteristic
58	C₃×C₄×C₈	Dic₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₃×C₄	1	1	yes	characteristic
60	C₃×C₄×C₈	C₃×C₄	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
65	#830	—	2	2	no	—
66	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	4	yes	—
67	#803	—	2	2	no	—
68	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	—	2	1	no	—
71	C₂×C₃²×C₄	—	2	1	no	—
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₃²×C₈	—	2	1	no	—
80	C₃²×C₈	—	2	1	no	—
81	C₃²×C₈	M₁₆	1	2	yes	—
82	C₃²×C₈	M₁₆	1	2	yes	—
83	C₃×C₃⋊C₈	—	4	4	no	—
84	#432	—	3	2	no	—
85	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
86	#335	—	3	2	no	—
87	#334	—	3	2	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
92	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₃×Q₈	1	1	yes	characteristic
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	4	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	6	4	no	—
108	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
116	C₃×C₄²	—	2	1	no	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	2	1	no	—
119	C₃×C₄²	—	2	1	no	—
120	C₃×C₄²	Dic₆	1	1	yes	characteristic
121	C₃×C₄²	D₁₂	1	1	yes	characteristic
122	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
123	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
124	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
125	C₃×C₄²	C₃×Q₈	1	1	yes	characteristic
126	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
127	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
128	C₂×C₃⋊C₈	—	2	4	no	—
129	C₂²×C₃²	—	2	1	no	—
130	C₂²×C₃²	—	2	1	no	—
131	C₂²×C₃²	#108	1	2	yes	—
132	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₃²×C₄	—	2	1	no	—
134	C₃²×C₄	—	2	1	no	—
135	C₃²×C₄	—	2	1	no	—
136	C₃²×C₄	—	2	1	no	—
137	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
138	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
139	C₃²×C₄	#135	1	2	yes	—
140	C₂²×C₈	C₃×Dic₃	1	1	yes	characteristic
141	C₂²×C₈	C₃×Dic₃	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
143	C₄⋊C₈	—	3	4	no	—
144	C₂²⋊C₈	—	6	2	no	—
145	#97	—	6	2	no	—
146	C₄×C₈	C₃×Dic₃	1	1	yes	characteristic
147	C₄×C₈	C₃×Dic₃	1	1	yes	characteristic
148	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
149	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
150	C₂³×C₃	—	2	1	no	—
151	C₂³×C₃	C₄⋊Dic₃	1	1	yes	characteristic
152	C₂³×C₃	C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	2	1	no	—
186	C₃×C₈	—	2	1	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	2	2	no	—
191	C₃×C₈	—	12	4	no	—
192	C₃×C₈	#208	1	2	yes	—
193	C₃×C₈	#208	1	2	yes	—
194	C₃×C₈	C₃×M₁₆	1	2	yes	—
195	C₃×C₈	C₃×M₁₆	1	2	yes	—
196	C₃⋊C₈	—	4	4	no	—
197	C₂×C₃²	—	2	2	no	—
198	C₂×C₃²	#423	1	1	yes	characteristic
199	C₂×C₃²	#427	1	1	yes	characteristic
200	C₂×C₃²	C₂×#108	1	1	yes	characteristic
201	C₂²×C₄	C₃×Dic₆	1	1	yes	characteristic
202	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	6	4	no	—
207	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
209	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
211	C₄²	C₃×Dic₆	1	1	yes	characteristic
212	C₄²	C₃×D₁₂	1	1	yes	characteristic
213	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
214	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	#819	1	2	yes	—
224	C₂²×C₃	C₃×#108	1	2	yes	—
225	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
226	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	#904	1	2	yes	—
243	C₃×C₄	C₂×#208	1	1	yes	characteristic
244	C₃×C₄	C₂×#208	1	1	yes	characteristic
245	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
246	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
247	C₃×C₄	C₃×#135	1	2	yes	—
248	C₃²	#1568	1	1	yes	characteristic, complemented, Sylow
249	C₂³	C₃×C₄⋊Dic₃	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
255	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
256	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	12	4	no	—
263	C₈	C₃×#208	1	2	yes	—
264	C₈	C₃×#208	1	2	yes	—
265	C₂×C₃	—	2	4	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	#4924	1	1	yes	characteristic
272	C₂×C₃	#5106	1	1	yes	characteristic
273	C₂×C₃	C₂×#819	1	1	yes	characteristic
274	C₂×C₃	C₃×#423	1	1	yes	characteristic
275	C₂×C₃	C₃×#427	1	1	yes	characteristic
276	C₂×C₃	C₂×C₃×#108	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₃×#819	1	2	yes	—
280	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	C₃×#904	1	2	yes	—
286	C₄	C₂×C₃×#208	1	1	yes	characteristic
287	C₄	C₂×C₃×#208	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	#69732	1	1	yes	characteristic, direct product factor
290	C₃	C₃×#1568	1	1	yes	characteristic, complemented
291	C₂	—	2	2	no	—
292	C₂	C₃×#4924	1	1	yes	characteristic
293	C₂	C₃×#5106	1	1	yes	characteristic
294	C₂	C₂×C₃×#819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1568
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#4458	C₂	semidirect
4. C₃×#4457	C₂	semidirect
6. #69732	C₃	direct (N × H)
24. #4458	C₂×C₃	semidirect
25. #4457	C₂×C₃	semidirect
248. C₃²	#1568	semidirect
289. C₃	#69732	direct (N × H)
290. C₃	C₃×#1568	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₃×#4926	2
C₃×#4458	2
C₃×#4457	2
C₃×#1568	3
#69732	3

Subgroups — C3×#69732 #11776917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69732 #11776917