Subgroups — C₃×#68838 #11795187

Order 1152.

505 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#4984	C₂	1	2	yes	maximal
4	C₃²×#340	C₂	1	1	yes	maximal, characteristic
5	C₃×#4529	C₂	1	2	yes	maximal
6	C₃×#2086	—	3	1	no	maximal
7	#68838	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₃⋊#95	—	2	2	no	—
15	C₃×#818	—	2	2	no	—
16	C₃×Dic₃⋊C₈	—	2	2	no	—
17	C₂×C₃×#107	—	3	1	no	—
18	C₂×C₃×C₄⋊C₈	—	3	1	no	—
19	C₃×#389	—	3	2	no	—
20	C₂×#818	C₂×C₃	1	1	yes	characteristic
21	C₂×Dic₃⋊C₈	C₂×C₃	1	1	yes	characteristic
22	#4984	C₂×C₃	1	2	yes	—
23	C₃×#340	—	2	1	no	—
24	C₃×#340	—	3	2	no	—
25	C₃×#340	D₃	1	1	yes	characteristic
26	C₃×#340	C₂×C₃	1	1	yes	characteristic
27	#4529	C₂×C₃	1	2	yes	—
28	C₂²×C₃²×C₄	—	2	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₃²×C₄⋊C₄	—	2	2	no	—
36	C₃²×C₄⋊C₄	Q₈	1	4	yes	—
37	C₃×C₄⋊Dic₃	—	2	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	4	4	no	—
40	C₃×C₄×Dic₃	—	2	1	no	—
41	C₃×C₄×Dic₃	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	#2086	—	3	1	no	Sylow
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	—	3	1	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₃×C₄⋊C₄	—	3	2	no	—
51	C₂×C₃×C₄⋊C₄	—	6	2	no	—
52	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₄⋊Dic₃	—	2	2	no	—
56	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
57	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃×#107	—	6	2	no	—
60	C₃×C₄⋊C₈	—	6	2	no	—
61	C₃×#95	—	6	2	no	—
62	C₃⋊#95	—	2	2	no	—
63	#818	—	2	2	no	—
64	Dic₃⋊C₈	—	2	2	no	—
65	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	—	4	2	no	—
68	C₂×C₃²×C₄	—	4	2	no	—
69	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₃²×C₄	#41	1	2	yes	—
72	C₂×C₃²×C₄	#41	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	2	1	no	—
74	C₂×C₃×Dic₃	—	2	1	no	—
75	C₂×C₃×Dic₃	—	2	1	no	—
76	C₂×C₃×Dic₃	—	2	2	no	—
77	C₂×C₃×Dic₃	—	4	2	no	—
78	C₂×C₃×Dic₃	—	4	2	no	—
79	C₂×C₃×Dic₃	—	4	2	no	—
80	C₂×C₃×Dic₃	D₈	1	2	yes	—
81	C₃²×C₈	—	2	2	no	—
82	C₃×C₃⋊C₈	—	4	1	no	—
83	C₂×#107	—	3	1	no	—
84	C₂×C₄⋊C₈	—	3	1	no	—
85	#389	—	3	2	no	—
86	#340	—	3	2	no	—
87	#340	C₃×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	3	1	no	—
92	C₂²×C₃×C₄	—	3	1	no	—
93	C₂²×C₃×C₄	—	4	2	no	—
94	C₂²×C₃×C₄	—	6	2	no	—
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂²×Dic₃	—	2	2	no	—
98	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
99	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
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	—	4	2	no	—
104	C₂×C₃×C₈	—	6	1	no	—
105	C₂×C₃×C₈	—	6	2	no	—
106	C₂×C₃×C₈	D₁₂	1	2	yes	—
107	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	—	2	4	no	—
111	C₃×C₄⋊C₄	—	4	2	no	—
112	C₃×C₄⋊C₄	—	6	2	no	—
113	C₃×C₄⋊C₄	—	6	2	no	—
114	C₃×C₄⋊C₄	—	12	4	no	—
115	C₃×C₄⋊C₄	Dic₆	1	4	yes	—
116	C₃×C₄⋊C₄	C₃×Q₈	1	4	yes	—
117	C₃×C₄²	—	6	1	no	—
118	C₃×C₄²	—	6	1	no	—
119	C₄⋊Dic₃	—	2	2	no	—
120	C₄⋊Dic₃	—	2	2	no	—
121	C₄⋊Dic₃	—	4	4	no	—
122	C₄×Dic₃	—	2	1	no	—
123	C₄×Dic₃	—	2	1	no	—
124	C₂×C₃⋊C₈	—	2	1	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂²×C₃²	#121	1	1	yes	characteristic
127	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
128	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
129	C₂²×C₃²	#135	1	1	yes	characteristic
130	C₂²×C₃²	#137	1	2	yes	—
131	C₃²×C₄	—	4	4	no	—
132	C₃²×C₄	#123	1	1	yes	characteristic
133	C₃²×C₄	#125	1	2	yes	—
134	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
135	C₃×Dic₃	—	2	2	no	—
136	C₃×Dic₃	—	4	1	no	—
137	C₃×Dic₃	—	8	2	no	—
138	C₂²×C₈	—	3	1	no	—
139	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	3	2	no	—
141	C₂×C₄⋊C₄	—	6	2	no	—
142	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
143	C₂×C₄²	—	3	1	no	—
144	#107	—	6	2	no	—
145	C₄⋊C₈	—	6	2	no	—
146	#95	—	6	2	no	—
147	C₂³×C₃	—	2	1	no	—
148	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
149	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	3	2	no	—
157	C₂×C₃×C₄	—	4	4	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	—	4	2	no	—
162	C₂×C₃×C₄	—	6	1	no	—
163	C₂×C₃×C₄	—	6	1	no	—
164	C₂×C₃×C₄	—	6	1	no	—
165	C₂×C₃×C₄	—	6	2	no	—
166	C₂×C₃×C₄	—	8	2	no	—
167	C₂×C₃×C₄	—	8	2	no	—
168	C₂×C₃×C₄	—	12	2	no	—
169	C₂×C₃×C₄	—	12	2	no	—
170	C₂×C₃×C₄	—	12	2	no	—
171	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
172	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₃×C₄	#237	1	2	yes	—
174	C₂×C₃×C₄	#239	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
177	C₂×C₃×C₄	C₃×#41	1	2	yes	—
178	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	—	2	1	no	—
182	C₂×Dic₃	—	2	2	no	—
183	C₂×Dic₃	—	4	2	no	—
184	C₂×Dic₃	—	4	2	no	—
185	C₂×Dic₃	—	4	2	no	—
186	C₂×Dic₃	C₃×D₈	1	2	yes	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	4	2	no	—
190	C₃×C₈	—	12	1	no	—
191	C₃⋊C₈	—	4	1	no	—
192	C₂×C₃²	#398	1	1	yes	characteristic
193	C₂×C₃²	#466	1	1	yes	characteristic
194	C₂×C₃²	#470	1	1	yes	characteristic
195	C₂×C₃²	#474	1	2	yes	—
196	C₂×C₃²	#479	1	2	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	6	2	no	—
201	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	6	1	no	—
204	C₂×C₈	—	6	2	no	—
205	C₂×C₈	C₃×D₁₂	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	6	2	no	—
208	C₄⋊C₄	—	6	2	no	—
209	C₄⋊C₄	—	12	4	no	—
210	C₄⋊C₄	C₃×Dic₆	1	4	yes	—
211	C₄²	—	6	1	no	—
212	C₄²	—	6	1	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	#878	1	2	yes	—
219	C₂²×C₃	#895	1	1	yes	characteristic
220	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
221	C₂²×C₃	#915	1	2	yes	—
222	C₂²×C₃	#919	1	1	yes	characteristic
223	C₂²×C₃	C₃×#121	1	1	yes	characteristic
224	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
225	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
226	C₂²×C₃	C₃×#135	1	1	yes	characteristic
227	C₂²×C₃	C₃×#137	1	2	yes	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	4	4	no	—
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	—	6	2	no	—
234	C₃×C₄	—	8	4	no	—
235	C₃×C₄	—	12	1	no	—
236	C₃×C₄	—	24	2	no	—
237	C₃×C₄	#869	1	1	yes	characteristic
238	C₃×C₄	#886	1	1	yes	characteristic
239	C₃×C₄	#890	1	2	yes	—
240	C₃×C₄	C₃×#123	1	1	yes	characteristic
241	C₃×C₄	C₃×#125	1	2	yes	—
242	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
243	Dic₃	—	2	2	no	—
244	Dic₃	—	4	1	no	—
245	Dic₃	—	8	2	no	—
246	C₃²	#2086	1	1	yes	characteristic, complemented, Sylow
247	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	3	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	6	1	no	—
253	C₂×C₄	—	6	1	no	—
254	C₂×C₄	—	6	1	no	—
255	C₂×C₄	—	6	2	no	—
256	C₂×C₄	—	12	2	no	—
257	C₂×C₄	—	12	2	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	C₂×C₃×Dic₆	1	2	yes	—
260	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
261	C₂×C₄	C₃×#237	1	2	yes	—
262	C₂×C₄	C₃×#239	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	12	1	no	—
265	C₂×C₃	—	2	1	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	#4661	1	1	yes	characteristic
271	C₂×C₃	#4759	1	2	yes	—
272	C₂×C₃	#4796	1	2	yes	—
273	C₂×C₃	#4867	1	1	yes	characteristic
274	C₂×C₃	#4883	1	1	yes	characteristic
275	C₂×C₃	C₃×#398	1	1	yes	characteristic
276	C₂×C₃	C₃×#466	1	1	yes	characteristic
277	C₂×C₃	C₃×#470	1	1	yes	characteristic
278	C₂×C₃	C₃×#474	1	2	yes	—
279	C₂×C₃	C₃×#479	1	2	yes	—
280	C₂²	C₃×#878	1	2	yes	—
281	C₂²	C₃×#895	1	1	yes	characteristic
282	C₂²	C₃×D₃×D₈	1	1	yes	characteristic
283	C₂²	C₃×#915	1	2	yes	—
284	C₂²	C₃×#919	1	1	yes	characteristic
285	C₄	—	4	4	no	—
286	C₄	—	6	2	no	—
287	C₄	—	12	1	no	—
288	C₄	—	24	2	no	—
289	C₄	C₃×#869	1	1	yes	characteristic
290	C₄	C₃×#886	1	1	yes	characteristic
291	C₄	C₃×#890	1	2	yes	—
292	C₃	—	2	1	no	—
293	C₃	#68838	1	1	yes	characteristic, direct product factor
294	C₃	C₃×#2086	1	1	yes	characteristic, complemented
295	C₂	C₃×#4661	1	1	yes	characteristic
296	C₂	C₃×#4759	1	2	yes	—
297	C₂	C₃×#4796	1	2	yes	—
298	C₂	C₃×#4867	1	1	yes	characteristic
299	C₂	C₃×#4883	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₃
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#340	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₃×#340	D₃
hypocenter	C₃	C₃×#2086
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#340

Complements

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

N	H ≅ G/N	type
7. #68838	C₃	direct (N × H)
246. C₃²	#2086	semidirect
293. C₃	#68838	direct (N × H)
294. C₃	C₃×#2086	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#818	2
C₂×C₃×Dic₃⋊C₈	2
C₃×#4984	2
C₃²×#340	2
C₃×#4529	2
C₃×#2086	3
#68838	3

Subgroups — C3×#68838 #11795187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68838 #11795187