Subgroups — C₃×#68939 #11785431

Order 1152.

629 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#5126	C₂	1	2	yes	maximal
3	C₂×C₃×#822	C₂	1	2	yes	maximal, complemented
4	C₃²×#341	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4552	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1810	—	3	1	no	maximal
7	#68939	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×D₃×C₈	—	2	4	no	—
14	C₂×C₃×D₃×C₈	C₄	1	4	yes	—
15	C₃×#822	—	2	4	no	—
16	C₃×(C₂×D₃)⋊C₈	—	2	4	no	—
17	C₂³×C₃×C₈	—	3	1	no	—
18	C₂²×D₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#108	—	3	2	no	—
20	C₃×#407	—	3	2	no	—
21	#5126	C₂×C₃	1	2	yes	—
22	C₂×#822	C₂×C₃	1	2	yes	complemented
23	C₃×#341	—	2	1	no	—
24	C₃×#341	—	3	1	no	—
25	C₃×#341	D₃	1	1	yes	characteristic, complemented
26	C₃×#341	C₂×C₃	1	1	yes	characteristic, complemented
27	#4552	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×D₃	Q₈	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	Q₈	1	1	yes	characteristic
31	C₂×C₃×C₄×D₃	—	2	2	no	—
32	C₂×C₃×C₄×D₃	—	2	2	no	—
33	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
35	C₃²×M₁₆	—	2	2	no	—
36	C₃²×M₁₆	—	2	2	no	—
37	C₂×C₃²×C₈	—	2	2	no	—
38	C₂×C₃²×C₈	—	2	2	no	—
39	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
40	C₃×#208	—	2	2	no	—
41	C₃×#208	—	2	2	no	—
42	C₂×C₃×C₃⋊C₈	—	2	2	no	—
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	D₄	1	4	yes	—
45	C₃×D₃×C₈	—	2	8	no	—
46	#1810	—	3	1	no	Sylow
47	C₂³×C₃×C₄	—	3	1	no	—
48	C₂²×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×M₁₆	—	2	2	no	—
50	C₂×C₃×M₁₆	—	3	2	no	—
51	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
52	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
53	C₂²×C₃×C₈	—	2	1	no	—
54	C₂²×C₃×C₈	—	3	1	no	—
55	C₂²×C₃×C₈	—	3	4	no	—
56	C₂²×C₃×C₈	—	6	4	no	—
57	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×#208	C₂²×C₃	1	2	yes	—
60	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×D₃×C₈	—	2	4	no	—
62	C₂×D₃×C₈	C₃×C₄	1	4	yes	—
63	C₃×#108	—	6	4	no	—
64	C₃×C₂²⋊C₈	—	6	4	no	—
65	#822	—	2	4	no	—
66	(C₂×D₃)⋊C₈	—	2	4	no	—
67	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₃×D₃	—	2	2	no	—
69	C₂²×C₃×D₃	—	2	4	no	—
70	C₂²×C₃×D₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	#41	1	2	yes	—
75	C₂×C₃×Dic₃	—	2	2	no	—
76	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
77	C₃×C₄×D₃	—	2	2	no	—
78	C₃×C₄×D₃	—	2	2	no	—
79	C₃×C₄×D₃	—	2	4	no	—
80	C₃²×C₈	—	2	2	no	—
81	C₃²×C₈	—	4	2	no	—
82	C₃×C₃⋊C₈	—	4	2	no	—
83	C₃×C₃⋊C₈	C₂²⋊C₄	1	4	yes	—
84	C₂³×C₈	—	3	1	no	—
85	C₂×#108	—	3	2	no	—
86	#407	—	3	2	no	—
87	#341	—	3	1	no	—
88	#341	C₃×D₃	1	1	yes	characteristic, complemented
89	C₂⁴×C₃	—	3	1	no	—
90	C₂³×D₃	C₃×Q₈	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	—	3	1	no	—
93	C₂²×C₃×C₄	—	3	2	no	—
94	C₂²×C₃×C₄	—	3	2	no	—
95	C₂²×C₃×C₄	—	6	2	no	—
96	C₂²×C₃×C₄	—	6	2	no	—
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂²×Dic₃	C₃×Q₈	1	1	yes	characteristic
100	C₂×C₄×D₃	—	2	2	no	—
101	C₂×C₄×D₃	—	2	2	no	—
102	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
104	C₃×M₁₆	—	2	4	no	—
105	C₃×M₁₆	—	2	2	no	—
106	C₃×M₁₆	—	2	2	no	—
107	C₃×M₁₆	—	2	2	no	—
108	C₃×M₁₆	—	2	2	no	—
109	C₃×M₁₆	—	4	2	no	—
110	C₃×M₁₆	—	6	2	no	—
111	C₃×M₁₆	—	6	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	—	3	4	no	—
119	C₂×C₃×C₈	—	4	2	no	—
120	C₂×C₃×C₈	—	4	2	no	—
121	C₂×C₃×C₈	—	6	8	no	—
122	C₂×C₃×C₈	—	6	2	no	—
123	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
125	#208	—	2	2	no	—
126	#208	—	2	2	no	—
127	C₂×C₃⋊C₈	—	2	2	no	—
128	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
129	C₂×C₃⋊C₈	C₃×D₄	1	4	yes	—
130	D₃×C₈	—	2	8	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₃×D₃	—	2	8	no	—
135	C₂×C₃×D₃	—	2	2	no	—
136	C₂×C₃×D₃	—	2	2	no	—
137	C₂×C₃×D₃	#108	1	4	yes	complemented
138	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	#117	1	1	yes	characteristic
140	C₃²×C₄	#121	1	2	yes	—
141	C₃×Dic₃	—	2	2	no	—
142	C₂³×C₄	—	3	1	no	—
143	C₂×M₁₆	—	3	2	no	—
144	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
145	C₂²×C₈	—	3	1	no	—
146	C₂²×C₈	—	3	4	no	—
147	C₂²×C₈	—	6	4	no	—
148	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
149	#108	—	6	4	no	—
150	C₂²⋊C₈	—	6	4	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	—	3	2	no	—
153	C₂³×C₃	—	6	4	no	—
154	C₂³×C₃	—	6	2	no	—
155	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
156	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
157	C₂²×D₃	—	2	2	no	—
158	C₂²×D₃	—	2	4	no	—
159	C₂²×D₃	C₃×C₄⋊C₄	1	2	yes	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	3	2	no	—
165	C₂×C₃×C₄	—	6	2	no	—
166	C₂×C₃×C₄	—	6	2	no	—
167	C₂×C₃×C₄	—	6	2	no	—
168	C₂×C₃×C₄	—	6	4	no	—
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	#235	1	2	yes	—
171	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
172	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×#41	1	2	yes	—
177	C₂×Dic₃	—	2	2	no	—
178	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
179	C₄×D₃	—	2	2	no	—
180	C₄×D₃	—	2	2	no	—
181	C₄×D₃	—	2	4	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	3	4	no	—
185	C₃×C₈	—	4	2	no	—
186	C₃×C₈	—	4	4	no	—
187	C₃×C₈	—	4	2	no	—
188	C₃×C₈	—	4	2	no	—
189	C₃×C₈	—	12	2	no	—
190	C₃⋊C₈	—	4	2	no	—
191	C₃⋊C₈	C₃×C₂²⋊C₄	1	4	yes	—
192	C₂×C₃²	—	2	2	no	—
193	C₂×C₃²	#380	1	1	yes	characteristic
194	C₂×C₃²	C₂×#108	1	2	yes	—
195	C₃×D₃	—	2	4	no	—
196	C₂⁴	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂²×C₄	—	6	2	no	—
201	C₂²×C₄	—	6	2	no	—
202	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	6	2	no	—
206	M₁₆	—	6	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	3	2	no	—
210	C₂×C₈	—	3	4	no	—
211	C₂×C₈	—	6	8	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	3	4	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	6	8	no	—
221	C₂²×C₃	—	6	2	no	—
222	C₂²×C₃	—	6	2	no	—
223	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
224	C₂²×C₃	#896	1	2	yes	—
225	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
226	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
227	C₂×D₃	—	2	8	no	—
228	C₂×D₃	—	2	2	no	—
229	C₂×D₃	—	2	2	no	—
230	C₂×D₃	C₃×#108	1	4	yes	complemented
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	6	2	no	—
235	C₃×C₄	#872	1	1	yes	characteristic
236	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₃×C₄	#884	1	2	yes	—
238	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
239	C₃×C₄	C₃×#117	1	1	yes	characteristic
240	C₃×C₄	C₃×#121	1	2	yes	—
241	Dic₃	—	2	2	no	—
242	C₃²	#1810	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	3	2	no	—
244	C₂³	—	6	4	no	—
245	C₂³	—	6	2	no	—
246	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	6	2	no	—
249	C₂×C₄	—	6	2	no	—
250	C₂×C₄	—	6	4	no	—
251	C₂×C₄	—	6	2	no	—
252	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
253	C₂×C₄	C₃×#235	1	2	yes	—
254	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
255	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	3	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	12	2	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	4	2	no	—
265	C₂×C₃	—	6	4	no	—
266	C₂×C₃	#4667	1	1	yes	characteristic
267	C₂×C₃	D₃×#108	1	2	yes	—
268	C₂×C₃	C₃×#380	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×#108	1	2	yes	—
270	D₃	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	3	4	no	—
273	C₂²	—	6	8	no	—
274	C₂²	—	6	2	no	—
275	C₂²	—	6	2	no	—
276	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
277	C₂²	C₃×#896	1	2	yes	—
278	C₄	—	6	2	no	—
279	C₄	C₃×#872	1	1	yes	characteristic
280	C₄	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
281	C₄	C₃×#884	1	2	yes	—
282	C₃	—	2	1	no	—
283	C₃	#68939	1	1	yes	characteristic, direct product factor
284	C₃	C₃×#1810	1	1	yes	characteristic, complemented
285	C₂	—	2	2	no	—
286	C₂	—	6	4	no	—
287	C₂	C₃×#4667	1	1	yes	characteristic
288	C₂	C₃×D₃×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#341	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃×D₃	Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#341	D₃
hypocenter	C₃	C₃×#1810
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#341

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#822	C₂	semidirect
4. C₃²×#341	C₂	semidirect
5. C₃×#4552	C₂	semidirect
7. #68939	C₃	direct (N × H)
22. C₂×#822	C₂×C₃	semidirect
25. C₃×#341	D₃	semidirect
26. C₃×#341	C₂×C₃	semidirect
27. #4552	C₂×C₃	semidirect
88. #341	C₃×D₃	semidirect
137. C₂×C₃×D₃	#108	semidirect
230. C₂×D₃	C₃×#108	semidirect
242. C₃²	#1810	semidirect
283. C₃	#68939	direct (N × H)
284. C₃	C₃×#1810	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₃×C₈	2
C₃×#5126	2
C₂×C₃×#822	2
C₃²×#341	2
C₃×#4552	2
C₃×#1810	3
#68939	3

Subgroups — C3×#68939 #11785431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68939 #11785431