Subgroups — C₃×#1877 #72592

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
5	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	#1877	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	—	2	1	no	—
12	C₂×C₃×SD₁₆	—	2	2	no	—
13	C₂×C₃×SD₁₆	—	2	2	no	—
14	C₂×C₃×SD₁₆	C₄	1	4	yes	complemented
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₄	—	2	2	no	—
18	C₃×C₄×D₄	—	2	2	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#103	—	2	2	no	—
24	C₃×#102	—	2	2	no	—
25	C₃×#95	—	2	1	no	—
26	C₃×#95	—	2	1	no	—
27	C₂²×SD₁₆	C₂×C₃	1	1	yes	characteristic
28	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
29	C₂×#103	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×#102	C₂×C₃	1	1	yes	characteristic
31	#391	C₂×C₃	1	1	yes	characteristic, complemented
32	#340	C₂×C₃	1	1	yes	characteristic, complemented
33	#336	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂⁴×C₃	—	2	1	no	—
35	C₂×C₃×Q₈	—	2	1	no	—
36	C₂×C₃×Q₈	—	2	1	no	—
37	C₂×C₃×Q₈	—	2	1	no	—
38	C₂×C₃×Q₈	—	2	1	no	—
39	C₂×C₃×Q₈	—	2	1	no	—
40	C₂×C₃×Q₈	C₂×C₄	1	2	yes	complemented
41	C₂×C₃×D₄	—	2	1	no	—
42	C₂×C₃×D₄	—	2	1	no	—
43	C₂×C₃×D₄	—	2	1	no	—
44	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
45	C₂×C₃×D₄	D₄	1	2	yes	—
46	C₂×C₃×D₄	D₄	1	2	yes	—
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	—	2	1	no	—
49	C₂²×C₃×C₄	—	2	1	no	—
50	C₂²×C₃×C₄	—	2	1	no	—
51	C₂²×C₃×C₄	—	4	2	no	—
52	C₂²×C₃×C₄	—	4	2	no	—
53	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₃×SD₁₆	—	2	8	no	—
57	C₂×C₃×C₈	—	2	1	no	—
58	C₂×C₃×C₈	—	2	1	no	—
59	C₂×C₃×C₈	—	2	1	no	—
60	C₂×C₃×C₈	—	2	1	no	—
61	C₂×C₃×C₈	—	2	1	no	—
62	C₂×C₃×C₈	C₂×C₄	1	2	yes	complemented
63	C₃×C₄⋊C₄	—	2	1	no	—
64	C₃×C₄⋊C₄	—	2	1	no	—
65	C₃×C₄⋊C₄	—	2	1	no	—
66	C₃×C₄⋊C₄	D₄	1	2	yes	—
67	C₃×C₂²⋊C₄	—	2	4	no	—
68	C₃×C₄²	—	2	1	no	—
69	C₃×C₄²	—	2	1	no	—
70	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
72	C₂³×C₄	—	2	1	no	—
73	C₂×SD₁₆	—	2	2	no	—
74	C₂×SD₁₆	—	2	2	no	—
75	C₂×SD₁₆	C₃×C₄	1	4	yes	complemented
76	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₄×D₄	—	2	2	no	—
79	C₄×D₄	—	2	2	no	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₄	—	2	1	no	—
83	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
84	#103	—	2	2	no	—
85	#102	—	2	2	no	—
86	#95	—	2	1	no	—
87	#95	—	2	1	no	—
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	—	2	2	no	—
90	C₂³×C₃	—	2	2	no	—
91	C₂³×C₃	—	4	1	no	—
92	C₂³×C₃	—	4	1	no	—
93	C₂³×C₃	—	4	1	no	—
94	C₂³×C₃	—	4	1	no	—
95	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
96	C₃×Q₈	—	2	2	no	—
97	C₃×Q₈	—	2	2	no	—
98	C₃×Q₈	—	4	1	no	—
99	C₃×Q₈	—	4	1	no	—
100	C₃×D₄	—	2	2	no	—
101	C₃×D₄	—	2	2	no	—
102	C₃×D₄	—	2	2	no	—
103	C₃×D₄	C₂²⋊C₄	1	4	yes	—
104	C₂×C₃×C₄	—	2	1	no	—
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	1	no	—
111	C₂×C₃×C₄	—	4	8	no	—
112	C₂×C₃×C₄	—	4	1	no	—
113	C₂×C₃×C₄	—	4	1	no	—
114	C₂×C₃×C₄	—	4	1	no	—
115	C₂×C₃×C₄	—	4	1	no	—
116	C₂×C₃×C₄	—	4	1	no	—
117	C₂×C₃×C₄	—	4	1	no	—
118	C₂×C₃×C₄	—	4	1	no	—
119	C₂×C₃×C₄	SD₁₆	1	2	yes	complemented
120	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	#41	1	1	yes	characteristic
125	C₂×C₃×C₄	#41	1	1	yes	characteristic
126	C₃×C₈	—	2	2	no	—
127	C₃×C₈	—	4	1	no	—
128	C₂⁴	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	C₂×C₃×C₄	1	2	yes	complemented
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	—	2	1	no	—
138	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
139	C₂×D₄	C₃×D₄	1	2	yes	—
140	C₂×D₄	C₃×D₄	1	2	yes	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
150	SD₁₆	—	2	8	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	C₂×C₃×C₄	1	2	yes	complemented
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	C₃×D₄	1	2	yes	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	2	no	—
171	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
172	C₂²×C₃	#120	1	1	yes	characteristic
173	C₂²×C₃	#121	1	1	yes	characteristic
174	C₂²×C₃	C₂×SD₁₆	1	1	yes	characteristic
175	C₂²×C₃	#135	1	1	yes	characteristic
176	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂²×C₃	#137	1	1	yes	characteristic
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	—	8	1	no	—
183	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₃×C₄	C₄×D₄	1	1	yes	characteristic
185	C₃×C₄	#121	1	1	yes	characteristic
186	C₃×C₄	#123	1	1	yes	characteristic
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	4	1	no	—
191	C₂³	—	4	1	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
195	Q₈	—	2	2	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	4	1	no	—
198	Q₈	—	4	1	no	—
199	D₄	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	C₃×C₂²⋊C₄	1	4	yes	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	C₃×SD₁₆	1	2	yes	complemented
219	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₄	C₃×#41	1	1	yes	characteristic
224	C₂×C₄	C₃×#41	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	4	1	no	—
227	C₂×C₃	—	4	4	no	—
228	C₂×C₃	#386	1	1	yes	characteristic
229	C₂×C₃	C₄×SD₁₆	1	1	yes	characteristic
230	C₂×C₃	#440	1	1	yes	characteristic
231	C₂×C₃	#450	1	1	yes	characteristic
232	C₂×C₃	#452	1	1	yes	characteristic
233	C₂×C₃	#461	1	1	yes	characteristic
234	C₂×C₃	#463	1	1	yes	characteristic
235	C₂²	—	2	4	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
243	C₂²	C₃×#120	1	1	yes	characteristic
244	C₂²	C₃×#121	1	1	yes	characteristic
245	C₂²	C₂×C₃×SD₁₆	1	1	yes	characteristic
246	C₂²	C₃×#135	1	1	yes	characteristic
247	C₂²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂²	C₃×#137	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	8	1	no	—
254	C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
255	C₄	C₃×C₄×D₄	1	1	yes	characteristic
256	C₄	C₃×#121	1	1	yes	characteristic
257	C₄	C₃×#123	1	1	yes	characteristic
258	C₃	#1877	1	1	yes	characteristic, direct product factor, Sylow
259	C₂	—	4	4	no	—
260	C₂	C₃×#386	1	1	yes	characteristic
261	C₂	C₃×C₄×SD₁₆	1	1	yes	characteristic
262	C₂	C₃×#440	1	1	yes	characteristic
263	C₂	C₃×#450	1	1	yes	characteristic
264	C₂	C₃×#452	1	1	yes	characteristic
265	C₂	C₃×#461	1	1	yes	characteristic
266	C₂	C₃×#463	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#103	C₂	semidirect
5. C₃×#391	C₂	semidirect
6. C₃×#340	C₂	semidirect
7. C₃×#336	C₂	semidirect
8. #1877	C₃	direct (N × H)
14. C₂×C₃×SD₁₆	C₄	semidirect
29. C₂×#103	C₂×C₃	semidirect
31. #391	C₂×C₃	semidirect
32. #340	C₂×C₃	semidirect
33. #336	C₂×C₃	semidirect
40. C₂×C₃×Q₈	C₂×C₄	semidirect
62. C₂×C₃×C₈	C₂×C₄	semidirect
75. C₂×SD₁₆	C₃×C₄	semidirect
119. C₂×C₃×C₄	SD₁₆	semidirect
134. C₂×Q₈	C₂×C₃×C₄	semidirect
156. C₂×C₈	C₂×C₃×C₄	semidirect
218. C₂×C₄	C₃×SD₁₆	semidirect
258. C₃	#1877	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×SD₁₆	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#103	2
C₂×C₃×#102	2
C₃×#391	2
C₃×#340	2
C₃×#336	2
#1877	3

Subgroups — C3×#1877 #72592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1877 #72592