Subgroups — C₃×#2507 #84371

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#398	C₂	1	2	yes	maximal
2	C₃×#398	C₂	1	2	yes	maximal
3	C₃×#389	C₂	1	2	yes	maximal
4	C₃×#389	C₂	1	1	yes	maximal, characteristic
5	C₃×#389	C₂	1	1	yes	maximal, characteristic
6	C₃×#389	C₂	1	1	yes	maximal, characteristic
7	C₃×#389	C₂	1	1	yes	maximal, characteristic
8	C₃×#387	C₂	1	2	yes	maximal
9	C₃×#384	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2507	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	2	yes	—
33	C₃×#95	C₂²	1	2	yes	—
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	#398	C₂×C₃	1	2	yes	—
37	#398	C₂×C₃	1	2	yes	—
38	#389	C₂×C₃	1	2	yes	—
39	#389	C₂×C₃	1	1	yes	characteristic
40	#389	C₂×C₃	1	1	yes	characteristic
41	#389	C₂×C₃	1	1	yes	characteristic
42	#389	C₂×C₃	1	1	yes	characteristic
43	#387	C₂×C₃	1	2	yes	—
44	#384	C₂×C₃	1	1	yes	characteristic
45	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
46	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₃×C₄⋊C₄	—	2	2	no	—
59	C₃×C₄⋊C₄	—	2	4	no	—
60	C₃×C₄⋊C₄	—	2	2	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	—	4	2	no	—
64	C₃×C₄⋊C₄	—	4	2	no	—
65	C₃×C₄⋊C₄	—	4	2	no	—
66	C₃×C₄⋊C₄	—	4	2	no	—
67	C₃×C₄⋊C₄	—	4	2	no	—
68	C₃×C₄⋊C₄	Q₈	1	4	yes	—
69	C₃×C₄⋊C₄	Q₈	1	4	yes	—
70	C₃×C₄²	—	2	2	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₃×C₄²	—	4	2	no	—
74	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₃	1	2	yes	—
90	#95	C₂²×C₃	1	1	yes	characteristic
91	#95	C₂²×C₃	1	1	yes	characteristic
92	#95	C₂²×C₃	1	2	yes	—
93	#95	C₂²×C₃	1	2	yes	—
94	#95	C₂²×C₃	1	2	yes	—
95	#95	C₂²×C₃	1	2	yes	—
96	#95	C₂²×C₃	1	2	yes	—
97	C₂³×C₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	1	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	1	no	—
101	C₂×C₃×C₄	—	2	1	no	—
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	1	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
124	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
125	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
126	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
127	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
128	C₂×C₃×C₄	#41	1	2	yes	—
129	C₂×C₃×C₄	#41	1	2	yes	—
130	C₂×C₃×C₄	#41	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	C₃×Q₈	1	4	yes	—
153	C₄⋊C₄	C₃×Q₈	1	4	yes	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
159	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₃	C₂×#41	1	1	yes	characteristic
161	C₂²×C₃	C₂×#41	1	1	yes	characteristic
162	C₂²×C₃	C₂×#41	1	1	yes	characteristic
163	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	4	no	—
170	C₃×C₄	—	4	1	no	—
171	C₃×C₄	—	4	2	no	—
172	C₃×C₄	—	4	2	no	—
173	C₃×C₄	—	4	1	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	#125	1	4	yes	—
176	C₂³	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
203	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
204	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
205	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
206	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
207	C₂×C₄	C₃×#41	1	2	yes	—
208	C₂×C₄	C₃×#41	1	2	yes	—
209	C₂×C₄	C₃×#41	1	2	yes	—
210	C₂×C₃	C₂×#125	1	1	yes	characteristic
211	C₂×C₃	#533	1	1	yes	characteristic
212	C₂×C₃	#540	1	1	yes	characteristic
213	C₂×C₃	#553	1	1	yes	characteristic
214	C₂×C₃	#554	1	1	yes	characteristic
215	C₂×C₃	#554	1	1	yes	characteristic
216	C₂×C₃	Q₈²	1	1	yes	characteristic
217	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
218	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
219	C₂²	C₂×C₃×#41	1	1	yes	characteristic
220	C₂²	C₂×C₃×#41	1	1	yes	characteristic
221	C₂²	C₂×C₃×#41	1	1	yes	characteristic
222	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
223	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	C₃×#125	1	4	yes	—
235	C₃	#2507	1	1	yes	characteristic, direct product factor, Sylow
236	C₂	C₂×C₃×#125	1	1	yes	characteristic
237	C₂	C₃×#533	1	1	yes	characteristic
238	C₂	C₃×#540	1	1	yes	characteristic
239	C₂	C₃×#553	1	1	yes	characteristic
240	C₂	C₃×#554	1	1	yes	characteristic
241	C₂	C₃×#554	1	1	yes	characteristic
242	C₂	C₃×Q₈²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. #2507	C₃	direct (N × H)
235. C₃	#2507	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#398	2
C₃×#389	2
C₃×#387	2
C₃×#384	2
C₃×C₄×C₄⋊C₄	2
#2507	3

Subgroups — C3×#2507 #84371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2507 #84371