Subgroups — C₃×#2497 #84361

Order 384.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#400	C₂	1	2	yes	maximal, complemented
2	C₃×#396	C₂	1	2	yes	maximal
3	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#388	C₂	1	2	yes	maximal
6	C₃×#387	C₂	1	2	yes	maximal, complemented
7	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2497	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₃×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	1	yes	characteristic
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×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	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	#400	C₂×C₃	1	2	yes	complemented
37	#396	C₂×C₃	1	2	yes	—
38	#389	C₂×C₃	1	1	yes	characteristic, complemented
39	#389	C₂×C₃	1	1	yes	characteristic, complemented
40	#388	C₂×C₃	1	2	yes	—
41	#387	C₂×C₃	1	2	yes	complemented
42	#384	C₂×C₃	1	1	yes	characteristic, complemented
43	#380	C₂×C₃	1	1	yes	characteristic
44	#380	C₂×C₃	1	1	yes	characteristic
45	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
47	C₂⁴×C₃	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	—	2	2	no	—
49	C₂²×C₃×C₄	—	2	2	no	—
50	C₂²×C₃×C₄	—	2	2	no	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	—	4	2	no	—
67	C₃×C₄⋊C₄	—	4	2	no	—
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₂²⋊C₄	—	4	2	no	—
71	C₃×C₂²⋊C₄	—	4	2	no	—
72	C₃×C₂²⋊C₄	Q₈	1	4	yes	—
73	C₃×C₄²	—	2	2	no	—
74	C₃×C₄²	—	2	2	no	—
75	C₃×C₄²	—	4	2	no	—
76	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₃	1	2	yes	—
92	#95	C₂²×C₃	1	1	yes	characteristic
93	#95	C₂²×C₃	1	1	yes	characteristic
94	#95	C₂²×C₃	1	2	yes	—
95	#95	C₂²×C₃	1	2	yes	—
96	#95	C₂²×C₃	1	2	yes	—
97	#95	C₂²×C₃	1	2	yes	—
98	#95	C₂²×C₃	1	2	yes	—
99	C₂³×C₃	—	2	1	no	—
100	C₂³×C₃	—	2	1	no	—
101	C₂³×C₃	—	2	1	no	—
102	C₂³×C₃	—	2	1	no	—
103	C₂³×C₃	—	2	1	no	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	C₂×Q₈	1	2	yes	—
106	C₂³×C₃	C₂⁴	1	1	yes	characteristic
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	1	no	—
111	C₂×C₃×C₄	—	2	1	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	4	4	no	—
132	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
133	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
134	C₂×C₃×C₄	#41	1	2	yes	—
135	C₂×C₃×C₄	#41	1	2	yes	—
136	C₂×C₃×C₄	#41	1	2	yes	—
137	C₂×C₃×C₄	#41	1	2	yes	—
138	C₂⁴	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	C₃×Q₈	1	4	yes	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	4	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	—	4	1	no	—
172	C₂²×C₃	—	4	1	no	—
173	C₂²×C₃	—	4	1	no	—
174	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
175	C₂²×C₃	C₂×#41	1	1	yes	characteristic
176	C₂²×C₃	C₂×#41	1	1	yes	characteristic
177	C₂²×C₃	C₂×#41	1	1	yes	characteristic
178	C₂²×C₃	C₂×#41	1	1	yes	characteristic
179	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	4	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	#126	1	4	yes	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂×C₃×Q₈	1	2	yes	—
198	C₂³	C₂⁴×C₃	1	1	yes	characteristic
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
225	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
226	C₂×C₄	C₃×#41	1	2	yes	—
227	C₂×C₄	C₃×#41	1	2	yes	—
228	C₂×C₄	C₃×#41	1	2	yes	—
229	C₂×C₄	C₃×#41	1	2	yes	—
230	C₂×C₃	—	4	2	no	—
231	C₂×C₃	C₂×#126	1	1	yes	characteristic
232	C₂×C₃	#533	1	1	yes	characteristic
233	C₂×C₃	#540	1	1	yes	characteristic
234	C₂×C₃	#553	1	1	yes	characteristic
235	C₂×C₃	#554	1	1	yes	characteristic
236	C₂×C₃	#554	1	1	yes	characteristic
237	C₂×C₃	#555	1	1	yes	characteristic
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
246	C₂²	C₂×C₃×#41	1	1	yes	characteristic
247	C₂²	C₂×C₃×#41	1	1	yes	characteristic
248	C₂²	C₂×C₃×#41	1	1	yes	characteristic
249	C₂²	C₂×C₃×#41	1	1	yes	characteristic
250	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
251	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	2	no	—
261	C₄	C₃×#126	1	4	yes	—
262	C₃	#2497	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	4	2	no	—
264	C₂	C₂×C₃×#126	1	1	yes	characteristic
265	C₂	C₃×#533	1	1	yes	characteristic
266	C₂	C₃×#540	1	1	yes	characteristic
267	C₂	C₃×#553	1	1	yes	characteristic
268	C₂	C₃×#554	1	1	yes	characteristic
269	C₂	C₃×#554	1	1	yes	characteristic
270	C₂	C₃×#555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
1. C₃×#400	C₂	semidirect
3. C₃×#389	C₂	semidirect
4. C₃×#389	C₂	semidirect
6. C₃×#387	C₂	semidirect
7. C₃×#384	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2497	C₃	direct (N × H)
36. #400	C₂×C₃	semidirect
38. #389	C₂×C₃	semidirect
39. #389	C₂×C₃	semidirect
41. #387	C₂×C₃	semidirect
42. #384	C₂×C₃	semidirect
45. C₄×C₄⋊C₄	C₂×C₃	semidirect
262. C₃	#2497	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×#400	2
C₃×#396	2
C₃×#389	2
C₃×#388	2
C₃×#387	2
C₃×#384	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2497	3

Subgroups — C3×#2497 #84361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2497 #84361