Subgroups — C₃×C₄×#137 #84809

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#137	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#441	C₂	1	2	yes	maximal, complemented
5	C₃×#440	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
8	C₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#417	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#137	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₃×#137	C₄	1	8	yes	direct product factor
16	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₃×C₄×Q₈	—	2	1	no	—
21	C₃×C₄×Q₈	—	2	1	no	—
22	C₃×C₄×Q₈	C₂²	1	2	yes	—
23	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×Q₈	C₂²	1	2	yes	—
25	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	—	2	1	no	—
27	C₃×C₄×D₄	—	2	1	no	—
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
30	C₃×#117	—	2	1	no	—
31	C₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄²	—	2	1	no	—
35	C₂×C₃×C₄²	—	2	1	no	—
36	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
37	C₃×#107	C₂²	1	2	yes	—
38	C₃×#106	C₂²	1	2	yes	—
39	C₃×#103	C₂²	1	2	yes	—
40	C₃×#103	C₂²	1	2	yes	—
41	C₃×#103	C₂²	1	2	yes	—
42	C₃×#102	C₂²	1	2	yes	—
43	C₃×#97	C₂²	1	2	yes	—
44	C₃×C₄×C₈	C₂²	1	2	yes	—
45	C₂×#137	C₂×C₃	1	1	yes	characteristic
46	C₄×#41	C₂×C₃	1	1	yes	characteristic
47	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
48	#441	C₂×C₃	1	2	yes	complemented
49	#440	C₂×C₃	1	2	yes	complemented
50	C₄×Q₁₆	C₂×C₃	1	2	yes	complemented
51	C₄×SD₁₆	C₂×C₃	1	2	yes	complemented
52	#428	C₂×C₃	1	1	yes	characteristic, complemented
53	#419	C₂×C₃	1	1	yes	characteristic, complemented
54	#417	C₂×C₃	1	1	yes	characteristic
55	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
56	C₃×#41	—	2	2	no	—
57	C₃×#41	C₂×C₄	1	4	yes	—
58	C₂×C₃×Q₈	—	2	1	no	—
59	C₂×C₃×Q₈	—	2	2	no	—
60	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
61	C₂×C₃×Q₈	C₂³	1	2	yes	—
62	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×D₄	—	2	1	no	—
64	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	1	no	—
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₃×Q₁₆	C₂×C₄	1	8	yes	complemented
74	C₃×SD₁₆	C₂×C₄	1	8	yes	complemented
75	C₃×M₁₆	—	2	2	no	—
76	C₃×M₁₆	C₂×C₄	1	4	yes	complemented
77	C₂×C₃×C₈	C₂³	1	2	yes	—
78	C₂×C₃×C₈	C₂³	1	2	yes	—
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	1	no	—
82	C₃×C₄⋊C₄	—	2	1	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	C₂³	1	2	yes	—
85	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₃×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₂²⋊C₄	—	2	1	no	—
89	C₃×C₂²⋊C₄	—	2	1	no	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	—	2	2	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	—	2	2	no	—
95	C₃×C₄²	D₄	1	1	yes	characteristic
96	C₃×C₄²	D₄	1	1	yes	characteristic
97	C₃×C₄²	C₂³	1	1	yes	characteristic
98	C₃×C₄²	C₂³	1	1	yes	characteristic
99	C₂×#41	C₂²×C₃	1	1	yes	characteristic
100	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
101	#137	C₃×C₄	1	8	yes	direct product factor
102	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
103	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
104	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
105	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
106	C₄×Q₈	—	2	1	no	—
107	C₄×Q₈	—	2	1	no	—
108	C₄×Q₈	C₂²×C₃	1	2	yes	—
109	C₄×Q₈	C₂²×C₃	1	2	yes	—
110	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
111	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
112	C₄×D₄	—	2	1	no	—
113	C₄×D₄	—	2	1	no	—
114	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
115	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
116	#117	—	2	1	no	—
117	#117	C₂²×C₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	—	2	1	no	—
122	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
123	#107	C₂²×C₃	1	2	yes	—
124	#106	C₂²×C₃	1	2	yes	—
125	#103	C₂²×C₃	1	2	yes	—
126	#103	C₂²×C₃	1	2	yes	—
127	#103	C₂²×C₃	1	2	yes	—
128	#102	C₂²×C₃	1	2	yes	—
129	#97	C₂²×C₃	1	2	yes	—
130	C₄×C₈	C₂²×C₃	1	2	yes	—
131	C₂³×C₃	—	2	1	no	—
132	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
133	C₃×Q₈	—	2	4	no	—
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	2	1	no	—
136	C₃×Q₈	C₂²×C₄	1	4	yes	—
137	C₃×Q₈	C₂²×C₄	1	2	yes	—
138	C₃×D₄	—	2	4	no	—
139	C₃×D₄	—	2	1	no	—
140	C₃×D₄	C₂²×C₄	1	2	yes	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	4	1	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	#41	1	2	yes	—
160	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
161	C₃×C₈	—	2	2	no	—
162	C₃×C₈	C₂²×C₄	1	4	yes	—
163	#41	—	2	2	no	—
164	#41	C₂×C₃×C₄	1	4	yes	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
168	C₂×Q₈	C₂³×C₃	1	2	yes	—
169	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
180	Q₁₆	C₂×C₃×C₄	1	8	yes	complemented
181	SD₁₆	C₂×C₃×C₄	1	8	yes	complemented
182	M₁₆	—	2	2	no	—
183	M₁₆	C₂×C₃×C₄	1	4	yes	complemented
184	C₂×C₈	C₂³×C₃	1	2	yes	—
185	C₂×C₈	C₂³×C₃	1	2	yes	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂³×C₃	1	2	yes	—
192	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₃×D₄	1	1	yes	characteristic
203	C₄²	C₃×D₄	1	1	yes	characteristic
204	C₄²	C₂³×C₃	1	1	yes	characteristic
205	C₄²	C₂³×C₃	1	1	yes	characteristic
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	C₄×D₄	1	2	yes	—
212	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	4	1	no	—
219	C₃×C₄	—	4	1	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	C₄×D₄	1	2	yes	—
222	C₃×C₄	#137	1	2	yes	direct product factor
223	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
224	C₃×C₄	C₂×#41	1	1	yes	characteristic
225	C₂³	—	2	1	no	—
226	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	2	2	no	—
229	Q₈	—	2	1	no	—
230	Q₈	C₂²×C₃×C₄	1	4	yes	—
231	Q₈	C₂²×C₃×C₄	1	2	yes	—
232	D₄	—	2	4	no	—
233	D₄	—	2	1	no	—
234	D₄	C₂²×C₃×C₄	1	2	yes	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
248	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
253	C₂×C₄	C₃×#41	1	2	yes	—
254	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	C₂²×C₃×C₄	1	4	yes	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂×C₃	C₂×#137	1	1	yes	characteristic
261	C₂×C₃	#575	1	1	yes	characteristic
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₃×C₄×D₄	1	2	yes	—
268	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	4	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₃×C₄×D₄	1	2	yes	—
278	C₄	C₃×#137	1	2	yes	direct product factor
279	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
280	C₄	C₂×C₃×#41	1	1	yes	characteristic
281	C₃	C₄×#137	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
285	C₂	C₂×C₃×#137	1	1	yes	characteristic
286	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄×Q₈	C₂	semidirect
4. C₃×#441	C₂	semidirect
5. C₃×#440	C₂	semidirect
6. C₃×C₄×Q₁₆	C₂	semidirect
7. C₃×C₄×SD₁₆	C₂	semidirect
8. C₃×#428	C₂	semidirect
9. C₃×#419	C₂	semidirect
11. C₃×C₄×M₁₆	C₂	semidirect
12. C₄×#137	C₃	direct (N × H)
15. C₃×#137	C₄	direct (N × H)
47. C₂×C₄×Q₈	C₂×C₃	semidirect
48. #441	C₂×C₃	semidirect
49. #440	C₂×C₃	semidirect
50. C₄×Q₁₆	C₂×C₃	semidirect
51. C₄×SD₁₆	C₂×C₃	semidirect
52. #428	C₂×C₃	semidirect
53. #419	C₂×C₃	semidirect
55. C₄×M₁₆	C₂×C₃	semidirect
60. C₂×C₃×Q₈	C₂×C₄	semidirect
73. C₃×Q₁₆	C₂×C₄	semidirect
74. C₃×SD₁₆	C₂×C₄	semidirect
76. C₃×M₁₆	C₂×C₄	semidirect
101. #137	C₃×C₄	direct (N × H)
167. C₂×Q₈	C₂×C₃×C₄	semidirect
180. Q₁₆	C₂×C₃×C₄	semidirect
181. SD₁₆	C₂×C₃×C₄	semidirect
183. M₁₆	C₂×C₃×C₄	semidirect
222. C₃×C₄	#137	direct (N × H)
278. C₄	C₃×#137	direct (N × H)
281. C₃	C₄×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#137	2
C₃×C₄×#41	2
C₂×C₃×C₄×Q₈	2
C₃×#441	2
C₃×#440	2
C₃×C₄×Q₁₆	2
C₃×C₄×SD₁₆	2
C₃×#428	2
C₃×#419	2
C₃×#417	2
C₃×C₄×M₁₆	2
C₄×#137	3

Subgroups — C3×C4×#137 #84809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#137 #84809