Subgroups — C₃²×#3058 #11770107

Order 1152.

1572 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#535	C₂	1	2	yes	maximal, complemented
4	C₃²×#469	C₂	1	4	yes	maximal, complemented
5	C₃²×#466	C₂	1	4	yes	maximal, complemented
6	C₃²×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#418	C₂	1	2	yes	maximal, complemented
8	C₃×#3058	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×D₄	C₂²	1	2	yes	—
11	C₂×C₃²×D₈	—	2	4	no	—
12	C₂×C₃²×D₈	C₂²	1	2	yes	—
13	C₂×C₃²×D₈	C₂²	1	2	yes	—
14	C₃²×#131	—	2	4	no	—
15	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃²×M₁₆	C₂²	1	2	yes	complemented
17	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃²×C₈	C₂²	1	2	yes	complemented
19	C₃²×#127	—	2	2	no	—
20	C₃²×#124	—	2	2	no	—
21	C₃²×#121	C₂²	1	8	yes	complemented
22	C₃²×#120	—	2	4	no	—
23	C₃²×#117	C₂²	1	2	yes	—
24	C₃²×#107	C₂²	1	2	yes	complemented
25	C₃²×#106	C₂²	1	2	yes	complemented
26	C₃²×#102	C₂²	1	8	yes	complemented
27	C₂²×C₃×D₈	C₂×C₃	1	4	yes	complemented
28	C₂×C₃×#131	C₂×C₃	1	4	yes	complemented
29	C₃×#535	C₂×C₃	1	8	yes	complemented
30	C₃×#469	C₂×C₃	1	16	yes	complemented
31	C₃×#466	C₂×C₃	1	16	yes	complemented
32	C₃×#427	C₂×C₃	1	4	yes	complemented
33	C₃×#418	C₂×C₃	1	8	yes	complemented
34	C₂⁴×C₃²	—	2	2	no	—
35	C₃²×#41	—	2	2	no	—
36	C₃²×#41	—	2	2	no	—
37	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×D₄	—	2	8	no	—
39	C₂×C₃²×D₄	—	2	4	no	—
40	C₂×C₃²×D₄	—	2	2	no	—
41	C₂×C₃²×D₄	—	2	4	no	—
42	C₂×C₃²×D₄	D₄	1	2	yes	—
43	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
44	C₂×C₃²×D₄	C₂³	1	4	yes	—
45	C₂²×C₃²×C₄	C₂³	1	2	yes	—
46	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
47	C₃²×D₈	—	2	4	no	—
48	C₃²×D₈	—	2	4	no	—
49	C₃²×M₁₆	—	2	2	no	—
50	C₃²×M₁₆	—	2	4	no	—
51	C₂×C₃²×C₈	—	2	4	no	—
52	C₂×C₃²×C₈	D₄	1	4	yes	complemented
53	C₂×C₃²×C₈	C₂³	1	2	yes	—
54	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
56	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
57	C₃²×C₂²⋊C₄	—	2	2	no	—
58	C₃²×C₂²⋊C₄	—	2	8	no	—
59	C₃²×C₄²	—	2	2	no	—
60	#3058	C₃²	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
62	C₂²×C₃×D₄	C₂²×C₃	1	8	yes	—
63	C₂×C₃×D₈	—	2	16	no	—
64	C₂×C₃×D₈	C₂²×C₃	1	8	yes	—
65	C₂×C₃×D₈	C₂²×C₃	1	8	yes	—
66	C₃×#131	—	2	16	no	—
67	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
68	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	complemented
69	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
70	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	complemented
71	C₃×#127	—	2	8	no	—
72	C₃×#124	—	2	8	no	—
73	C₃×#121	C₂²×C₃	1	32	yes	complemented
74	C₃×#120	—	2	16	no	—
75	C₃×#117	C₂²×C₃	1	8	yes	—
76	C₃×#107	C₂²×C₃	1	8	yes	complemented
77	C₃×#106	C₂²×C₃	1	8	yes	complemented
78	C₃×#102	C₂²×C₃	1	32	yes	complemented
79	C₂³×C₃²	—	2	4	no	—
80	C₂³×C₃²	—	4	4	no	—
81	C₂³×C₃²	—	8	4	no	—
82	C₂³×C₃²	C₂×D₄	1	2	yes	—
83	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
84	C₃²×Q₈	—	2	2	no	—
85	C₃²×D₄	—	2	4	no	—
86	C₃²×D₄	—	2	2	no	—
87	C₃²×D₄	—	2	4	no	—
88	C₃²×D₄	—	2	4	no	—
89	C₃²×D₄	—	4	4	no	—
90	C₃²×D₄	—	8	8	no	—
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	—	2	2	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
97	C₂×C₃²×C₄	#41	1	2	yes	—
98	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
99	C₃²×C₈	—	2	2	no	—
100	C₃²×C₈	C₂×D₄	1	2	yes	—
101	C₃²×C₈	C₂×D₄	1	2	yes	—
102	C₂²×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
103	C₂×#131	C₂×C₃²	1	1	yes	characteristic, complemented
104	#535	C₂×C₃²	1	2	yes	complemented
105	#469	C₂×C₃²	1	4	yes	complemented
106	#466	C₂×C₃²	1	4	yes	complemented
107	#427	C₂×C₃²	1	1	yes	characteristic, complemented
108	#418	C₂×C₃²	1	2	yes	complemented
109	C₂⁴×C₃	—	2	8	no	—
110	C₃×#41	—	2	8	no	—
111	C₃×#41	—	2	8	no	—
112	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
113	C₂×C₃×D₄	—	2	32	no	—
114	C₂×C₃×D₄	—	2	16	no	—
115	C₂×C₃×D₄	—	2	8	no	—
116	C₂×C₃×D₄	—	2	16	no	—
117	C₂×C₃×D₄	C₃×D₄	1	8	yes	—
118	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
119	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
122	C₃×D₈	—	2	16	no	—
123	C₃×D₈	—	2	16	no	—
124	C₃×M₁₆	—	2	8	no	—
125	C₃×M₁₆	—	2	16	no	—
126	C₂×C₃×C₈	—	2	16	no	—
127	C₂×C₃×C₈	C₃×D₄	1	16	yes	complemented
128	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
129	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
130	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
131	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
132	C₃×C₂²⋊C₄	—	2	8	no	—
133	C₃×C₂²⋊C₄	—	2	32	no	—
134	C₃×C₄²	—	2	8	no	—
135	C₂²×C₃²	—	2	2	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	4	4	no	—
139	C₂²×C₃²	—	4	8	no	—
140	C₂²×C₃²	—	8	8	no	—
141	C₂²×C₃²	#121	1	2	yes	—
142	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
143	C₃²×C₄	—	2	2	no	—
144	C₃²×C₄	—	4	4	no	—
145	C₃²×C₄	#121	1	2	yes	—
146	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₃²×C₄	C₂×#41	1	1	yes	characteristic
148	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
149	C₂²×D₄	C₂²×C₃²	1	2	yes	—
150	C₂×D₈	—	2	4	no	—
151	C₂×D₈	C₂²×C₃²	1	2	yes	—
152	C₂×D₈	C₂²×C₃²	1	2	yes	—
153	#131	—	2	4	no	—
154	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
155	C₂×M₁₆	C₂²×C₃²	1	2	yes	complemented
156	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₃²	1	2	yes	complemented
158	#127	—	2	2	no	—
159	#124	—	2	2	no	—
160	#121	C₂²×C₃²	1	8	yes	complemented
161	#120	—	2	4	no	—
162	#117	C₂²×C₃²	1	2	yes	—
163	#107	C₂²×C₃²	1	2	yes	complemented
164	#106	C₂²×C₃²	1	2	yes	complemented
165	#102	C₂²×C₃²	1	8	yes	complemented
166	C₂³×C₃	—	2	16	no	—
167	C₂³×C₃	—	4	16	no	—
168	C₂³×C₃	—	8	16	no	—
169	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
170	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
171	C₃×Q₈	—	2	8	no	—
172	C₃×D₄	—	2	16	no	—
173	C₃×D₄	—	2	8	no	—
174	C₃×D₄	—	2	16	no	—
175	C₃×D₄	—	2	16	no	—
176	C₃×D₄	—	4	16	no	—
177	C₃×D₄	—	8	32	no	—
178	C₂×C₃×C₄	—	2	8	no	—
179	C₂×C₃×C₄	—	2	8	no	—
180	C₂×C₃×C₄	—	2	16	no	—
181	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
184	C₂×C₃×C₄	C₃×#41	1	8	yes	—
185	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
186	C₃×C₈	—	2	8	no	—
187	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
188	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
189	C₂×C₃²	—	2	2	no	—
190	C₂×C₃²	—	4	2	no	—
191	C₂×C₃²	—	8	4	no	—
192	C₂×C₃²	C₂×#121	1	1	yes	characteristic
193	C₂×C₃²	#576	1	2	yes	—
194	C₂⁴	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	C₃²×D₄	1	2	yes	—
203	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₃²	1	4	yes	—
205	C₂²×C₄	C₂³×C₃²	1	2	yes	—
206	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
207	D₈	—	2	4	no	—
208	D₈	—	2	4	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₃²×D₄	1	4	yes	complemented
213	C₂×C₈	C₂³×C₃²	1	2	yes	—
214	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
215	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
216	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	8	no	—
219	C₄²	—	2	2	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	2	16	no	—
222	C₂²×C₃	—	2	8	no	—
223	C₂²×C₃	—	4	32	no	—
224	C₂²×C₃	—	4	16	no	—
225	C₂²×C₃	—	8	32	no	—
226	C₂²×C₃	C₃×#121	1	8	yes	—
227	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
228	C₃×C₄	—	2	8	no	—
229	C₃×C₄	—	4	16	no	—
230	C₃×C₄	C₃×#121	1	8	yes	—
231	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
232	C₃×C₄	C₂×C₃×#41	1	4	yes	—
233	C₃²	#3058	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	2	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	8	4	no	—
237	C₂³	C₂×C₃²×D₄	1	2	yes	—
238	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	4	no	—
245	D₄	—	8	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
250	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
251	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
252	C₂×C₄	C₃²×#41	1	2	yes	—
253	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	C₂×C₃²×D₄	1	2	yes	—
256	C₈	C₂×C₃²×D₄	1	2	yes	—
257	C₂×C₃	—	2	8	no	—
258	C₂×C₃	—	4	8	no	—
259	C₂×C₃	—	8	16	no	—
260	C₂×C₃	C₂×C₃×#121	1	4	yes	—
261	C₂×C₃	C₃×#576	1	8	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	4	no	—
267	C₂²	—	8	8	no	—
268	C₂²	C₃²×#121	1	2	yes	—
269	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	4	4	no	—
272	C₄	C₃²×#121	1	2	yes	—
273	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
274	C₄	C₂×C₃²×#41	1	1	yes	characteristic
275	C₃	C₃×#3058	1	4	yes	direct product factor
276	C₂	—	2	2	no	—
277	C₂	—	4	2	no	—
278	C₂	—	8	4	no	—
279	C₂	C₂×C₃²×#121	1	1	yes	characteristic
280	C₂	C₃²×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×D₈	C₂	semidirect
2. C₂×C₃²×#131	C₂	semidirect
3. C₃²×#535	C₂	semidirect
4. C₃²×#469	C₂	semidirect
5. C₃²×#466	C₂	semidirect
6. C₃²×#427	C₂	semidirect
7. C₃²×#418	C₂	semidirect
8. C₃×#3058	C₃	direct (N × H)
16. C₂×C₃²×M₁₆	C₂²	semidirect
18. C₂²×C₃²×C₈	C₂²	semidirect
21. C₃²×#121	C₂²	semidirect
24. C₃²×#107	C₂²	semidirect
25. C₃²×#106	C₂²	semidirect
26. C₃²×#102	C₂²	semidirect
27. C₂²×C₃×D₈	C₂×C₃	semidirect
28. C₂×C₃×#131	C₂×C₃	semidirect
29. C₃×#535	C₂×C₃	semidirect
30. C₃×#469	C₂×C₃	semidirect
31. C₃×#466	C₂×C₃	semidirect
32. C₃×#427	C₂×C₃	semidirect
33. C₃×#418	C₂×C₃	semidirect
52. C₂×C₃²×C₈	D₄	semidirect
60. #3058	C₃²	direct (N × H)
68. C₂×C₃×M₁₆	C₂²×C₃	semidirect
70. C₂²×C₃×C₈	C₂²×C₃	semidirect
73. C₃×#121	C₂²×C₃	semidirect
76. C₃×#107	C₂²×C₃	semidirect
77. C₃×#106	C₂²×C₃	semidirect
78. C₃×#102	C₂²×C₃	semidirect
102. C₂²×D₈	C₂×C₃²	semidirect
103. C₂×#131	C₂×C₃²	semidirect
104. #535	C₂×C₃²	semidirect
105. #469	C₂×C₃²	semidirect
106. #466	C₂×C₃²	semidirect
107. #427	C₂×C₃²	semidirect
108. #418	C₂×C₃²	semidirect
127. C₂×C₃×C₈	C₃×D₄	semidirect
155. C₂×M₁₆	C₂²×C₃²	semidirect
157. C₂²×C₈	C₂²×C₃²	semidirect
160. #121	C₂²×C₃²	semidirect
163. #107	C₂²×C₃²	semidirect
164. #106	C₂²×C₃²	semidirect
165. #102	C₂²×C₃²	semidirect
212. C₂×C₈	C₃²×D₄	semidirect
233. C₃²	#3058	direct (N × H)
275. C₃	C₃×#3058	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×C₃²×D₈	2
C₂×C₃²×#131	2
C₃²×#535	2
C₃²×#469	2
C₃²×#466	2
C₃²×#427	2
C₃²×#418	2
C₃×#3058	3

Subgroups — C32×#3058 #11770107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3058 #11770107