Subgroups — C₃²×#3059 #11770108

Order 1152.

1380 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
3	C₃²×#536	C₂	1	2	yes	maximal
4	C₃²×#470	C₂	1	4	yes	maximal, complemented
5	C₃²×#467	C₂	1	4	yes	maximal, complemented
6	C₃²×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#419	C₂	1	2	yes	maximal, complemented
8	C₃×#3059	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
11	C₂×C₃²×Q₁₆	—	2	4	no	—
12	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
14	C₃²×#131	—	2	4	no	—
15	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
16	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃²×C₈	C₂²	1	2	yes	—
18	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
19	C₃²×C₄⋊Q₈	—	2	2	no	—
20	C₃²×#124	—	2	2	no	—
21	C₃²×#123	—	2	4	no	—
22	C₃²×C₂²⋊Q₈	C₂²	1	8	yes	—
23	C₃²×#117	C₂²	1	2	yes	—
24	C₃²×#107	C₂²	1	2	yes	—
25	C₃²×#106	C₂²	1	2	yes	—
26	C₃²×#103	C₂²	1	8	yes	—
27	C₂²×C₃×Q₁₆	C₂×C₃	1	4	yes	complemented
28	C₂×C₃×#131	C₂×C₃	1	4	yes	—
29	C₃×#536	C₂×C₃	1	8	yes	—
30	C₃×#470	C₂×C₃	1	16	yes	complemented
31	C₃×#467	C₂×C₃	1	16	yes	complemented
32	C₃×#427	C₂×C₃	1	4	yes	complemented
33	C₃×#419	C₂×C₃	1	8	yes	complemented
34	C₃²×#41	—	2	2	no	—
35	C₃²×#41	—	2	2	no	—
36	C₂×C₃²×Q₈	—	2	2	no	—
37	C₂×C₃²×Q₈	—	2	4	no	—
38	C₂×C₃²×Q₈	—	2	4	no	—
39	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
40	C₂×C₃²×Q₈	C₂³	1	4	yes	—
41	C₂×C₃²×D₄	D₄	1	2	yes	—
42	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
43	C₂²×C₃²×C₄	—	2	2	no	—
44	C₂²×C₃²×C₄	C₂³	1	2	yes	—
45	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
46	C₃²×Q₁₆	—	2	4	no	—
47	C₃²×Q₁₆	—	2	4	no	—
48	C₃²×M₁₆	—	2	2	no	—
49	C₃²×M₁₆	—	2	4	no	—
50	C₂×C₃²×C₈	—	2	4	no	—
51	C₂×C₃²×C₈	D₄	1	4	yes	—
52	C₂×C₃²×C₈	C₂³	1	2	yes	—
53	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	—	2	8	no	—
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	#3059	C₃²	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
62	C₂²×C₃×Q₈	C₂²×C₃	1	8	yes	—
63	C₂×C₃×Q₁₆	—	2	16	no	—
64	C₂×C₃×Q₁₆	C₂²×C₃	1	8	yes	—
65	C₂×C₃×Q₁₆	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	—
69	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
70	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
71	C₃×C₄⋊Q₈	—	2	8	no	—
72	C₃×#124	—	2	8	no	—
73	C₃×#123	—	2	16	no	—
74	C₃×C₂²⋊Q₈	C₂²×C₃	1	32	yes	—
75	C₃×#117	C₂²×C₃	1	8	yes	—
76	C₃×#107	C₂²×C₃	1	8	yes	—
77	C₃×#106	C₂²×C₃	1	8	yes	—
78	C₃×#103	C₂²×C₃	1	32	yes	—
79	C₂³×C₃²	C₂×D₄	1	2	yes	—
80	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
81	C₃²×Q₈	—	2	2	no	—
82	C₃²×Q₈	—	2	4	no	—
83	C₃²×Q₈	—	2	4	no	—
84	C₃²×Q₈	—	4	4	no	—
85	C₃²×D₄	—	2	4	no	—
86	C₃²×D₄	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	4	4	no	—
92	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
95	C₂×C₃²×C₄	#41	1	2	yes	—
96	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
97	C₃²×C₈	—	2	2	no	—
98	C₃²×C₈	C₂×D₄	1	2	yes	—
99	C₃²×C₈	C₂×D₄	1	2	yes	—
100	C₂²×Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
101	C₂×#131	C₂×C₃²	1	1	yes	characteristic
102	#536	C₂×C₃²	1	2	yes	—
103	#470	C₂×C₃²	1	4	yes	complemented
104	#467	C₂×C₃²	1	4	yes	complemented
105	#427	C₂×C₃²	1	1	yes	characteristic, complemented
106	#419	C₂×C₃²	1	2	yes	complemented
107	C₃×#41	—	2	8	no	—
108	C₃×#41	—	2	8	no	—
109	C₂×C₃×Q₈	—	2	8	no	—
110	C₂×C₃×Q₈	—	2	16	no	—
111	C₂×C₃×Q₈	—	2	16	no	—
112	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
113	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
114	C₂×C₃×D₄	C₃×D₄	1	8	yes	—
115	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
116	C₂²×C₃×C₄	—	2	8	no	—
117	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
118	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
119	C₃×Q₁₆	—	2	16	no	—
120	C₃×Q₁₆	—	2	16	no	—
121	C₃×M₁₆	—	2	8	no	—
122	C₃×M₁₆	—	2	16	no	—
123	C₂×C₃×C₈	—	2	16	no	—
124	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
125	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
126	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
127	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
128	C₃×C₄⋊C₄	—	2	32	no	—
129	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
130	C₃×C₂²⋊C₄	—	2	8	no	—
131	C₃×C₂²⋊C₄	—	2	32	no	—
132	C₃×C₄²	—	2	8	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	—	2	2	no	—
135	C₂²×C₃²	—	2	4	no	—
136	C₂²×C₃²	#121	1	2	yes	—
137	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
138	C₃²×C₄	—	2	2	no	—
139	C₃²×C₄	—	4	4	no	—
140	C₃²×C₄	—	4	4	no	—
141	C₃²×C₄	#121	1	2	yes	—
142	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₃²×C₄	C₂×#41	1	1	yes	characteristic
144	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
145	C₂²×Q₈	C₂²×C₃²	1	2	yes	—
146	C₂×Q₁₆	—	2	4	no	—
147	C₂×Q₁₆	C₂²×C₃²	1	2	yes	—
148	C₂×Q₁₆	C₂²×C₃²	1	2	yes	—
149	#131	—	2	4	no	—
150	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
151	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
152	C₂²×C₈	C₂²×C₃²	1	2	yes	—
153	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
154	C₄⋊Q₈	—	2	2	no	—
155	#124	—	2	2	no	—
156	#123	—	2	4	no	—
157	C₂²⋊Q₈	C₂²×C₃²	1	8	yes	—
158	#117	C₂²×C₃²	1	2	yes	—
159	#107	C₂²×C₃²	1	2	yes	—
160	#106	C₂²×C₃²	1	2	yes	—
161	#103	C₂²×C₃²	1	8	yes	—
162	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
163	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
164	C₃×Q₈	—	2	8	no	—
165	C₃×Q₈	—	2	16	no	—
166	C₃×Q₈	—	2	16	no	—
167	C₃×Q₈	—	4	16	no	—
168	C₃×D₄	—	2	16	no	—
169	C₃×D₄	—	2	8	no	—
170	C₂×C₃×C₄	—	2	8	no	—
171	C₂×C₃×C₄	—	2	8	no	—
172	C₂×C₃×C₄	—	2	16	no	—
173	C₂×C₃×C₄	—	2	16	no	—
174	C₂×C₃×C₄	—	4	16	no	—
175	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
176	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
177	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
178	C₂×C₃×C₄	C₃×#41	1	8	yes	—
179	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
180	C₃×C₈	—	2	8	no	—
181	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
182	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
183	C₂×C₃²	—	2	2	no	—
184	C₂×C₃²	—	4	2	no	—
185	C₂×C₃²	C₂×#121	1	1	yes	characteristic
186	C₂×C₃²	#578	1	2	yes	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
193	C₂×Q₈	C₂³×C₃²	1	4	yes	—
194	C₂×D₄	C₃²×D₄	1	2	yes	—
195	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂³×C₃²	1	2	yes	—
198	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₃²×D₄	1	4	yes	—
205	C₂×C₈	C₂³×C₃²	1	2	yes	—
206	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
207	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	8	no	—
212	C₄²	—	2	2	no	—
213	C₂²×C₃	—	2	8	no	—
214	C₂²×C₃	—	2	16	no	—
215	C₂²×C₃	—	2	8	no	—
216	C₂²×C₃	C₃×#121	1	8	yes	—
217	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
218	C₃×C₄	—	2	8	no	—
219	C₃×C₄	—	4	16	no	—
220	C₃×C₄	—	4	16	no	—
221	C₃×C₄	C₃×#121	1	8	yes	—
222	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
223	C₃×C₄	C₂×C₃×#41	1	4	yes	—
224	C₃²	#3059	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	C₂×C₃²×D₄	1	2	yes	—
226	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	2	4	no	—
229	Q₈	—	2	4	no	—
230	Q₈	—	4	4	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
239	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
240	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
241	C₂×C₄	C₃²×#41	1	2	yes	—
242	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	C₂×C₃²×D₄	1	2	yes	—
245	C₈	C₂×C₃²×D₄	1	2	yes	—
246	C₂×C₃	—	2	8	no	—
247	C₂×C₃	—	4	8	no	—
248	C₂×C₃	C₂×C₃×#121	1	4	yes	—
249	C₂×C₃	C₃×#578	1	8	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	C₃²×#121	1	2	yes	—
254	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	4	4	no	—
258	C₄	C₃²×#121	1	2	yes	—
259	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
260	C₄	C₂×C₃²×#41	1	1	yes	characteristic
261	C₃	C₃×#3059	1	4	yes	direct product factor
262	C₂	—	2	2	no	—
263	C₂	—	4	2	no	—
264	C₂	C₂×C₃²×#121	1	1	yes	characteristic
265	C₂	C₃²×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
prime radical	C₂×C₃²×#41	C₂²
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×Q₁₆	C₂	semidirect
4. C₃²×#470	C₂	semidirect
5. C₃²×#467	C₂	semidirect
6. C₃²×#427	C₂	semidirect
7. C₃²×#419	C₂	semidirect
8. C₃×#3059	C₃	direct (N × H)
27. C₂²×C₃×Q₁₆	C₂×C₃	semidirect
30. C₃×#470	C₂×C₃	semidirect
31. C₃×#467	C₂×C₃	semidirect
32. C₃×#427	C₂×C₃	semidirect
33. C₃×#419	C₂×C₃	semidirect
60. #3059	C₃²	direct (N × H)
100. C₂²×Q₁₆	C₂×C₃²	semidirect
103. #470	C₂×C₃²	semidirect
104. #467	C₂×C₃²	semidirect
105. #427	C₂×C₃²	semidirect
106. #419	C₂×C₃²	semidirect
224. C₃²	#3059	direct (N × H)
261. C₃	C₃×#3059	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×Q₁₆	2
C₂×C₃²×#131	2
C₃²×#536	2
C₃²×#470	2
C₃²×#467	2
C₃²×#427	2
C₃²×#419	2
C₃×#3059	3

Subgroups — C32×#3059 #11770108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3059 #11770108