Subgroups — C₃²×#3283 #11770332

Order 1152.

1452 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×C₃²×#135	C₂	1	2	yes	maximal, complemented
2	C₃²×#547	C₂	1	2	yes	maximal, complemented
3	C₃²×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#467	C₂	1	2	yes	maximal, complemented
5	C₃²×#461	C₂	1	2	yes	maximal, complemented
6	C₃²×#452	C₂	1	4	yes	maximal, complemented
7	C₃²×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3283	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×#41	C₂²	1	4	yes	—
11	C₃²×#135	—	2	8	no	—
12	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃²×SD₁₆	C₂²	1	4	yes	—
15	C₂×C₃²×D₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃²×C₈	C₂²	1	2	yes	—
17	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₃²×#123	—	2	4	no	—
19	C₃²×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₃²×#121	—	2	2	no	—
21	C₃²×C₄×D₄	C₂²	1	2	yes	—
22	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
23	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
24	C₃²×#107	C₂²	1	1	yes	characteristic
25	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₃²×#103	C₂²	1	4	yes	—
27	C₃²×#102	C₂²	1	2	yes	—
28	C₃²×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃×#135	C₂×C₃	1	8	yes	complemented
31	C₃×#547	C₂×C₃	1	8	yes	complemented
32	C₃×#494	C₂×C₃	1	4	yes	complemented
33	C₃×#467	C₂×C₃	1	8	yes	complemented
34	C₃×#461	C₂×C₃	1	8	yes	complemented
35	C₃×#452	C₂×C₃	1	16	yes	complemented
36	C₃×C₄×D₈	C₂×C₃	1	4	yes	complemented
37	C₃×C₈×D₄	C₂×C₃	1	4	yes	complemented
38	C₃²×#41	—	2	8	no	—
39	C₃²×#41	—	4	4	no	—
40	C₂×C₃²×Q₈	C₂³	1	4	yes	—
41	C₂×C₃²×D₄	—	2	4	no	—
42	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
43	C₂×C₃²×D₄	C₂³	1	2	yes	—
44	C₂²×C₃²×C₄	—	2	4	no	—
45	C₂²×C₃²×C₄	—	2	2	no	—
46	C₂²×C₃²×C₄	C₂³	1	2	yes	—
47	C₃²×Q₁₆	—	2	4	no	—
48	C₃²×Q₁₆	—	4	2	no	—
49	C₃²×SD₁₆	—	2	8	no	—
50	C₃²×D₈	D₄	1	4	yes	complemented
51	C₂×C₃²×C₈	—	2	4	no	—
52	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂³	1	2	yes	—
54	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	—	2	2	no	—
56	C₃²×C₄⋊C₄	—	2	4	no	—
57	C₃²×C₄⋊C₄	—	2	2	no	—
58	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
60	C₃²×C₂²⋊C₄	—	2	4	no	—
61	C₃²×C₂²⋊C₄	—	2	2	no	—
62	C₃²×C₂²⋊C₄	D₄	1	2	yes	—
63	C₃²×C₄²	C₂³	1	1	yes	characteristic
64	#3283	C₃²	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃×#41	C₂²×C₃	1	16	yes	—
66	C₃×#135	—	2	32	no	—
67	C₂×C₃×Q₁₆	C₂²×C₃	1	8	yes	—
68	C₂×C₃×Q₁₆	C₂²×C₃	1	8	yes	—
69	C₂×C₃×SD₁₆	C₂²×C₃	1	16	yes	—
70	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
71	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
72	C₃×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
73	C₃×#123	—	2	16	no	—
74	C₃×C₂²⋊Q₈	C₂²×C₃	1	16	yes	—
75	C₃×#121	—	2	8	no	—
76	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
77	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
78	C₂×C₃×C₄⋊C₄	—	2	8	no	—
79	C₃×#107	C₂²×C₃	1	4	yes	—
80	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
81	C₃×#103	C₂²×C₃	1	16	yes	—
82	C₃×#102	C₂²×C₃	1	8	yes	—
83	C₃×C₂²⋊C₈	C₂²×C₃	1	8	yes	—
84	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
85	C₂³×C₃²	—	2	2	no	—
86	C₂³×C₃²	C₂×D₄	1	2	yes	—
87	C₃²×Q₈	—	2	4	no	—
88	C₃²×Q₈	—	2	4	no	—
89	C₃²×D₄	—	2	2	no	—
90	C₃²×D₄	—	2	2	no	—
91	C₃²×D₄	—	4	8	no	—
92	C₃²×D₄	C₂×D₄	1	4	yes	—
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	—	2	2	no	—
95	C₂×C₃²×C₄	—	2	2	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	4	8	no	—
98	C₂×C₃²×C₄	—	4	4	no	—
99	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
101	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
103	C₃²×C₈	—	2	2	no	—
104	C₃²×C₈	—	2	1	no	—
105	C₃²×C₈	C₂×D₄	1	2	yes	—
106	C₂×#135	C₂×C₃²	1	2	yes	complemented
107	#547	C₂×C₃²	1	2	yes	complemented
108	#494	C₂×C₃²	1	1	yes	characteristic, complemented
109	#467	C₂×C₃²	1	2	yes	complemented
110	#461	C₂×C₃²	1	2	yes	complemented
111	#452	C₂×C₃²	1	4	yes	complemented
112	C₄×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
113	C₈×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
114	C₃×#41	—	2	32	no	—
115	C₃×#41	—	4	16	no	—
116	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
117	C₂×C₃×D₄	—	2	16	no	—
118	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
119	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
120	C₂²×C₃×C₄	—	2	8	no	—
121	C₂²×C₃×C₄	—	2	16	no	—
122	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
123	C₃×Q₁₆	—	2	16	no	—
124	C₃×Q₁₆	—	4	8	no	—
125	C₃×SD₁₆	—	2	32	no	—
126	C₃×D₈	C₃×D₄	1	16	yes	complemented
127	C₂×C₃×C₈	—	2	16	no	—
128	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
129	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
130	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
131	C₃×C₄⋊C₄	—	2	8	no	—
132	C₃×C₄⋊C₄	—	2	16	no	—
133	C₃×C₄⋊C₄	—	2	8	no	—
134	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
135	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
136	C₃×C₂²⋊C₄	—	2	16	no	—
137	C₃×C₂²⋊C₄	—	2	8	no	—
138	C₃×C₂²⋊C₄	C₃×D₄	1	8	yes	—
139	C₃×C₄²	C₂³×C₃	1	4	yes	—
140	C₂²×C₃²	—	2	2	no	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	—	2	2	no	—
143	C₂²×C₃²	—	2	2	no	—
144	C₂²×C₃²	—	4	2	no	—
145	C₂²×C₃²	—	4	2	no	—
146	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
147	C₃²×C₄	—	2	2	no	—
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	—	4	4	no	—
150	C₃²×C₄	—	4	2	no	—
151	C₃²×C₄	#135	1	2	yes	complemented
152	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₃²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×#41	C₂²×C₃²	1	4	yes	—
155	#135	—	2	8	no	—
156	C₂×Q₁₆	C₂²×C₃²	1	2	yes	—
157	C₂×Q₁₆	C₂²×C₃²	1	2	yes	—
158	C₂×SD₁₆	C₂²×C₃²	1	4	yes	—
159	C₂×D₈	C₂²×C₃²	1	1	yes	characteristic
160	C₂²×C₈	C₂²×C₃²	1	2	yes	—
161	C₄⋊Q₈	C₂²×C₃²	1	2	yes	—
162	#123	—	2	4	no	—
163	C₂²⋊Q₈	C₂²×C₃²	1	4	yes	—
164	#121	—	2	2	no	—
165	C₄×D₄	C₂²×C₃²	1	2	yes	—
166	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	2	no	—
168	#107	C₂²×C₃²	1	1	yes	characteristic
169	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic
170	#103	C₂²×C₃²	1	4	yes	—
171	#102	C₂²×C₃²	1	2	yes	—
172	C₂²⋊C₈	C₂²×C₃²	1	2	yes	—
173	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
174	C₂³×C₃	—	2	8	no	—
175	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
176	C₃×Q₈	—	2	16	no	—
177	C₃×Q₈	—	2	16	no	—
178	C₃×D₄	—	2	8	no	—
179	C₃×D₄	—	2	8	no	—
180	C₃×D₄	—	4	32	no	—
181	C₃×D₄	C₂×C₃×D₄	1	16	yes	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	8	no	—
184	C₂×C₃×C₄	—	2	8	no	—
185	C₂×C₃×C₄	—	2	16	no	—
186	C₂×C₃×C₄	—	4	32	no	—
187	C₂×C₃×C₄	—	4	16	no	—
188	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
189	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
190	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
191	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
192	C₃×C₈	—	2	8	no	—
193	C₃×C₈	—	2	4	no	—
194	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
195	C₂×C₃²	—	4	2	no	—
196	C₂×C₃²	—	4	4	no	—
197	C₂×C₃²	D₄²	1	1	yes	characteristic
198	C₂×C₃²	C₂×#135	1	1	yes	characteristic
199	C₂×C₃²	#578	1	1	yes	characteristic
200	#41	—	2	8	no	—
201	#41	—	4	4	no	—
202	C₂×Q₈	C₂³×C₃²	1	4	yes	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
205	C₂×D₄	C₂³×C₃²	1	2	yes	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂³×C₃²	1	2	yes	—
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	4	2	no	—
211	SD₁₆	—	2	8	no	—
212	D₈	C₃²×D₄	1	4	yes	complemented
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
215	C₂×C₈	C₂³×C₃²	1	2	yes	—
216	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₃²×D₄	1	1	yes	characteristic
221	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	C₃²×D₄	1	2	yes	—
225	C₄²	C₂³×C₃²	1	1	yes	characteristic
226	C₂²×C₃	—	2	8	no	—
227	C₂²×C₃	—	2	8	no	—
228	C₂²×C₃	—	2	8	no	—
229	C₂²×C₃	—	2	16	no	—
230	C₂²×C₃	—	4	8	no	—
231	C₂²×C₃	—	4	8	no	—
232	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	8	no	—
235	C₃×C₄	—	4	8	no	—
236	C₃×C₄	—	4	16	no	—
237	C₃×C₄	C₃×#135	1	8	yes	complemented
238	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
239	C₃×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
240	C₃²	#3283	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	C₂×C₃²×D₄	1	2	yes	—
243	Q₈	—	2	4	no	—
244	Q₈	—	2	4	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	8	no	—
248	D₄	C₂×C₃²×D₄	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
257	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	1	no	—
261	C₈	C₂×C₃²×D₄	1	2	yes	—
262	C₂×C₃	—	4	8	no	—
263	C₂×C₃	—	4	16	no	—
264	C₂×C₃	C₃×D₄²	1	4	yes	—
265	C₂×C₃	C₂×C₃×#135	1	4	yes	—
266	C₂×C₃	C₃×#578	1	4	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	4	no	—
278	C₄	C₃²×#135	1	2	yes	complemented
279	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
280	C₄	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
281	C₃	C₃×#3283	1	4	yes	direct product factor
282	C₂	—	4	2	no	—
283	C₂	—	4	4	no	—
284	C₂	C₃²×D₄²	1	1	yes	characteristic
285	C₂	C₂×C₃²×#135	1	1	yes	characteristic
286	C₂	C₃²×#578	1	1	yes	characteristic

	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 ◁ C₃²×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#135	C₂	semidirect
2. C₃²×#547	C₂	semidirect
3. C₃²×#494	C₂	semidirect
4. C₃²×#467	C₂	semidirect
5. C₃²×#461	C₂	semidirect
6. C₃²×#452	C₂	semidirect
7. C₃²×C₄×D₈	C₂	semidirect
8. C₃²×C₈×D₄	C₂	semidirect
9. C₃×#3283	C₃	direct (N × H)
30. C₂×C₃×#135	C₂×C₃	semidirect
31. C₃×#547	C₂×C₃	semidirect
32. C₃×#494	C₂×C₃	semidirect
33. C₃×#467	C₂×C₃	semidirect
34. C₃×#461	C₂×C₃	semidirect
35. C₃×#452	C₂×C₃	semidirect
36. C₃×C₄×D₈	C₂×C₃	semidirect
37. C₃×C₈×D₄	C₂×C₃	semidirect
50. C₃²×D₈	D₄	semidirect
64. #3283	C₃²	direct (N × H)
106. C₂×#135	C₂×C₃²	semidirect
107. #547	C₂×C₃²	semidirect
108. #494	C₂×C₃²	semidirect
109. #467	C₂×C₃²	semidirect
110. #461	C₂×C₃²	semidirect
111. #452	C₂×C₃²	semidirect
112. C₄×D₈	C₂×C₃²	semidirect
113. C₈×D₄	C₂×C₃²	semidirect
126. C₃×D₈	C₃×D₄	semidirect
151. C₃²×C₄	#135	semidirect
212. D₈	C₃²×D₄	semidirect
237. C₃×C₄	C₃×#135	semidirect
240. C₃²	#3283	direct (N × H)
278. C₄	C₃²×#135	semidirect
281. C₃	C₃×#3283	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×C₃²×#135	2
C₃²×#547	2
C₃²×#494	2
C₃²×#467	2
C₃²×#461	2
C₃²×#452	2
C₃²×C₄×D₈	2
C₃²×C₈×D₄	2
C₃×#3283	3

Subgroups — C32×#3283 #11770332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3283 #11770332