Subgroups — C₃×#33772 #10652379

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2248	C₂	1	2	yes	maximal, complemented
4	C₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₃×#2208	C₂	1	4	yes	maximal, complemented
6	C₃×#2185	C₂	1	4	yes	maximal, complemented
7	C₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	#33772	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₁₆	—	2	1	no	—
10	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
11	C₃×#532	—	2	1	no	—
12	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃×SD₃₂	C₂²	1	8	yes	—
14	C₃×#500	C₂²	1	1	yes	characteristic
15	C₃×#494	—	2	1	no	—
16	C₃×#494	—	2	1	no	—
17	C₃×#494	C₂²	1	2	yes	—
18	C₃×#494	C₂²	1	2	yes	—
19	C₃×#493	C₂²	1	1	yes	characteristic
20	C₃×#486	—	2	1	no	—
21	C₃×#466	—	2	1	no	—
22	C₃×#466	C₂²	1	2	yes	—
23	C₃×C₄×D₈	—	2	1	no	—
24	C₃×C₄×D₈	C₂²	1	2	yes	—
25	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₃×#358	C₂²	1	8	yes	—
28	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₃×C₄×C₁₆	C₂²	1	2	yes	—
30	#3169	C₂×C₃	1	1	yes	characteristic
31	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
32	#2248	C₂×C₃	1	2	yes	complemented
33	#2241	C₂×C₃	1	2	yes	complemented
34	#2208	C₂×C₃	1	4	yes	complemented
35	#2185	C₂×C₃	1	4	yes	complemented
36	#2162	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₃×#41	—	4	1	no	—
38	C₂²×C₃×Q₈	—	2	2	no	—
39	C₂×C₃×Q₁₆	—	2	2	no	—
40	C₂×C₃×Q₁₆	—	2	2	no	—
41	C₂×C₃×Q₁₆	—	2	2	no	—
42	C₂×C₃×Q₁₆	—	4	4	no	—
43	C₂×C₃×Q₁₆	—	4	4	no	—
44	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
45	C₂×C₃×D₈	—	2	1	no	—
46	C₂×C₃×D₈	C₂³	1	2	yes	—
47	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊Q₈	—	2	1	no	—
50	C₃×C₄⋊Q₈	—	2	2	no	—
51	C₃×C₄⋊Q₈	—	2	2	no	—
52	C₃×C₄⋊Q₈	—	4	1	no	—
53	C₃×C₄⋊Q₈	—	4	1	no	—
54	C₃×#127	—	2	1	no	—
55	C₃×#124	—	4	1	no	—
56	C₃×#121	—	2	2	no	—
57	C₃×#121	—	4	1	no	—
58	C₃×C₄×D₄	—	2	2	no	—
59	C₃×C₄×D₄	—	4	1	no	—
60	C₂×C₃×C₄⋊C₄	—	4	1	no	—
61	C₂×C₃×C₄⋊C₄	—	4	1	no	—
62	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×SD₃₂	—	4	8	no	—
64	C₂×C₃×C₁₆	C₂³	1	4	yes	—
65	C₃×#107	—	2	1	no	—
66	C₃×#107	C₂³	1	2	yes	—
67	C₃×#102	—	2	2	no	—
68	C₃×#102	—	2	2	no	—
69	C₃×C₄×C₈	D₄	1	1	yes	characteristic
70	C₃×C₄×C₈	D₄	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂²×Q₁₆	—	2	1	no	—
74	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
75	#532	—	2	1	no	—
76	C₂×C₄⋊Q₈	—	2	1	no	—
77	C₂×SD₃₂	C₂²×C₃	1	8	yes	—
78	#500	C₂²×C₃	1	1	yes	characteristic
79	#494	—	2	1	no	—
80	#494	—	2	1	no	—
81	#494	C₂²×C₃	1	2	yes	—
82	#494	C₂²×C₃	1	2	yes	—
83	#493	C₂²×C₃	1	1	yes	characteristic
84	#486	—	2	1	no	—
85	#466	—	2	1	no	—
86	#466	C₂²×C₃	1	2	yes	—
87	C₄×D₈	—	2	1	no	—
88	C₄×D₈	C₂²×C₃	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
91	#358	C₂²×C₃	1	8	yes	—
92	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
93	C₄×C₁₆	C₂²×C₃	1	2	yes	—
94	C₃×#41	—	8	4	no	—
95	C₂×C₃×Q₈	—	2	4	no	—
96	C₂×C₃×Q₈	—	4	4	no	—
97	C₂×C₃×Q₈	—	4	4	no	—
98	C₂×C₃×Q₈	—	4	1	no	—
99	C₂×C₃×Q₈	—	4	2	no	—
100	C₂×C₃×D₄	—	2	2	no	—
101	C₂×C₃×D₄	—	4	2	no	—
102	C₂×C₃×D₄	—	4	1	no	—
103	C₂×C₃×D₄	—	4	1	no	—
104	C₂²×C₃×C₄	—	4	2	no	—
105	C₂²×C₃×C₄	—	4	2	no	—
106	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
107	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₃×Q₁₆	—	2	8	no	—
110	C₃×Q₁₆	—	4	4	no	—
111	C₃×Q₁₆	—	4	4	no	—
112	C₃×Q₁₆	—	4	4	no	—
113	C₃×D₈	—	2	4	no	—
114	C₃×D₈	—	4	2	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	4	2	no	—
123	C₃×C₄⋊C₄	—	4	2	no	—
124	C₃×C₄⋊C₄	—	4	2	no	—
125	C₃×C₄⋊C₄	—	4	2	no	—
126	C₃×C₄⋊C₄	—	4	1	no	—
127	C₃×C₂²⋊C₄	—	4	2	no	—
128	C₃×C₂²⋊C₄	—	4	2	no	—
129	C₃×C₄²	D₈	1	1	yes	characteristic
130	C₃×C₄²	D₈	1	1	yes	characteristic
131	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₃×C₁₆	—	2	4	no	—
134	C₂×#41	—	4	1	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	4	4	no	—
140	C₂×Q₁₆	—	4	4	no	—
141	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
142	C₂×D₈	—	2	1	no	—
143	C₂×D₈	C₂³×C₃	1	2	yes	—
144	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	1	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	1	no	—
150	C₄⋊Q₈	—	4	1	no	—
151	#127	—	2	1	no	—
152	#124	—	4	1	no	—
153	#121	—	2	2	no	—
154	#121	—	4	1	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	4	1	no	—
157	C₂×C₄⋊C₄	—	4	1	no	—
158	C₂×C₄⋊C₄	—	4	1	no	—
159	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
160	SD₃₂	—	4	8	no	—
161	C₂×C₁₆	C₂³×C₃	1	4	yes	—
162	#107	—	2	1	no	—
163	#107	C₂³×C₃	1	2	yes	—
164	#102	—	2	2	no	—
165	#102	—	2	2	no	—
166	C₄×C₈	C₃×D₄	1	1	yes	characteristic
167	C₄×C₈	C₃×D₄	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
170	C₂³×C₃	—	4	2	no	—
171	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
172	C₃×Q₈	—	4	4	no	—
173	C₃×Q₈	—	4	4	no	—
174	C₃×Q₈	—	8	4	no	—
175	C₃×Q₈	—	8	2	no	—
176	C₃×D₄	—	4	2	no	—
177	C₃×D₄	—	4	2	no	—
178	C₃×D₄	—	8	4	no	—
179	C₃×D₄	—	8	2	no	—
180	C₃×D₄	—	8	2	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	4	no	—
186	C₂×C₃×C₄	—	8	4	no	—
187	C₂×C₃×C₄	—	8	4	no	—
188	C₂×C₃×C₄	SD₃₂	1	4	yes	complemented
189	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
198	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
199	#41	—	8	4	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
212	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	Q₁₆	—	2	8	no	—
215	Q₁₆	—	4	4	no	—
216	Q₁₆	—	4	4	no	—
217	Q₁₆	—	4	4	no	—
218	D₈	—	2	4	no	—
219	D₈	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₄²	C₃×D₈	1	1	yes	characteristic
235	C₄²	C₃×D₈	1	1	yes	characteristic
236	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
237	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
238	C₁₆	—	2	4	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	8	2	no	—
242	C₂²×C₃	—	8	2	no	—
243	C₂²×C₃	—	8	2	no	—
244	C₂²×C₃	C₂×SD₃₂	1	2	yes	—
245	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	8	2	no	—
249	C₃×C₄	—	8	4	no	—
250	C₃×C₄	C₂×SD₃₂	1	2	yes	—
251	C₃×C₄	C₂×SD₃₂	1	2	yes	—
252	C₃×C₄	#510	1	2	yes	—
253	C₃×C₄	#535	1	1	yes	characteristic
254	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₂³	—	4	2	no	—
256	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	Q₈	—	8	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	8	4	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	C₃×SD₃₂	1	4	yes	complemented
274	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
275	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
276	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
283	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	—	16	2	no	—
286	C₂×C₃	#3205	1	1	yes	characteristic
287	C₂×C₃	C₂²×SD₃₂	1	1	yes	characteristic
288	C₂×C₃	C₂×#510	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂×C₃×SD₃₂	1	2	yes	—
295	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	4	no	—
300	C₄	C₂×C₃×SD₃₂	1	2	yes	—
301	C₄	C₂×C₃×SD₃₂	1	2	yes	—
302	C₄	C₃×#510	1	2	yes	—
303	C₄	C₃×#535	1	1	yes	characteristic
304	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₃	#33772	1	1	yes	characteristic, direct product factor, Sylow
306	C₂	—	2	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₃×#3205	1	1	yes	characteristic
309	C₂	C₂²×C₃×SD₃₂	1	1	yes	characteristic
310	C₂	C₂×C₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₃×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#494	C₂	semidirect
3. C₃×#2248	C₂	semidirect
4. C₃×#2241	C₂	semidirect
5. C₃×#2208	C₂	semidirect
6. C₃×#2185	C₂	semidirect
7. C₃×#2162	C₂	semidirect
8. #33772	C₃	direct (N × H)
31. C₂×#494	C₂×C₃	semidirect
32. #2248	C₂×C₃	semidirect
33. #2241	C₂×C₃	semidirect
34. #2208	C₂×C₃	semidirect
35. #2185	C₂×C₃	semidirect
36. #2162	C₂×C₃	semidirect
188. C₂×C₃×C₄	SD₃₂	semidirect
273. C₂×C₄	C₃×SD₃₂	semidirect
305. C₃	#33772	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×#3169	2
C₂×C₃×#494	2
C₃×#2248	2
C₃×#2241	2
C₃×#2208	2
C₃×#2185	2
C₃×#2162	2
#33772	3

Subgroups — C3×#33772 #10652379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33772 #10652379