Subgroups — C₃×#33776 #10652383

Order 768.

592 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2211	C₂	1	4	yes	maximal, complemented
8	C₃×#2189	C₂	1	4	yes	maximal, complemented
9	C₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	#33776	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	—	2	1	no	—
12	C₂×C₃×#135	C₂²	1	2	yes	—
13	C₃×#533	—	2	1	no	—
14	C₃×#532	—	2	1	no	—
15	C₂×C₃×Q₃₂	C₂²	1	4	yes	—
16	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
17	C₃×#500	C₂²	1	1	yes	characteristic
18	C₃×#495	—	2	1	no	—
19	C₃×#495	C₂²	1	2	yes	—
20	C₃×#494	C₂²	1	1	yes	characteristic
21	C₃×#494	C₂²	1	1	yes	characteristic
22	C₃×#493	C₂²	1	1	yes	characteristic
23	C₃×#492	—	2	1	no	—
24	C₃×#487	—	2	1	no	—
25	C₃×#467	—	2	1	no	—
26	C₃×#467	C₂²	1	2	yes	—
27	C₃×C₄×Q₁₆	—	2	1	no	—
28	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃×#358	C₂²	1	4	yes	—
32	C₃×#357	C₂²	1	4	yes	—
33	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
36	#3170	C₂×C₃	1	1	yes	characteristic, complemented
37	#3147	C₂×C₃	1	1	yes	characteristic
38	#2248	C₂×C₃	1	1	yes	characteristic, complemented
39	#2247	C₂×C₃	1	1	yes	characteristic, complemented
40	#2241	C₂×C₃	1	1	yes	characteristic, complemented
41	#2240	C₂×C₃	1	1	yes	characteristic, complemented
42	#2211	C₂×C₃	1	4	yes	complemented
43	#2189	C₂×C₃	1	4	yes	complemented
44	#2162	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	2	2	no	—
46	C₃×#135	—	4	4	no	—
47	C₃×#135	—	4	4	no	—
48	C₂×C₃×Q₁₆	—	2	1	no	—
49	C₂×C₃×Q₁₆	—	2	1	no	—
50	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
51	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
52	C₂×C₃×SD₁₆	—	2	2	no	—
53	C₂×C₃×SD₁₆	—	2	2	no	—
54	C₂×C₃×D₈	—	2	1	no	—
55	C₂×C₃×D₈	C₂³	1	2	yes	—
56	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	—	2	1	no	—
59	C₃×C₄⋊Q₈	—	2	1	no	—
60	C₃×C₄⋊Q₈	—	2	1	no	—
61	C₃×#127	—	2	1	no	—
62	C₃×#125	—	4	1	no	—
63	C₃×#124	—	2	2	no	—
64	C₃×C₂²⋊Q₈	—	2	2	no	—
65	C₃×C₂²⋊Q₈	—	4	1	no	—
66	C₃×#121	—	4	1	no	—
67	C₃×#121	—	4	1	no	—
68	C₃×C₄×Q₈	—	2	2	no	—
69	C₃×C₄×Q₈	—	4	1	no	—
70	C₃×C₄×D₄	—	4	1	no	—
71	C₃×C₄×D₄	—	4	1	no	—
72	C₃×#117	—	4	1	no	—
73	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
74	C₃×Q₃₂	—	4	4	no	—
75	C₃×SD₃₂	—	4	4	no	—
76	C₂×C₃×C₁₆	C₂³	1	2	yes	—
77	C₂×C₃×C₁₆	C₂³	1	2	yes	—
78	C₃×#107	—	2	1	no	—
79	C₃×#107	C₂³	1	2	yes	—
80	C₃×#103	—	2	2	no	—
81	C₃×#103	—	2	2	no	—
82	C₃×C₄×C₈	D₄	1	1	yes	characteristic
83	C₃×C₄×C₈	D₄	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×#135	—	2	1	no	—
87	C₂×#135	C₂²×C₃	1	2	yes	—
88	#533	—	2	1	no	—
89	#532	—	2	1	no	—
90	C₂×Q₃₂	C₂²×C₃	1	4	yes	—
91	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
92	#500	C₂²×C₃	1	1	yes	characteristic
93	#495	—	2	1	no	—
94	#495	C₂²×C₃	1	2	yes	—
95	#494	C₂²×C₃	1	1	yes	characteristic
96	#494	C₂²×C₃	1	1	yes	characteristic
97	#493	C₂²×C₃	1	1	yes	characteristic
98	#492	—	2	1	no	—
99	#487	—	2	1	no	—
100	#467	—	2	1	no	—
101	#467	C₂²×C₃	1	2	yes	—
102	C₄×Q₁₆	—	2	1	no	—
103	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
105	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
106	#358	C₂²×C₃	1	4	yes	—
107	#357	C₂²×C₃	1	4	yes	—
108	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
109	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
110	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
111	C₃×#41	—	4	4	no	—
112	C₃×#41	—	4	4	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×Q₈	—	4	1	no	—
116	C₂×C₃×D₄	—	2	2	no	—
117	C₂×C₃×D₄	—	4	2	no	—
118	C₂×C₃×D₄	—	4	1	no	—
119	C₂×C₃×D₄	—	4	1	no	—
120	C₂²×C₃×C₄	—	4	2	no	—
121	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
122	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₃×Q₁₆	—	2	4	no	—
125	C₃×Q₁₆	—	2	4	no	—
126	C₃×Q₁₆	—	4	2	no	—
127	C₃×Q₁₆	—	4	2	no	—
128	C₃×SD₁₆	—	4	4	no	—
129	C₃×SD₁₆	—	4	4	no	—
130	C₃×D₈	—	2	4	no	—
131	C₃×D₈	—	4	2	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	4	1	no	—
140	C₃×C₄⋊C₄	—	4	1	no	—
141	C₃×C₄⋊C₄	—	4	1	no	—
142	C₃×C₄⋊C₄	—	4	1	no	—
143	C₃×C₄⋊C₄	—	4	2	no	—
144	C₃×C₄⋊C₄	—	4	2	no	—
145	C₃×C₄⋊C₄	—	4	1	no	—
146	C₃×C₂²⋊C₄	—	4	2	no	—
147	C₃×C₂²⋊C₄	—	4	2	no	—
148	C₃×C₂²⋊C₄	—	4	2	no	—
149	C₃×C₄²	—	4	2	no	—
150	C₃×C₄²	D₈	1	1	yes	characteristic
151	C₃×C₄²	D₈	1	1	yes	characteristic
152	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
153	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
154	C₃×C₁₆	—	2	2	no	—
155	C₃×C₁₆	—	2	2	no	—
156	C₂×#41	—	2	2	no	—
157	#135	—	4	4	no	—
158	#135	—	4	4	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
162	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂³×C₃	1	2	yes	—
167	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	—	2	1	no	—
172	#127	—	2	1	no	—
173	#125	—	4	1	no	—
174	#124	—	2	2	no	—
175	C₂²⋊Q₈	—	2	2	no	—
176	C₂²⋊Q₈	—	4	1	no	—
177	#121	—	4	1	no	—
178	#121	—	4	1	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	4	1	no	—
181	C₄×D₄	—	4	1	no	—
182	C₄×D₄	—	4	1	no	—
183	#117	—	4	1	no	—
184	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
185	Q₃₂	—	4	4	no	—
186	SD₃₂	—	4	4	no	—
187	C₂×C₁₆	C₂³×C₃	1	2	yes	—
188	C₂×C₁₆	C₂³×C₃	1	2	yes	—
189	#107	—	2	1	no	—
190	#107	C₂³×C₃	1	2	yes	—
191	#103	—	2	2	no	—
192	#103	—	2	2	no	—
193	C₄×C₈	C₃×D₄	1	1	yes	characteristic
194	C₄×C₈	C₃×D₄	1	1	yes	characteristic
195	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
197	C₂³×C₃	—	4	2	no	—
198	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
199	C₃×Q₈	—	4	2	no	—
200	C₃×Q₈	—	4	2	no	—
201	C₃×Q₈	—	4	2	no	—
202	C₃×Q₈	—	4	2	no	—
203	C₃×Q₈	—	8	2	no	—
204	C₃×D₄	—	4	2	no	—
205	C₃×D₄	—	4	2	no	—
206	C₃×D₄	—	8	4	no	—
207	C₃×D₄	—	8	2	no	—
208	C₃×D₄	—	8	2	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	8	4	no	—
217	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	—	2	1	no	—
225	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
226	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
227	#41	—	4	4	no	—
228	#41	—	4	4	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
238	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	Q₁₆	—	2	4	no	—
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	2	no	—
243	Q₁₆	—	4	2	no	—
244	SD₁₆	—	4	4	no	—
245	SD₁₆	—	4	4	no	—
246	D₈	—	2	4	no	—
247	D₈	—	4	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
252	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₃×D₈	1	1	yes	characteristic
267	C₄²	C₃×D₈	1	1	yes	characteristic
268	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
269	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	8	2	no	—
276	C₂²×C₃	—	8	2	no	—
277	C₂²×C₃	—	8	2	no	—
278	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	8	2	no	—
283	C₃×C₄	—	8	2	no	—
284	C₃×C₄	—	8	2	no	—
285	C₃×C₄	#508	1	2	yes	—
286	C₃×C₄	#510	1	2	yes	—
287	C₃×C₄	#535	1	1	yes	characteristic
288	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	4	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
310	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
318	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
319	C₂×C₃	—	4	1	no	—
320	C₂×C₃	—	16	2	no	—
321	C₂×C₃	#3205	1	1	yes	characteristic
322	C₂×C₃	C₂×#508	1	1	yes	characteristic
323	C₂×C₃	C₂×#510	1	1	yes	characteristic
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₃×#508	1	2	yes	—
338	C₄	C₃×#510	1	2	yes	—
339	C₄	C₃×#535	1	1	yes	characteristic
340	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
341	C₃	#33776	1	1	yes	characteristic, direct product factor, Sylow
342	C₂	—	4	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₃×#3205	1	1	yes	characteristic
345	C₂	C₂×C₃×#508	1	1	yes	characteristic
346	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×#3147	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
1. C₃×#3170	C₂	semidirect
3. C₃×#2248	C₂	semidirect
4. C₃×#2247	C₂	semidirect
5. C₃×#2241	C₂	semidirect
6. C₃×#2240	C₂	semidirect
7. C₃×#2211	C₂	semidirect
8. C₃×#2189	C₂	semidirect
9. C₃×#2162	C₂	semidirect
10. #33776	C₃	direct (N × H)
36. #3170	C₂×C₃	semidirect
38. #2248	C₂×C₃	semidirect
39. #2247	C₂×C₃	semidirect
40. #2241	C₂×C₃	semidirect
41. #2240	C₂×C₃	semidirect
42. #2211	C₂×C₃	semidirect
43. #2189	C₂×C₃	semidirect
44. #2162	C₂×C₃	semidirect
341. C₃	#33776	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×#3170	2
C₃×#3147	2
C₃×#2248	2
C₃×#2247	2
C₃×#2241	2
C₃×#2240	2
C₃×#2211	2
C₃×#2189	2
C₃×#2162	2
#33776	3

Subgroups — C3×#33776 #10652383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33776 #10652383