Subgroups — C₃×#33784 #10652391

Order 768.

584 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₃×#2245	C₂	1	2	yes	maximal, complemented
5	C₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₃×#2208	C₂	1	2	yes	maximal, complemented
7	C₃×#2188	C₂	1	2	yes	maximal, complemented
8	C₃×#2185	C₂	1	2	yes	maximal
9	C₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	#33784	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₁₆	—	2	1	no	—
12	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
14	C₃×#529	—	2	1	no	—
15	C₂×C₃×Q₃₂	C₂²	1	4	yes	—
16	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
17	C₃×#499	C₂²	1	1	yes	characteristic
18	C₃×#495	C₂²	1	1	yes	characteristic
19	C₃×#494	—	2	1	no	—
20	C₃×#494	—	2	1	no	—
21	C₃×#494	C₂²	1	2	yes	—
22	C₃×#494	C₂²	1	2	yes	—
23	C₃×#488	—	2	1	no	—
24	C₃×#467	C₂²	1	1	yes	characteristic
25	C₃×#466	C₂²	1	1	yes	characteristic
26	C₃×#465	—	2	1	no	—
27	C₃×C₄×Q₁₆	C₂²	1	1	yes	characteristic
28	C₃×C₄×SD₁₆	—	2	1	no	—
29	C₃×C₄×D₈	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₃×#358	C₂²	1	4	yes	—
33	C₃×#358	C₂²	1	4	yes	—
34	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₃×#346	C₂²	1	2	yes	—
36	#3168	C₂×C₃	1	1	yes	characteristic
37	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
38	#2251	C₂×C₃	1	2	yes	complemented
39	#2245	C₂×C₃	1	2	yes	complemented
40	#2209	C₂×C₃	1	2	yes	complemented
41	#2208	C₂×C₃	1	2	yes	complemented
42	#2188	C₂×C₃	1	2	yes	complemented
43	#2185	C₂×C₃	1	2	yes	—
44	#2163	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂²×C₃×Q₈	—	2	2	no	—
46	C₂×C₃×Q₁₆	—	2	2	no	—
47	C₂×C₃×Q₁₆	—	2	2	no	—
48	C₂×C₃×Q₁₆	—	2	2	no	—
49	C₂×C₃×Q₁₆	—	4	4	no	—
50	C₂×C₃×Q₁₆	—	4	4	no	—
51	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
52	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₃×SD₁₆	—	2	1	no	—
54	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄⋊Q₈	—	2	2	no	—
58	C₃×C₄⋊Q₈	—	2	2	no	—
59	C₃×C₄⋊Q₈	—	4	1	no	—
60	C₃×C₄⋊Q₈	—	4	1	no	—
61	C₃×#126	—	4	1	no	—
62	C₃×#125	—	2	1	no	—
63	C₃×#124	—	2	1	no	—
64	C₃×#123	—	4	1	no	—
65	C₃×C₂²⋊Q₈	—	2	1	no	—
66	C₃×#121	—	2	1	no	—
67	C₃×C₄×Q₈	—	2	1	no	—
68	C₃×C₄×D₄	—	2	1	no	—
69	C₃×C₄×D₄	—	4	1	no	—
70	C₃×#117	—	4	1	no	—
71	C₂×C₃×C₄⋊C₄	—	4	1	no	—
72	C₂×C₃×C₄⋊C₄	—	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	4	yes	—
77	C₃×#107	C₂³	1	1	yes	characteristic
78	C₃×#107	C₂³	1	1	yes	characteristic
79	C₃×#106	—	2	1	no	—
80	C₃×#103	—	2	1	no	—
81	C₃×#103	—	2	1	no	—
82	C₃×#102	—	2	1	no	—
83	C₃×#102	—	2	1	no	—
84	C₃×C₄×C₈	D₄	1	1	yes	characteristic
85	C₃×C₄×C₈	D₄	1	1	yes	characteristic
86	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
87	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
88	C₂²×Q₁₆	—	2	1	no	—
89	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
90	C₂×C₄⋊Q₈	—	2	1	no	—
91	#529	—	2	1	no	—
92	C₂×Q₃₂	C₂²×C₃	1	4	yes	—
93	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
94	#499	C₂²×C₃	1	1	yes	characteristic
95	#495	C₂²×C₃	1	1	yes	characteristic
96	#494	—	2	1	no	—
97	#494	—	2	1	no	—
98	#494	C₂²×C₃	1	2	yes	—
99	#494	C₂²×C₃	1	2	yes	—
100	#488	—	2	1	no	—
101	#467	C₂²×C₃	1	1	yes	characteristic
102	#466	C₂²×C₃	1	1	yes	characteristic
103	#465	—	2	1	no	—
104	C₄×Q₁₆	C₂²×C₃	1	1	yes	characteristic
105	C₄×SD₁₆	—	2	1	no	—
106	C₄×D₈	C₂²×C₃	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
109	#358	C₂²×C₃	1	4	yes	—
110	#358	C₂²×C₃	1	4	yes	—
111	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
112	#346	C₂²×C₃	1	2	yes	—
113	C₂×C₃×Q₈	—	2	1	no	—
114	C₂×C₃×Q₈	—	2	4	no	—
115	C₂×C₃×Q₈	—	4	4	no	—
116	C₂×C₃×Q₈	—	4	4	no	—
117	C₂×C₃×Q₈	—	4	2	no	—
118	C₂×C₃×D₄	—	2	1	no	—
119	C₂×C₃×D₄	—	4	1	no	—
120	C₂²×C₃×C₄	—	4	2	no	—
121	C₂²×C₃×C₄	—	4	1	no	—
122	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₃×Q₁₆	—	2	4	no	—
126	C₃×Q₁₆	—	2	2	no	—
127	C₃×Q₁₆	—	2	4	no	—
128	C₃×Q₁₆	—	4	4	no	—
129	C₃×Q₁₆	—	4	4	no	—
130	C₃×Q₁₆	—	4	4	no	—
131	C₃×SD₁₆	—	4	2	no	—
132	C₃×D₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	4	2	no	—
142	C₃×C₄⋊C₄	—	4	2	no	—
143	C₃×C₄⋊C₄	—	4	2	no	—
144	C₃×C₄⋊C₄	—	4	2	no	—
145	C₃×C₄⋊C₄	—	4	1	no	—
146	C₃×C₄⋊C₄	—	4	1	no	—
147	C₃×C₄⋊C₄	—	4	1	no	—
148	C₃×C₄⋊C₄	—	4	1	no	—
149	C₃×C₂²⋊C₄	—	4	1	no	—
150	C₃×C₂²⋊C₄	—	4	1	no	—
151	C₃×C₂²⋊C₄	—	4	1	no	—
152	C₃×C₂²⋊C₄	—	4	1	no	—
153	C₃×C₂²⋊C₄	—	4	1	no	—
154	C₃×C₄²	—	4	1	no	—
155	C₃×C₄²	D₈	1	1	yes	characteristic
156	C₃×C₄²	D₈	1	1	yes	characteristic
157	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
159	C₃×C₁₆	—	2	4	no	—
160	C₂²×Q₈	—	2	2	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×Q₁₆	—	4	4	no	—
166	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
167	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
170	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	4	1	no	—
175	C₄⋊Q₈	—	4	1	no	—
176	#126	—	4	1	no	—
177	#125	—	2	1	no	—
178	#124	—	2	1	no	—
179	#123	—	4	1	no	—
180	C₂²⋊Q₈	—	2	1	no	—
181	#121	—	2	1	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×D₄	—	2	1	no	—
184	C₄×D₄	—	4	1	no	—
185	#117	—	4	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
189	Q₃₂	—	4	4	no	—
190	SD₃₂	—	4	4	no	—
191	C₂×C₁₆	C₂³×C₃	1	4	yes	—
192	#107	C₂³×C₃	1	1	yes	characteristic
193	#107	C₂³×C₃	1	1	yes	characteristic
194	#106	—	2	1	no	—
195	#103	—	2	1	no	—
196	#103	—	2	1	no	—
197	#102	—	2	1	no	—
198	#102	—	2	1	no	—
199	C₄×C₈	C₃×D₄	1	1	yes	characteristic
200	C₄×C₈	C₃×D₄	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
203	C₂³×C₃	—	4	1	no	—
204	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
205	C₃×Q₈	—	4	4	no	—
206	C₃×Q₈	—	4	1	no	—
207	C₃×Q₈	—	4	1	no	—
208	C₃×Q₈	—	4	4	no	—
209	C₃×Q₈	—	8	4	no	—
210	C₃×D₄	—	4	1	no	—
211	C₃×D₄	—	4	1	no	—
212	C₃×D₄	—	8	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	—	8	4	no	—
222	C₂×C₃×C₄	—	8	2	no	—
223	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
232	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
243	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
245	Q₁₆	—	2	4	no	—
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	4	4	no	—
249	Q₁₆	—	4	4	no	—
250	Q₁₆	—	4	4	no	—
251	SD₁₆	—	4	2	no	—
252	D₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
258	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₃×D₈	1	1	yes	characteristic
276	C₄²	C₃×D₈	1	1	yes	characteristic
277	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
278	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	2	1	no	—
282	C₂²×C₃	—	8	1	no	—
283	C₂²×C₃	—	8	1	no	—
284	C₂²×C₃	—	8	1	no	—
285	C₂²×C₃	#510	1	2	yes	—
286	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	8	1	no	—
292	C₃×C₄	—	8	1	no	—
293	C₃×C₄	—	8	4	no	—
294	C₃×C₄	—	8	1	no	—
295	C₃×C₄	#510	1	2	yes	—
296	C₃×C₄	#535	1	1	yes	characteristic
297	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂³	—	4	1	no	—
299	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
300	Q₈	—	4	1	no	—
301	Q₈	—	4	4	no	—
302	Q₈	—	4	4	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	8	4	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	1	no	—
307	D₄	—	8	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
321	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
327	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
328	C₂×C₃	—	2	2	no	—
329	C₂×C₃	—	16	1	no	—
330	C₂×C₃	#3205	1	1	yes	characteristic
331	C₂×C₃	C₂×#510	1	1	yes	characteristic
332	C₂×C₃	C₂×#510	1	1	yes	characteristic
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	C₃×#510	1	2	yes	—
339	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	4	no	—
347	C₄	—	8	1	no	—
348	C₄	C₃×#510	1	2	yes	—
349	C₄	C₃×#535	1	1	yes	characteristic
350	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
351	C₃	#33784	1	1	yes	characteristic, direct product factor, Sylow
352	C₂	—	2	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₃×#3205	1	1	yes	characteristic
355	C₂	C₂×C₃×#510	1	1	yes	characteristic
356	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×#466	C₂²
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×#2251	C₂	semidirect
4. C₃×#2245	C₂	semidirect
5. C₃×#2209	C₂	semidirect
6. C₃×#2208	C₂	semidirect
7. C₃×#2188	C₂	semidirect
9. C₃×#2163	C₂	semidirect
10. #33784	C₃	direct (N × H)
37. C₂×#494	C₂×C₃	semidirect
38. #2251	C₂×C₃	semidirect
39. #2245	C₂×C₃	semidirect
40. #2209	C₂×C₃	semidirect
41. #2208	C₂×C₃	semidirect
42. #2188	C₂×C₃	semidirect
44. #2163	C₂×C₃	semidirect
351. C₃	#33784	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×#3168	2
C₂×C₃×#494	2
C₃×#2251	2
C₃×#2245	2
C₃×#2209	2
C₃×#2208	2
C₃×#2188	2
C₃×#2185	2
C₃×#2163	2
#33784	3

Subgroups — C3×#33784 #10652391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33784 #10652391