Subgroups — C₃×#27693 #10646300

Order 768.

634 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2102	C₂	1	2	yes	maximal, complemented
4	C₃×#2098	C₂	1	2	yes	maximal, complemented
5	C₃×#1945	C₂	1	2	yes	maximal, complemented
6	C₃×#1884	C₂	1	2	yes	maximal
7	C₃×#1868	C₂	1	2	yes	maximal, complemented
8	C₃×#1814	C₂	1	2	yes	maximal, complemented
9	C₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
10	#27693	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#568	C₂²	1	2	yes	—
12	C₂×C₃×#131	C₂²	1	1	yes	characteristic
13	C₂×C₃×#131	C₂²	1	1	yes	characteristic
14	C₃×#532	C₂²	1	1	yes	characteristic
15	C₃×#436	—	2	2	no	—
16	C₃×C₈×D₄	—	2	2	no	—
17	C₃×#430	—	2	2	no	—
18	C₃×#430	C₂²	1	2	yes	—
19	C₂×C₃×#108	—	2	2	no	—
20	C₂×C₃×#108	C₂²	1	2	yes	—
21	C₃×#427	C₂²	1	2	yes	—
22	C₃×#424	C₂²	1	4	yes	—
23	C₃×#423	C₂²	1	1	yes	characteristic
24	C₃×#421	C₂²	1	4	yes	—
25	C₃×#410	C₂²	1	2	yes	—
26	C₃×#408	C₂²	1	1	yes	characteristic
27	C₃×#408	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×#344	C₂²	1	4	yes	—
30	C₃×#339	C₂²	1	4	yes	—
31	C₃×H(3,ℤ/4)	C₂²	1	2	yes	—
32	#2931	C₂×C₃	1	1	yes	characteristic
33	#2916	C₂×C₃	1	1	yes	characteristic, complemented
34	#2102	C₂×C₃	1	2	yes	complemented
35	#2098	C₂×C₃	1	2	yes	complemented
36	#1945	C₂×C₃	1	2	yes	complemented
37	#1884	C₂×C₃	1	2	yes	—
38	#1868	C₂×C₃	1	2	yes	complemented
39	#1814	C₂×C₃	1	2	yes	complemented
40	#1812	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃×#41	C₂³	1	1	yes	characteristic
42	C₂×C₃×#41	C₂³	1	1	yes	characteristic
43	C₃×#131	—	2	2	no	—
44	C₃×#131	—	2	2	no	—
45	C₃×#131	—	2	2	no	—
46	C₃×#131	—	2	2	no	—
47	C₃×#131	—	4	4	no	—
48	C₂×C₃×M₁₆	—	2	1	no	—
49	C₂×C₃×M₁₆	—	2	4	no	—
50	C₂×C₃×M₁₆	D₄	1	2	yes	—
51	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₃×M₁₆	C₂³	1	4	yes	—
53	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₈	—	2	1	no	—
55	C₂²×C₃×C₈	D₄	1	2	yes	—
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
59	C₃×#127	Q₈	1	1	yes	characteristic
60	C₃×#124	Q₈	1	1	yes	characteristic
61	C₃×#124	Q₈	1	1	yes	characteristic
62	C₃×#121	—	2	2	no	—
63	C₃×C₄×D₄	—	2	2	no	—
64	C₃×#117	C₂³	1	4	yes	—
65	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
66	C₃×#108	—	2	2	no	—
67	C₃×#108	—	2	4	no	—
68	C₃×#108	—	2	2	no	—
69	C₃×#107	—	2	2	no	—
70	C₃×#106	—	2	2	no	—
71	C₃×C₄⋊C₈	—	2	2	no	—
72	C₃×C₄⋊C₈	—	4	4	no	—
73	C₃×#104	—	4	4	no	—
74	C₃×#99	—	2	4	no	—
75	C₃×C₂²⋊C₈	—	2	2	no	—
76	C₃×C₂²⋊C₈	—	2	2	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	—	2	1	no	—
79	#568	C₂²×C₃	1	2	yes	—
80	C₂×#131	C₂²×C₃	1	1	yes	characteristic
81	C₂×#131	C₂²×C₃	1	1	yes	characteristic
82	#532	C₂²×C₃	1	1	yes	characteristic
83	#436	—	2	2	no	—
84	C₈×D₄	—	2	2	no	—
85	#430	—	2	2	no	—
86	#430	C₂²×C₃	1	2	yes	—
87	C₂×#108	—	2	2	no	—
88	C₂×#108	C₂²×C₃	1	2	yes	—
89	#427	C₂²×C₃	1	2	yes	—
90	#424	C₂²×C₃	1	4	yes	—
91	#423	C₂²×C₃	1	1	yes	characteristic
92	#421	C₂²×C₃	1	4	yes	—
93	#410	C₂²×C₃	1	2	yes	—
94	#408	C₂²×C₃	1	1	yes	characteristic
95	#408	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
97	#344	C₂²×C₃	1	4	yes	—
98	#339	C₂²×C₃	1	4	yes	—
99	H(3,ℤ/4)	C₂²×C₃	1	2	yes	—
100	C₃×#41	—	2	2	no	—
101	C₃×#41	—	2	2	no	—
102	C₃×#41	—	2	2	no	—
103	C₃×#41	—	2	2	no	—
104	C₂×C₃×Q₈	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₃×Q₈	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₃×D₄	—	2	2	no	—
107	C₂×C₃×D₄	—	2	1	no	—
108	C₂×C₃×D₄	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₃×D₄	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₃×D₄	#41	1	2	yes	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
115	C₃×M₁₆	—	2	2	no	—
116	C₃×M₁₆	—	2	4	no	—
117	C₃×M₁₆	—	2	4	no	—
118	C₃×M₁₆	—	2	2	no	—
119	C₃×M₁₆	—	2	4	no	—
120	C₃×M₁₆	—	4	4	no	—
121	C₃×M₁₆	—	4	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	—	4	4	no	—
129	C₂×C₃×C₈	—	4	1	no	—
130	C₂×C₃×C₈	—	4	1	no	—
131	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	#41	1	2	yes	—
135	C₂×C₃×C₈	#41	1	2	yes	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	4	no	—
138	C₃×C₂²⋊C₄	—	2	2	no	—
139	C₃×C₂²⋊C₄	—	2	2	no	—
140	C₃×C₂²⋊C₄	—	2	4	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	—	2	4	no	—
143	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
144	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
145	C₂×#41	C₂³×C₃	1	1	yes	characteristic
146	C₂×#41	C₂³×C₃	1	1	yes	characteristic
147	#131	—	2	2	no	—
148	#131	—	2	2	no	—
149	#131	—	2	2	no	—
150	#131	—	2	2	no	—
151	#131	—	4	4	no	—
152	C₂×M₁₆	—	2	1	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	C₃×D₄	1	2	yes	—
155	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
156	C₂×M₁₆	C₂³×C₃	1	4	yes	—
157	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₃×D₄	1	2	yes	—
160	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
162	C₄⋊Q₈	C₃×Q₈	1	1	yes	characteristic
163	#127	C₃×Q₈	1	1	yes	characteristic
164	#124	C₃×Q₈	1	1	yes	characteristic
165	#124	C₃×Q₈	1	1	yes	characteristic
166	#121	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	#117	C₂³×C₃	1	4	yes	—
169	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
170	#108	—	2	2	no	—
171	#108	—	2	4	no	—
172	#108	—	2	2	no	—
173	#107	—	2	2	no	—
174	#106	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	4	4	no	—
177	#104	—	4	4	no	—
178	#99	—	2	4	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₂³×C₃	—	2	1	no	—
184	C₂³×C₃	C₂²⋊Q₈	1	2	yes	—
185	C₂³×C₃	C₂×#41	1	1	yes	characteristic
186	C₃×Q₈	—	2	2	no	—
187	C₃×Q₈	—	2	2	no	—
188	C₃×D₄	—	2	4	no	—
189	C₃×D₄	—	2	2	no	—
190	C₃×D₄	—	2	2	no	—
191	C₃×D₄	—	4	4	no	—
192	C₃×D₄	—	4	2	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	4	2	no	—
199	C₂×C₃×C₄	—	4	1	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	—	4	4	no	—
202	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
203	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×#41	1	2	yes	—
206	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃×C₈	—	4	1	no	—
210	C₃×C₈	—	4	4	no	—
211	C₃×C₈	#123	1	1	yes	characteristic
212	C₃×C₈	#123	1	1	yes	characteristic
213	C₃×C₈	#123	1	2	yes	—
214	#41	—	2	2	no	—
215	#41	—	2	2	no	—
216	#41	—	2	2	no	—
217	#41	—	2	2	no	—
218	C₂×Q₈	C₂×C₃×Q₈	1	1	yes	characteristic
219	C₂×Q₈	C₂×C₃×Q₈	1	1	yes	characteristic
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	C₂×C₃×Q₈	1	1	yes	characteristic
223	C₂×D₄	C₂×C₃×Q₈	1	1	yes	characteristic
224	C₂×D₄	C₃×#41	1	2	yes	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
246	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
247	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
248	C₂×C₈	C₃×#41	1	2	yes	—
249	C₂×C₈	C₃×#41	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	4	no	—
257	C₄²	C₂×C₃×Q₈	1	1	yes	characteristic
258	C₄²	C₂×C₃×Q₈	1	1	yes	characteristic
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	4	1	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	4	1	no	—
264	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
265	C₂²×C₃	#553	1	2	yes	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	4	1	no	—
270	C₃×C₄	—	8	4	no	—
271	C₃×C₄	C₂×#123	1	1	yes	characteristic
272	C₃×C₄	#535	1	1	yes	characteristic
273	C₃×C₄	#554	1	2	yes	—
274	C₂³	—	2	1	no	—
275	C₂³	C₃×C₂²⋊Q₈	1	2	yes	—
276	C₂³	C₂×C₃×#41	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	2	2	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
294	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×#41	1	2	yes	—
297	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
298	C₈	—	2	2	no	—
299	C₈	—	4	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	4	no	—
302	C₈	C₃×#123	1	1	yes	characteristic
303	C₈	C₃×#123	1	1	yes	characteristic
304	C₈	C₃×#123	1	2	yes	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	4	2	no	—
308	C₂×C₃	—	8	1	no	—
309	C₂×C₃	#2487	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
316	C₂²	C₃×#553	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	8	4	no	—
322	C₄	C₂×C₃×#123	1	1	yes	characteristic
323	C₄	C₃×#535	1	1	yes	characteristic
324	C₄	C₃×#554	1	2	yes	—
325	C₃	#27693	1	1	yes	characteristic, direct product factor, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	4	2	no	—
329	C₂	—	8	1	no	—
330	C₂	C₃×#2487	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	#2487
prime radical	C₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2916	C₂	semidirect
3. C₃×#2102	C₂	semidirect
4. C₃×#2098	C₂	semidirect
5. C₃×#1945	C₂	semidirect
7. C₃×#1868	C₂	semidirect
8. C₃×#1814	C₂	semidirect
9. C₃×#1812	C₂	semidirect
10. #27693	C₃	direct (N × H)
33. #2916	C₂×C₃	semidirect
34. #2102	C₂×C₃	semidirect
35. #2098	C₂×C₃	semidirect
36. #1945	C₂×C₃	semidirect
38. #1868	C₂×C₃	semidirect
39. #1814	C₂×C₃	semidirect
40. #1812	C₂×C₃	semidirect
325. C₃	#27693	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×#2931	2
C₃×#2916	2
C₃×#2102	2
C₃×#2098	2
C₃×#1945	2
C₃×#1884	2
C₃×#1868	2
C₃×#1814	2
C₃×#1812	2
#27693	3

Subgroups — C3×#27693 #10646300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#27693 #10646300