Subgroups — C₃×#3196 #85060

Order 384.

414 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#485	C₂	1	2	yes	maximal, complemented
4	C₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#470	C₂	1	2	yes	maximal, complemented
7	C₃×#468	C₂	1	2	yes	maximal, complemented
8	C₃×#452	C₂	1	2	yes	maximal, complemented
9	C₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
12	#3196	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×Q₁₆	C₂²	1	2	yes	complemented
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
18	C₂×C₃×M₁₆	C₂²	1	2	yes	—
19	C₃×#123	—	2	1	no	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×C₂²⋊Q₈	—	2	1	no	—
22	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₃×#121	C₂²	1	1	yes	characteristic
27	C₃×#121	C₂²	1	1	yes	characteristic
28	C₃×#120	—	2	1	no	—
29	C₃×C₄×D₄	—	2	1	no	—
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
35	C₃×#107	C₂²	1	2	yes	complemented
36	C₃×#106	C₂²	1	2	yes	complemented
37	C₃×#103	C₂²	1	2	yes	complemented
38	C₃×#103	C₂²	1	2	yes	complemented
39	C₃×#103	C₂²	1	2	yes	complemented
40	C₃×#102	C₂²	1	2	yes	complemented
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
44	#525	C₂×C₃	1	1	yes	characteristic
45	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
46	#485	C₂×C₃	1	2	yes	complemented
47	#484	C₂×C₃	1	1	yes	characteristic, complemented
48	#481	C₂×C₃	1	1	yes	characteristic, complemented
49	#470	C₂×C₃	1	2	yes	complemented
50	#468	C₂×C₃	1	2	yes	complemented
51	#452	C₂×C₃	1	2	yes	complemented
52	#451	C₂×C₃	1	1	yes	characteristic, complemented
53	#448	C₂×C₃	1	1	yes	characteristic, complemented
54	#407	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃	D₄	1	1	yes	characteristic
56	C₃×#41	—	4	2	no	—
57	C₂×C₃×Q₈	—	2	1	no	—
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×Q₈	C₂³	1	2	yes	—
60	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	—	2	1	no	—
63	C₂×C₃×D₄	—	2	1	no	—
64	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	1	no	—
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	4	2	no	—
69	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
75	C₃×Q₁₆	—	4	2	no	—
76	C₃×SD₁₆	—	4	2	no	—
77	C₃×M₁₆	—	4	2	no	—
78	C₂×C₃×C₈	C₂³	1	2	yes	—
79	C₂×C₃×C₈	C₂³	1	2	yes	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	2	1	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	C₂³	1	2	yes	—
86	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₂²⋊C₄	—	2	2	no	—
89	C₃×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	—	2	1	no	—
92	C₃×C₂²⋊C₄	—	2	1	no	—
93	C₃×C₂²⋊C₄	—	2	1	no	—
94	C₃×C₂²⋊C₄	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₂×#41	C₂²×C₃	1	1	yes	characteristic
97	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
98	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×Q₁₆	C₂²×C₃	1	2	yes	complemented
100	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
101	C₂×M₁₆	C₂²×C₃	1	2	yes	—
102	#123	—	2	1	no	—
103	C₂²⋊Q₈	—	2	1	no	—
104	C₂²⋊Q₈	—	2	1	no	—
105	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
106	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
107	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
108	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
109	#121	C₂²×C₃	1	1	yes	characteristic
110	#121	C₂²×C₃	1	1	yes	characteristic
111	#120	—	2	1	no	—
112	C₄×D₄	—	2	1	no	—
113	C₄×D₄	—	2	1	no	—
114	#117	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
117	C₂×C₂²⋊C₄	—	2	1	no	—
118	#107	C₂²×C₃	1	2	yes	complemented
119	#106	C₂²×C₃	1	2	yes	complemented
120	#103	C₂²×C₃	1	2	yes	complemented
121	#103	C₂²×C₃	1	2	yes	complemented
122	#103	C₂²×C₃	1	2	yes	complemented
123	#102	C₂²×C₃	1	2	yes	complemented
124	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
125	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
126	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
127	C₂³×C₃	—	2	2	no	—
128	C₂³×C₃	—	2	1	no	—
129	C₂³×C₃	—	4	1	no	—
130	C₂³×C₃	—	4	1	no	—
131	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
132	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
133	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	2	2	no	—
136	C₃×Q₈	—	2	2	no	—
137	C₃×Q₈	—	4	2	no	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	4	2	no	—
140	C₃×D₄	—	4	2	no	—
141	C₃×D₄	—	4	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	4	1	no	—
151	C₂×C₃×C₄	—	4	1	no	—
152	C₂×C₃×C₄	—	4	1	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	2	2	no	—
162	C₂⁴	C₃×D₄	1	1	yes	characteristic
163	#41	—	4	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	C₂³×C₃	1	2	yes	—
167	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
182	Q₁₆	—	4	2	no	—
183	SD₁₆	—	4	2	no	—
184	M₁₆	—	4	2	no	—
185	C₂×C₈	C₂³×C₃	1	2	yes	—
186	C₂×C₈	C₂³×C₃	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	C₂³×C₃	1	2	yes	—
193	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	4	4	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	#137	1	2	yes	—
216	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	4	1	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	—	4	1	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	1	no	—
230	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	4	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	4	1	no	—
263	C₂×C₃	—	4	1	no	—
264	C₂×C₃	—	8	1	no	—
265	C₂×C₃	#534	1	1	yes	characteristic
266	C₂×C₃	C₂×#137	1	1	yes	characteristic
267	C₂×C₃	#575	1	1	yes	characteristic
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	C₃×#137	1	2	yes	—
281	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	2	no	—
290	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
291	C₃	#3196	1	1	yes	characteristic, direct product factor, Sylow
292	C₂	—	2	2	no	—
293	C₂	—	4	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	8	1	no	—
296	C₂	C₃×#534	1	1	yes	characteristic
297	C₂	C₂×C₃×#137	1	1	yes	characteristic
298	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
3. C₃×#485	C₂	semidirect
4. C₃×#484	C₂	semidirect
5. C₃×#481	C₂	semidirect
6. C₃×#470	C₂	semidirect
7. C₃×#468	C₂	semidirect
8. C₃×#452	C₂	semidirect
9. C₃×#451	C₂	semidirect
10. C₃×#448	C₂	semidirect
11. C₃×#407	C₂	semidirect
12. #3196	C₃	direct (N × H)
16. C₂×C₃×Q₁₆	C₂²	semidirect
17. C₂×C₃×SD₁₆	C₂²	semidirect
35. C₃×#107	C₂²	semidirect
36. C₃×#106	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
38. C₃×#103	C₂²	semidirect
39. C₃×#103	C₂²	semidirect
40. C₃×#102	C₂²	semidirect
45. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
46. #485	C₂×C₃	semidirect
47. #484	C₂×C₃	semidirect
48. #481	C₂×C₃	semidirect
49. #470	C₂×C₃	semidirect
50. #468	C₂×C₃	semidirect
51. #452	C₂×C₃	semidirect
52. #451	C₂×C₃	semidirect
53. #448	C₂×C₃	semidirect
54. #407	C₂×C₃	semidirect
99. C₂×Q₁₆	C₂²×C₃	semidirect
100. C₂×SD₁₆	C₂²×C₃	semidirect
118. #107	C₂²×C₃	semidirect
119. #106	C₂²×C₃	semidirect
120. #103	C₂²×C₃	semidirect
121. #103	C₂²×C₃	semidirect
122. #103	C₂²×C₃	semidirect
123. #102	C₂²×C₃	semidirect
291. C₃	#3196	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×#525	2
C₂×C₃×C₂²⋊Q₈	2
C₃×#485	2
C₃×#484	2
C₃×#481	2
C₃×#470	2
C₃×#468	2
C₃×#452	2
C₃×#451	2
C₃×#448	2
C₃×#407	2
#3196	3

Subgroups — C3×#3196 #85060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3196 #85060