Subgroups — C₃×#3298 #85162

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#483	C₂	1	2	yes	maximal, complemented
5	C₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#465	C₂	1	2	yes	maximal, complemented
7	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#448	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#103	C₂	1	2	yes	maximal, complemented
12	#3298	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	—	2	1	no	—
14	C₂²×C₃×Q₈	C₂²	1	2	yes	—
15	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
17	C₂²×C₃×C₈	C₂²	1	2	yes	—
18	C₃×C₄⋊Q₈	—	2	1	no	—
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃×#123	—	2	2	no	—
22	C₃×C₂²⋊Q₈	—	2	1	no	—
23	C₃×C₂²⋊Q₈	—	2	2	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₃×#121	C₂²	1	2	yes	—
26	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
27	C₃×C₄×D₄	—	2	1	no	—
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
31	C₃×#106	C₂²	1	1	yes	characteristic
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×#103	C₂²	1	1	yes	characteristic
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	2	yes	—
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	1	yes	characteristic
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
43	#547	C₂×C₃	1	1	yes	characteristic
44	#487	C₂×C₃	1	1	yes	characteristic, complemented
45	#483	C₂×C₃	1	2	yes	complemented
46	#475	C₂×C₃	1	1	yes	characteristic, complemented
47	#465	C₂×C₃	1	2	yes	complemented
48	#461	C₂×C₃	1	1	yes	characteristic, complemented
49	#448	C₂×C₃	1	2	yes	complemented
50	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#103	C₂×C₃	1	2	yes	complemented
53	C₃×#41	—	4	4	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	—	4	2	no	—
58	C₂×C₃×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	D₄	1	1	yes	characteristic
62	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₂²×C₃×C₄	—	2	1	no	—
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₃×SD₁₆	—	2	2	no	—
68	C₃×SD₁₆	—	4	2	no	—
69	C₂×C₃×C₈	—	4	2	no	—
70	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂³	1	2	yes	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	2	no	—
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₄⋊C₄	C₂³	1	2	yes	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₂²⋊C₄	D₄	1	2	yes	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	C₂³	1	1	yes	characteristic
89	C₂×#41	—	2	1	no	—
90	C₂²×Q₈	C₂²×C₃	1	2	yes	—
91	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
93	C₂²×C₈	C₂²×C₃	1	2	yes	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
96	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
97	#123	—	2	2	no	—
98	C₂²⋊Q₈	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
101	#121	C₂²×C₃	1	2	yes	—
102	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
103	C₄×D₄	—	2	1	no	—
104	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
105	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
107	#106	C₂²×C₃	1	1	yes	characteristic
108	#106	C₂²×C₃	1	2	yes	—
109	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
110	#103	C₂²×C₃	1	1	yes	characteristic
111	#103	C₂²×C₃	1	2	yes	—
112	#103	C₂²×C₃	1	2	yes	—
113	#103	C₂²×C₃	1	2	yes	—
114	#103	C₂²×C₃	1	2	yes	—
115	#102	C₂²×C₃	1	1	yes	characteristic
116	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
117	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	C₂×D₄	1	2	yes	—
120	C₃×Q₈	—	2	1	no	—
121	C₃×Q₈	—	2	2	no	—
122	C₃×Q₈	—	2	2	no	—
123	C₃×Q₈	—	4	4	no	—
124	C₃×Q₈	—	4	2	no	—
125	C₃×Q₈	#41	1	2	yes	complemented
126	C₃×D₄	—	2	1	no	—
127	C₃×D₄	—	2	1	no	—
128	C₃×D₄	—	4	4	no	—
129	C₃×D₄	SD₁₆	1	4	yes	complemented
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	2	no	—
147	#41	—	4	4	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	C₂³×C₃	1	2	yes	—
153	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₃×D₄	1	1	yes	characteristic
156	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	C₂³×C₃	1	2	yes	—
161	SD₁₆	—	2	2	no	—
162	SD₁₆	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
166	C₂×C₈	C₂³×C₃	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
174	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
175	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
176	C₄⋊C₄	C₂³×C₃	1	2	yes	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	C₃×D₄	1	2	yes	—
181	C₄²	—	2	1	no	—
182	C₄²	C₂³×C₃	1	1	yes	characteristic
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	4	1	no	—
186	C₂²×C₃	—	4	1	no	—
187	C₂²×C₃	—	4	1	no	—
188	C₂²×C₃	C₂×SD₁₆	1	4	yes	—
189	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	C₂×SD₁₆	1	2	yes	—
198	C₃×C₄	C₂×#41	1	1	yes	characteristic
199	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂³	—	2	1	no	—
201	C₂³	C₂×C₃×D₄	1	2	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	2	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	C₃×#41	1	2	yes	complemented
208	D₄	—	2	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	4	4	no	—
211	D₄	C₃×SD₁₆	1	4	yes	complemented
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
224	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	2	2	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	8	1	no	—
231	C₂×C₃	#546	1	1	yes	characteristic
232	C₂×C₃	C₂²×SD₁₆	1	1	yes	characteristic
233	C₂×C₃	#578	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₂×C₃×SD₁₆	1	4	yes	—
240	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	C₂×C₃×SD₁₆	1	2	yes	—
249	C₄	C₂×C₃×#41	1	1	yes	characteristic
250	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
251	C₃	#3298	1	1	yes	characteristic, direct product factor, Sylow
252	C₂	—	2	4	no	—
253	C₂	—	8	1	no	—
254	C₂	C₃×#546	1	1	yes	characteristic
255	C₂	C₂²×C₃×SD₁₆	1	1	yes	characteristic
256	C₂	C₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×D₄×Q₈	C₂	semidirect
3. C₃×#487	C₂	semidirect
4. C₃×#483	C₂	semidirect
5. C₃×#475	C₂	semidirect
6. C₃×#465	C₂	semidirect
7. C₃×#461	C₂	semidirect
8. C₃×#448	C₂	semidirect
9. C₃×C₄×SD₁₆	C₂	semidirect
10. C₃×C₈×D₄	C₂	semidirect
11. C₂×C₃×#103	C₂	semidirect
12. #3298	C₃	direct (N × H)
42. D₄×Q₈	C₂×C₃	semidirect
44. #487	C₂×C₃	semidirect
45. #483	C₂×C₃	semidirect
46. #475	C₂×C₃	semidirect
47. #465	C₂×C₃	semidirect
48. #461	C₂×C₃	semidirect
49. #448	C₂×C₃	semidirect
50. C₄×SD₁₆	C₂×C₃	semidirect
51. C₈×D₄	C₂×C₃	semidirect
52. C₂×#103	C₂×C₃	semidirect
125. C₃×Q₈	#41	semidirect
129. C₃×D₄	SD₁₆	semidirect
207. Q₈	C₃×#41	semidirect
211. D₄	C₃×SD₁₆	semidirect
251. C₃	#3298	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×D₄×Q₈	2
C₃×#547	2
C₃×#487	2
C₃×#483	2
C₃×#475	2
C₃×#465	2
C₃×#461	2
C₃×#448	2
C₃×C₄×SD₁₆	2
C₃×C₈×D₄	2
C₂×C₃×#103	2
#3298	3

Subgroups — C3×#3298 #85162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3298 #85162