Subgroups — C₄×#912 #79585

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#912	C₂	1	1	yes	maximal, characteristic
2	C₄×#237	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#214	C₂	1	2	yes	maximal, complemented
6	#4880	C₂	1	1	yes	maximal, characteristic, complemented
7	#4873	C₂	1	1	yes	maximal, characteristic, complemented
8	#4772	C₂	1	2	yes	maximal, complemented
9	#4758	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#135	—	3	1	no	maximal, Sylow
13	C₂×#237	C₂²	1	2	yes	—
14	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₄	C₂²	1	2	yes	—
16	D₄×Dic₃	C₂²	1	2	yes	—
17	C₂×#214	C₂²	1	2	yes	—
18	C₄×#67	—	2	2	no	—
19	C₂×C₄×Dic₃	—	2	2	no	—
20	#912	C₄	1	8	yes	direct product factor
21	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
22	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
23	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
24	#892	C₂²	1	2	yes	—
25	#887	C₂²	1	2	yes	—
26	#67⋊C₄	—	2	2	no	—
27	#877	—	2	2	no	—
28	C₄²×D₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₆	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#102	C₂²	1	2	yes	—
32	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
33	#832	C₂²	1	2	yes	—
34	#818	C₂²	1	1	yes	characteristic
35	#816	C₂²	1	2	yes	—
36	C₈×Dic₃	C₂²	1	1	yes	characteristic
37	C₈×Dic₃	C₂²	1	1	yes	characteristic
38	#810	C₂²	1	2	yes	—
39	#808	C₂²	1	2	yes	—
40	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×#135	—	3	1	no	—
42	C₄×#41	—	3	2	no	—
43	C₄×Q₁₆	—	3	1	no	—
44	C₄×Q₁₆	—	3	1	no	—
45	C₄×SD₁₆	—	3	2	no	—
46	C₄×SD₁₆	—	3	2	no	—
47	C₄×D₈	—	3	1	no	—
48	C₄×D₈	D₃	1	1	yes	characteristic, complemented
49	#427	—	3	1	no	—
50	#416	—	3	2	no	—
51	C₂×C₄×C₈	—	3	1	no	—
52	C₂×C₃×D₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂×#67	—	2	2	no	—
55	C₂²×Dic₃	—	2	2	no	—
56	C₂²×Dic₃	—	2	2	no	—
57	#237	—	2	4	no	—
58	#237	C₂×C₄	1	8	yes	complemented
59	C₂×C₄×D₃	D₄	1	1	yes	characteristic
60	C₂×C₄×D₃	D₄	1	1	yes	characteristic
61	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
62	C₂×Dic₆	C₂³	1	2	yes	—
63	C₃×D₈	C₂×C₄	1	4	yes	complemented
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	#217	—	2	2	no	—
70	#214	C₂×C₄	1	8	yes	complemented
71	(C₂×D₃)⋊C₄	—	2	2	no	—
72	C₄⋊Dic₃	C₂³	1	2	yes	—
73	Dic₃⋊C₄	—	2	2	no	—
74	Dic₃⋊C₄	—	2	2	no	—
75	C₄×Dic₃	—	2	2	no	—
76	C₄×Dic₃	—	2	2	no	—
77	C₄×Dic₃	—	2	2	no	—
78	C₄×Dic₃	D₄	1	1	yes	characteristic
79	C₄×Dic₃	D₄	1	1	yes	characteristic
80	C₄×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	Dic₁₂	C₂×C₄	1	4	yes	complemented
84	D₃×C₈	—	2	2	no	—
85	D₃×C₈	C₂×C₄	1	4	yes	complemented
86	C₂×#41	—	3	2	no	—
87	#135	—	3	8	no	—
88	C₂×Q₁₆	—	3	1	no	—
89	C₂×SD₁₆	—	3	2	no	—
90	C₂×D₈	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₈	—	3	1	no	—
92	C₂²×C₈	—	3	1	no	—
93	C₄×Q₈	—	3	2	no	—
94	C₄×Q₈	—	3	2	no	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	6	2	no	—
97	C₄×D₄	—	6	2	no	—
98	C₄×D₄	C₂×D₃	1	2	yes	—
99	#117	—	3	2	no	—
100	#117	—	6	2	no	—
101	C₂×C₄²	—	3	1	no	—
102	C₂×C₄²	—	6	2	no	—
103	#107	—	3	1	no	—
104	#107	C₂×D₃	1	1	yes	characteristic
105	#106	—	3	2	no	—
106	#103	—	3	2	no	—
107	#103	—	3	2	no	—
108	#102	—	3	2	no	—
109	#102	C₂×D₃	1	2	yes	—
110	C₄×C₈	—	3	1	no	—
111	C₄×C₈	—	3	1	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	C₂×D₃	1	1	yes	characteristic
114	C₂³×C₃	—	2	2	no	—
115	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
116	C₃×D₄	—	2	2	no	—
117	C₃×D₄	C₂²×C₄	1	4	yes	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	4	4	no	—
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
123	#67	—	2	8	no	—
124	C₂×Dic₃	—	2	2	no	—
125	C₂×Dic₃	—	2	8	no	—
126	C₂×Dic₃	—	2	2	no	—
127	C₂×Dic₃	—	4	4	no	—
128	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
131	C₄×D₃	—	2	2	no	—
132	C₄×D₃	—	2	2	no	—
133	C₄×D₃	C₂²×C₄	1	2	yes	—
134	C₄×D₃	#41	1	2	yes	—
135	Dic₆	—	2	2	no	—
136	Dic₆	C₂²×C₄	1	4	yes	—
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	C₂²×C₄	1	2	yes	—
139	C₃⋊C₈	—	2	1	no	—
140	C₃⋊C₈	C₂²×C₄	1	2	yes	—
141	#41	—	3	8	no	—
142	#41	—	6	4	no	—
143	C₂×Q₈	—	3	2	no	—
144	C₂×D₄	—	6	2	no	—
145	C₂×D₄	C₂²×D₃	1	2	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	3	1	no	—
148	C₂²×C₄	—	3	1	no	—
149	C₂²×C₄	—	3	1	no	—
150	C₂²×C₄	—	6	2	no	—
151	C₂²×C₄	—	6	2	no	—
152	Q₁₆	—	3	4	no	—
153	SD₁₆	—	3	8	no	—
154	D₈	C₄×D₃	1	4	yes	complemented
155	C₂×C₈	—	3	1	no	—
156	C₂×C₈	—	3	4	no	—
157	C₂×C₈	—	3	1	no	—
158	C₂×C₈	—	6	2	no	—
159	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
161	C₄⋊C₄	—	3	2	no	—
162	C₄⋊C₄	—	6	2	no	—
163	C₄⋊C₄	—	6	2	no	—
164	C₄⋊C₄	C₂²×D₃	1	2	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	6	2	no	—
167	C₂²⋊C₄	—	6	2	no	—
168	C₄²	—	3	1	no	—
169	C₄²	—	3	1	no	—
170	C₄²	—	3	1	no	—
171	C₄²	—	6	2	no	—
172	C₄²	—	6	2	no	—
173	C₄²	—	6	2	no	—
174	C₄²	C₂²×D₃	1	1	yes	characteristic
175	C₂²×C₃	—	2	4	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×D₃	—	2	1	no	—
180	C₂×D₃	—	2	1	no	—
181	C₂×D₃	C₄×D₄	1	2	yes	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	#135	1	2	yes	complemented
185	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
186	C₃×C₄	C₂×#41	1	1	yes	characteristic
187	Dic₃	—	2	4	no	—
188	Dic₃	—	2	1	no	—
189	Dic₃	—	4	2	no	—
190	Dic₃	C₄×D₄	1	2	yes	—
191	Dic₃	#135	1	2	yes	complemented
192	C₂³	—	2	2	no	—
193	C₂³	—	3	1	no	—
194	Q₈	—	3	4	no	—
195	Q₈	—	6	2	no	—
196	D₄	—	2	2	no	—
197	D₄	—	6	8	no	—
198	D₄	C₂×C₄×D₃	1	4	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	3	1	no	—
201	C₂×C₄	—	3	1	no	—
202	C₂×C₄	—	3	2	no	—
203	C₂×C₄	—	3	2	no	—
204	C₂×C₄	—	3	1	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	6	2	no	—
207	C₂×C₄	—	6	8	no	—
208	C₂×C₄	—	6	2	no	—
209	C₂×C₄	—	6	2	no	—
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	—	12	4	no	—
212	C₂×C₄	D₃×D₄	1	1	yes	characteristic
213	C₂×C₄	D₃×D₄	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	3	2	no	—
217	C₈	—	6	1	no	—
218	C₈	C₂×C₄×D₃	1	2	yes	—
219	C₂×C₃	—	4	4	no	—
220	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂×C₃	C₂×#135	1	1	yes	characteristic
222	C₂×C₃	C₂×#135	1	1	yes	characteristic
223	D₃	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	3	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	6	1	no	—
229	C₂²	—	6	1	no	—
230	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	3	2	no	—
233	C₄	—	3	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	6	1	no	—
236	C₄	—	6	4	no	—
237	C₄	—	12	2	no	—
238	C₄	#912	1	2	yes	direct product factor
239	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
240	C₄	D₃×#41	1	1	yes	characteristic
241	C₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
242	C₂	—	4	4	no	—
243	C₂	—	6	2	no	—
244	C₂	C₄×D₃×D₄	1	1	yes	characteristic
245	C₂	C₂×#912	1	1	yes	characteristic
246	C₂	D₃×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×D₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#912	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×D₈	D₃
hypocenter	C₃	C₄×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#237	C₂	semidirect
3. C₃×C₄×D₈	C₂	semidirect
4. Dic₃×D₈	C₂	semidirect
5. C₄×#214	C₂	semidirect
6. #4880	C₂	semidirect
7. #4873	C₂	semidirect
8. #4772	C₂	semidirect
9. #4758	C₂	semidirect
10. C₄×Dic₁₂	C₂	semidirect
11. C₄×D₃×C₈	C₂	semidirect
20. #912	C₄	direct (N × H)
48. C₄×D₈	D₃	semidirect
58. #237	C₂×C₄	semidirect
63. C₃×D₈	C₂×C₄	semidirect
70. #214	C₂×C₄	semidirect
83. Dic₁₂	C₂×C₄	semidirect
85. D₃×C₈	C₂×C₄	semidirect
154. D₈	C₄×D₃	semidirect
184. C₃×C₄	#135	semidirect
191. Dic₃	#135	semidirect
238. C₄	#912	direct (N × H)
241. C₃	C₄×#135	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#912	2
C₄×#237	2
C₃×C₄×D₈	2
Dic₃×D₈	2
C₄×#214	2
#4880	2
#4873	2
#4772	2
#4758	2
C₄×Dic₁₂	2
C₄×D₃×C₈	2
C₄×#135	3

Subgroups — C4×#912 #79585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#912 #79585