Subgroups — C₄×#63368 #11783970

Order 1152.

498 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#63368	C₂	1	2	yes	maximal
4	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#10588206	C₂	1	2	yes	maximal
6	C₄×#810	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₉	—	2	1	no	—
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₉⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₉	—	2	1	no	—
14	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
15	C₄×Dic₁₈	—	2	2	no	—
16	C₄×Dic₁₈	C₄	1	4	yes	complemented
17	C₉×#95	—	2	1	no	—
18	#63368	C₄	1	8	yes	direct product factor
19	C₄×C₉⋊C₈	—	2	2	no	—
20	C₂×C₄×Dic₆	—	3	1	no	—
21	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₂×#810	—	3	2	no	—
23	C₂×C₄×C₃⋊C₈	—	3	1	no	—
24	#4529	—	3	2	no	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₉	—	2	1	no	—
30	C₂²×Dic₉	—	2	1	no	—
31	C₂×Dic₁₈	—	2	1	no	—
32	C₂×Dic₁₈	—	2	2	no	—
33	C₂×Dic₁₈	C₂×C₄	1	4	yes	—
34	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₄²×C₉	—	4	2	no	—
38	C₄²×C₉	C₂×C₄	1	2	yes	—
39	C₄²×C₉	D₄	1	2	yes	—
40	C₄⋊Dic₉	—	2	1	no	—
41	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
42	Dic₉⋊C₄	—	2	4	no	—
43	C₄×Dic₉	—	2	4	no	—
44	C₂×C₉⋊C₈	—	2	2	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×#103	—	9	1	no	Sylow
48	C₂²×Dic₆	—	3	1	no	—
49	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₃×C₄²	—	2	1	no	—
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₃	—	3	1	no	—
53	C₂×Dic₃⋊C₄	—	6	1	no	—
54	C₂×C₄×Dic₃	—	6	1	no	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₄×Dic₆	—	3	4	no	—
57	C₄×Dic₆	—	6	2	no	—
58	C₃×#95	—	2	1	no	—
59	#810	—	3	8	no	—
60	C₄×C₃⋊C₈	—	6	2	no	—
61	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	4	2	no	—
65	C₂×C₄×C₉	—	4	2	no	—
66	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₉	SD₁₆	1	2	yes	—
69	C₂×C₄×C₉	Q₁₆	1	2	yes	—
70	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	2	yes	—
74	C₂×Dic₉	—	2	2	no	—
75	C₂×Dic₉	—	2	4	no	—
76	C₂×Dic₉	—	2	2	no	—
77	C₂×Dic₉	—	4	2	no	—
78	Dic₁₈	—	2	2	no	—
79	Dic₁₈	—	2	4	no	—
80	Dic₁₈	C₄²	1	4	yes	complemented
81	C₉⋊C₈	—	2	4	no	—
82	C₂×C₄×Q₈	—	9	1	no	—
83	C₂×#103	—	9	2	no	—
84	C₂×C₄×C₈	—	9	1	no	—
85	C₄×C₄⋊C₄	D₉	1	1	yes	characteristic
86	#340	—	9	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	#67	1	1	yes	characteristic
89	C₂²×C₃×C₄	#67	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂²×Dic₃	—	6	1	no	—
92	C₂²×Dic₃	—	6	1	no	—
93	C₂×Dic₆	—	3	4	no	—
94	C₂×Dic₆	—	3	2	no	—
95	C₂×Dic₆	—	6	1	no	—
96	C₂×Dic₆	—	6	2	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	C₄×D₃	1	4	yes	—
99	C₃×C₄²	—	4	2	no	—
100	C₃×C₄²	C₄×D₃	1	2	yes	—
101	C₃×C₄²	D₁₂	1	2	yes	—
102	C₄⋊Dic₃	—	3	2	no	—
103	C₄⋊Dic₃	—	6	1	no	—
104	Dic₃⋊C₄	—	6	4	no	—
105	C₄×Dic₃	—	6	4	no	—
106	C₂×C₃⋊C₈	—	3	4	no	—
107	C₂×C₃⋊C₈	—	6	2	no	—
108	C₂×C₃⋊C₈	—	6	2	no	—
109	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₉	C₄×D₄	1	2	yes	—
111	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
113	C₂²×C₉	#135	1	2	yes	—
114	C₄×C₉	—	2	2	no	—
115	C₄×C₉	—	4	4	no	—
116	C₄×C₉	#103	1	4	yes	complemented
117	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
118	C₄×C₉	#117	1	1	yes	characteristic
119	C₄×C₉	C₄×D₄	1	2	yes	—
120	Dic₉	—	2	4	no	—
121	Dic₉	—	4	2	no	—
122	C₂²×Q₈	—	9	1	no	—
123	C₂²×C₈	—	9	2	no	—
124	C₄×Q₈	—	9	4	no	—
125	C₄×Q₈	—	18	2	no	—
126	C₂×C₄⋊C₄	—	9	1	no	—
127	C₂×C₄⋊C₄	—	18	1	no	—
128	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	—	18	1	no	—
131	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
132	#103	—	9	8	no	—
133	C₄×C₈	—	18	2	no	—
134	#95	—	2	1	no	—
135	C₂³×C₃	C₂×#67	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
141	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₂×C₃×C₄	#214	1	2	yes	—
143	C₂×C₃×C₄	#216	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
147	C₂×C₃×C₄	#235	1	2	yes	—
148	C₂×Dic₃	—	6	2	no	—
149	C₂×Dic₃	—	6	4	no	—
150	C₂×Dic₃	—	6	2	no	—
151	C₂×Dic₃	—	12	2	no	—
152	Dic₆	—	3	4	no	—
153	Dic₆	—	6	2	no	—
154	Dic₆	—	6	4	no	—
155	C₃⋊C₈	—	6	4	no	—
156	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₉	C₂×#103	1	1	yes	characteristic
158	C₂×C₉	#416	1	1	yes	characteristic
159	C₂×C₉	C₄×SD₁₆	1	2	yes	—
160	C₂×C₉	C₄×Q₁₆	1	2	yes	—
161	C₂×Q₈	—	9	4	no	—
162	C₂×Q₈	—	9	2	no	—
163	C₂×Q₈	—	18	1	no	—
164	C₂×Q₈	—	18	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	18	1	no	—
167	C₂²×C₄	—	18	1	no	—
168	C₂²×C₄	#611	1	1	yes	characteristic
169	C₂²×C₄	#611	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
171	C₂×C₈	—	9	4	no	—
172	C₂×C₈	—	18	2	no	—
173	C₂×C₈	—	18	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	9	2	no	—
176	C₄⋊C₄	—	18	4	no	—
177	C₄⋊C₄	—	18	1	no	—
178	C₄⋊C₄	C₄×D₉	1	4	yes	—
179	C₄²	—	4	2	no	—
180	C₄²	—	18	4	no	—
181	C₄²	C₄×D₉	1	2	yes	—
182	C₄²	D₃₆	1	2	yes	—
183	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂²×C₃	C₄×#67	1	2	yes	—
185	C₂²×C₃	C₂×#214	1	1	yes	characteristic
186	C₂²×C₃	C₂×#216	1	1	yes	characteristic
187	C₂²×C₃	#950	1	2	yes	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	#810	1	4	yes	—
191	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
192	C₃×C₄	#872	1	1	yes	characteristic
193	C₃×C₄	C₄×D₁₂	1	2	yes	—
194	Dic₃	—	6	4	no	—
195	Dic₃	—	12	2	no	—
196	C₉	C₄×#103	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×#611	1	1	yes	characteristic
198	Q₈	—	9	4	no	—
199	Q₈	—	18	2	no	—
200	Q₈	—	18	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	18	2	no	—
206	C₂×C₄	—	18	4	no	—
207	C₂×C₄	—	18	2	no	—
208	C₂×C₄	—	36	2	no	—
209	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
210	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
211	C₂×C₄	#3672	1	2	yes	—
212	C₂×C₄	#3674	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
215	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
216	C₂×C₄	#3697	1	2	yes	—
217	C₈	—	18	4	no	—
218	C₂×C₃	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
219	C₂×C₃	C₂×#810	1	1	yes	characteristic
220	C₂×C₃	#5002	1	1	yes	characteristic
221	C₂×C₃	C₄×#214	1	2	yes	—
222	C₂×C₃	C₄×#216	1	2	yes	—
223	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
224	C₂²	C₄×#611	1	2	yes	—
225	C₂²	C₂×#3672	1	1	yes	characteristic
226	C₂²	C₂×#3674	1	1	yes	characteristic
227	C₂²	#63513	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	4	4	no	—
230	C₄	—	18	4	no	—
231	C₄	—	36	2	no	—
232	C₄	#63368	1	4	yes	direct product factor
233	C₄	C₄²×D₉	1	1	yes	characteristic
234	C₄	#63434	1	1	yes	characteristic
235	C₄	C₄×D₃₆	1	2	yes	—
236	C₃	C₄×#810	1	1	yes	characteristic
237	C₂	C₄×(C₂×D₉)⋊C₄	1	1	yes	characteristic
238	C₂	C₂×#63368	1	1	yes	characteristic
239	C₂	#10588683	1	1	yes	characteristic
240	C₂	C₄×#3672	1	2	yes	—
241	C₂	C₄×#3674	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#611
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₉
hypocenter	C₉	C₄×#103
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. C₄×Dic₁₈	C₄	semidirect
18. #63368	C₄	direct (N × H)
80. Dic₁₈	C₄²	semidirect
116. C₄×C₉	#103	semidirect
196. C₉	C₄×#103	semidirect
232. C₄	#63368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈	2
C₄×C₉×C₄⋊C₄	2
C₂×#63368	2
C₂×C₄×C₉⋊C₈	2
#10588206	2
C₄×#810	3

Subgroups — C4×#63368 #11783970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#63368 #11783970