Subgroups — #11775078

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#533	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	#10592624	C₂	1	1	yes	maximal, characteristic
4	#10589404	C₂	1	4	yes	maximal
5	#71835	—	3	4	no	maximal
6	C₃²×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₃²×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₃²×#125	—	2	1	no	—
9	C₃²×C₂²⋊Q₈	—	2	1	no	—
10	C₃²×C₂²⋊Q₈	C₂²	1	2	yes	—
11	C₃²×C₄×Q₈	—	2	1	no	—
12	C₃²×C₄×Q₈	C₄	1	2	yes	complemented
13	C₃²×#117	—	2	1	no	—
14	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
16	C₂×C₄×C₃⋊Dic₃	—	2	1	no	—
17	#63658	C₂²	1	2	yes	—
18	#63657	C₂²	1	2	yes	—
19	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
20	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
21	C₃×#533	D₃	1	4	yes	—
22	C₄×C₄⋊Dic₃	—	3	4	no	—
23	#4925	—	3	4	no	—
24	#4538	—	3	16	no	—
25	C₂×C₃²×Q₈	—	2	1	no	—
26	C₂×C₃²×Q₈	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
31	C₃²×C₄⋊C₄	—	2	1	no	—
32	C₃²×C₄⋊C₄	—	2	1	no	—
33	C₃²×C₄⋊C₄	—	2	1	no	—
34	C₃²×C₄⋊C₄	—	2	2	no	—
35	C₃²×C₄⋊C₄	—	2	2	no	—
36	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₃²×C₂²⋊C₄	—	2	2	no	—
38	C₃²×C₄²	—	2	2	no	—
39	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₄²	D₄	1	1	yes	characteristic
42	C₃²×C₄²	D₄	1	1	yes	characteristic
43	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
44	C₄×C₃⋊Dic₃	—	4	2	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
46	#1503	—	9	1	no	Sylow
47	C₃×C₄⋊Q₈	Dic₃	1	4	yes	complemented
48	C₃×C₄⋊Q₈	Dic₃	1	4	yes	complemented
49	C₃×#125	—	2	4	no	—
50	C₃×C₂²⋊Q₈	—	2	4	no	—
51	C₃×C₂²⋊Q₈	C₂×D₃	1	8	yes	—
52	C₃×C₄×Q₈	—	2	4	no	—
53	C₃×C₄×Q₈	Dic₃	1	8	yes	complemented
54	C₃×#117	—	2	4	no	—
55	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
56	C₂×C₄⋊Dic₃	—	6	4	no	—
57	C₂×C₄×Dic₃	—	6	4	no	—
58	C₃⋊#95	—	3	8	no	—
59	#830	—	3	8	no	—
60	C₄⋊(C₃⋊C₈)	—	6	4	no	—
61	C₄×C₃⋊C₈	—	6	4	no	—
62	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
63	C₃²×Q₈	—	4	2	no	—
64	C₃²×Q₈	—	4	2	no	—
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	—	2	2	no	—
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	—	4	1	no	—
69	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
74	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃⋊Dic₃	—	2	4	no	—
78	C₂×C₃⋊Dic₃	—	4	2	no	—
79	C₂×C₃⋊Dic₃	—	4	2	no	—
80	C₃⋊(C₃⋊C₈)	—	4	2	no	—
81	#533	C₃⋊D₃	1	1	yes	characteristic
82	#431	—	9	1	no	—
83	C₄×C₄⋊C₄	—	9	1	no	—
84	#328	—	9	4	no	—
85	C₂×C₃×Q₈	—	2	4	no	—
86	C₂×C₃×Q₈	C₂×Dic₃	1	8	yes	—
87	C₂²×C₃×C₄	#67	1	4	yes	—
88	C₂²×C₃×C₄	#67	1	4	yes	—
89	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
90	C₂²×Dic₃	—	6	8	no	—
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	—	2	4	no	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	8	no	—
95	C₃×C₄⋊C₄	—	2	8	no	—
96	C₃×C₄⋊C₄	C₂×Dic₃	1	8	yes	—
97	C₃×C₂²⋊C₄	—	2	8	no	—
98	C₃×C₄²	—	2	8	no	—
99	C₃×C₄²	C₂×Dic₃	1	4	yes	—
100	C₃×C₄²	C₂×Dic₃	1	4	yes	—
101	C₃×C₄²	#67	1	4	yes	—
102	C₃×C₄²	#67	1	4	yes	—
103	C₄⋊Dic₃	—	6	16	no	—
104	C₄×Dic₃	—	12	8	no	—
105	C₂×C₃⋊C₈	—	6	8	no	—
106	C₂²×C₃²	—	2	1	no	—
107	C₂²×C₃²	—	2	1	no	—
108	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
110	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
111	C₃²×C₄	—	2	1	no	—
112	C₃²×C₄	—	2	1	no	—
113	C₃²×C₄	—	2	1	no	—
114	C₃²×C₄	—	4	2	no	—
115	C₃²×C₄	—	4	2	no	—
116	C₃²×C₄	#103	1	2	yes	—
117	C₃²×C₄	#103	1	2	yes	—
118	C₃²×C₄	#104	1	2	yes	—
119	C₃⋊Dic₃	—	4	4	no	—
120	C₄⋊Q₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
121	C₄⋊Q₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
122	#125	—	2	1	no	—
123	C₂²⋊Q₈	—	2	1	no	—
124	C₂²⋊Q₈	C₂×C₃⋊D₃	1	2	yes	—
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	C₃⋊Dic₃	1	2	yes	complemented
127	#117	—	2	1	no	—
128	C₂×C₄⋊C₄	—	18	1	no	—
129	C₂×C₄²	—	18	1	no	—
130	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
131	C₄⋊C₈	—	18	1	no	—
132	C₂²⋊C₈	—	9	2	no	—
133	C₄×C₈	—	18	1	no	—
134	#95	—	9	2	no	—
135	C₂³×C₃	C₂×#67	1	4	yes	—
136	C₃×Q₈	—	4	8	no	—
137	C₃×Q₈	—	4	8	no	—
138	C₂×C₃×C₄	—	2	8	no	—
139	C₂×C₃×C₄	—	2	8	no	—
140	C₂×C₃×C₄	—	2	8	no	—
141	C₂×C₃×C₄	—	4	4	no	—
142	C₂×C₃×C₄	#215	1	8	yes	—
143	C₂×C₃×C₄	#216	1	8	yes	—
144	C₂×C₃×C₄	#217	1	4	yes	—
145	C₂×C₃×C₄	#217	1	4	yes	—
146	C₂×C₃×C₄	#217	1	4	yes	—
147	C₂×C₃×C₄	#217	1	4	yes	—
148	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
149	C₂×C₃×C₄	C₂×#67	1	4	yes	—
150	C₂×Dic₃	—	6	16	no	—
151	C₂×Dic₃	—	12	8	no	—
152	C₂×Dic₃	—	12	8	no	—
153	C₃⋊C₈	—	12	8	no	—
154	C₂×C₃²	—	2	1	no	—
155	C₂×C₃²	—	2	1	no	—
156	C₂×C₃²	#410	1	1	yes	characteristic
157	C₂×C₃²	C₂×#103	1	1	yes	characteristic
158	C₂×C₃²	C₂×#104	1	1	yes	characteristic
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
161	C₂²×C₄	—	18	2	no	—
162	C₂²×C₄	#638	1	1	yes	characteristic
163	C₂²×C₄	#638	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
165	C₂×C₈	—	18	2	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	18	4	no	—
172	C₄⋊C₄	C₂×C₃⋊Dic₃	1	2	yes	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	36	2	no	—
176	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
177	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
178	C₄²	#638	1	1	yes	characteristic
179	C₄²	#638	1	1	yes	characteristic
180	C₂²×C₃	—	2	4	no	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	C₂×#215	1	4	yes	—
183	C₂²×C₃	C₂×#216	1	4	yes	—
184	C₂²×C₃	C₂×#217	1	4	yes	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	2	4	no	—
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	4	8	no	—
189	C₃×C₄	—	4	8	no	—
190	C₃×C₄	#835	1	8	yes	—
191	C₃×C₄	#835	1	8	yes	—
192	C₃×C₄	#837	1	8	yes	—
193	Dic₃	—	12	16	no	—
194	C₃²	#1503	1	1	yes	characteristic, complemented, Sylow
195	C₂³	C₂×#638	1	1	yes	characteristic
196	Q₈	—	4	2	no	—
197	Q₈	—	4	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	18	4	no	—
203	C₂×C₄	—	36	2	no	—
204	C₂×C₄	—	36	2	no	—
205	C₂×C₄	#3755	1	2	yes	—
206	C₂×C₄	#3756	1	2	yes	—
207	C₂×C₄	#3757	1	1	yes	characteristic
208	C₂×C₄	#3757	1	1	yes	characteristic
209	C₂×C₄	#3757	1	1	yes	characteristic
210	C₂×C₄	#3757	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
212	C₂×C₄	C₂×#638	1	1	yes	characteristic
213	C₈	—	36	2	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	C₂×#835	1	4	yes	—
217	C₂×C₃	C₂×#837	1	4	yes	—
218	C₂×C₃	#5240	1	4	yes	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#3755	1	1	yes	characteristic
222	C₂²	C₂×#3756	1	1	yes	characteristic
223	C₂²	C₂×#3757	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	36	4	no	—
230	C₄	#63662	1	2	yes	—
231	C₄	#63662	1	2	yes	—
232	C₄	#63664	1	2	yes	—
233	C₃	#71835	1	4	yes	complemented
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	C₂×#63662	1	1	yes	characteristic
237	C₂	C₂×#63664	1	1	yes	characteristic
238	C₂	#10592939	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3757
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#533	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#533	C₃⋊D₃
hypocenter	C₃²	#1503
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
6. C₃²×C₄⋊Q₈	C₄	semidirect
7. C₃²×C₄⋊Q₈	C₄	semidirect
12. C₃²×C₄×Q₈	C₄	semidirect
47. C₃×C₄⋊Q₈	Dic₃	semidirect
48. C₃×C₄⋊Q₈	Dic₃	semidirect
53. C₃×C₄×Q₈	Dic₃	semidirect
120. C₄⋊Q₈	C₃⋊Dic₃	semidirect
121. C₄⋊Q₈	C₃⋊Dic₃	semidirect
126. C₄×Q₈	C₃⋊Dic₃	semidirect
194. C₃²	#1503	semidirect
233. C₃	#71835	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#533	2
C₄×C₄⋊(C₃⋊Dic₃)	2
#10592624	2
#10589404	2
#71835	3