Subgroups — #11790028

Order 1152.

780 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	2	yes	maximal
4	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#10589390	C₂	1	2	yes	maximal
6	#69484	—	3	4	no	maximal
7	C₂×C₃²×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃²×#95	C₄	1	2	yes	—
15	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
17	C₄×C₃⋊(C₃⋊C₈)	—	2	2	no	—
18	C₃×C₄×C₄⋊C₄	D₃	1	4	yes	—
19	C₂×C₄⋊(C₃⋊C₈)	—	3	8	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
21	C₂×C₄×C₃⋊C₈	—	3	4	no	—
22	#4524	—	3	8	no	—
23	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	—	2	2	no	—
29	C₃²×C₄⋊C₄	C₈	1	4	yes	complemented
30	C₃²×C₄²	—	2	2	no	—
31	C₃²×C₄²	—	2	2	no	—
32	C₃²×C₄²	D₄	1	2	yes	—
33	C₃²×C₄²	Q₈	1	2	yes	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
41	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
42	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
43	#1939	—	9	1	no	Sylow
44	C₂×C₃×C₄⋊C₄	Dic₃	1	4	yes	—
45	C₂×C₃×C₄⋊C₄	Dic₃	1	4	yes	—
46	C₂×C₃×C₄²	C₂×D₃	1	8	yes	—
47	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
48	C₂²×C₃⋊C₈	—	3	4	no	—
49	C₂²×C₃⋊C₈	—	3	4	no	—
50	C₂²×C₃⋊C₈	—	3	8	no	—
51	C₃×#95	Dic₃	1	8	yes	—
52	C₄⋊(C₃⋊C₈)	—	6	16	no	—
53	C₄⋊(C₃⋊C₈)	—	6	8	no	—
54	C₄×C₃⋊C₈	—	6	8	no	—
55	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	2	no	—
57	C₂×C₃²×C₄	—	2	1	no	—
58	C₂×C₃²×C₄	—	2	1	no	—
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	—	2	2	no	—
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	—	2	2	no	—
63	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
64	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
65	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
66	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
67	C₂×C₃²×C₄	M₁₆	1	2	yes	—
68	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	#41	1	1	yes	characteristic
73	C₂×C₃²×C₄	#41	1	1	yes	characteristic
74	C₃⋊(C₃⋊C₈)	—	2	2	no	—
75	C₃⋊(C₃⋊C₈)	—	2	2	no	—
76	C₃⋊(C₃⋊C₈)	—	4	2	no	—
77	C₂×C₄⋊C₈	—	9	2	no	—
78	C₂×C₄⋊C₈	—	9	1	no	—
79	C₂×C₄×C₈	—	9	1	no	—
80	C₄×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
81	#336	—	9	2	no	—
82	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
83	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
84	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
85	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
87	C₃×C₄⋊C₄	—	2	8	no	—
88	C₃×C₄⋊C₄	C₃⋊C₈	1	16	yes	complemented
89	C₃×C₄²	—	2	8	no	—
90	C₃×C₄²	—	2	8	no	—
91	C₃×C₄²	Dic₆	1	8	yes	—
92	C₃×C₄²	D₁₂	1	8	yes	—
93	C₂×C₃⋊C₈	—	3	8	no	—
94	C₂×C₃⋊C₈	—	3	8	no	—
95	C₂×C₃⋊C₈	—	6	4	no	—
96	C₂×C₃⋊C₈	—	6	4	no	—
97	C₂×C₃⋊C₈	—	6	4	no	—
98	C₂×C₃⋊C₈	—	6	4	no	—
99	C₂×C₃⋊C₈	—	6	8	no	—
100	C₂×C₃⋊C₈	—	6	8	no	—
101	C₂×C₃⋊C₈	—	6	8	no	—
102	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
105	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
106	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
107	C₂²×C₃²	#131	1	1	yes	characteristic
108	C₂²×C₃²	#131	1	1	yes	characteristic
109	C₃²×C₄	—	2	2	no	—
110	C₃²×C₄	—	2	4	no	—
111	C₃²×C₄	—	4	2	no	—
112	C₃²×C₄	C₄⋊C₈	1	4	yes	complemented
113	C₃²×C₄	#121	1	1	yes	characteristic
114	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
115	C₃²×C₄	#125	1	1	yes	characteristic
116	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₈	—	9	1	no	—
118	C₂²×C₈	—	9	1	no	—
119	C₂²×C₈	—	9	2	no	—
120	C₂×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₃⋊D₃	1	2	yes	—
123	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
124	C₄⋊C₈	—	18	4	no	—
125	C₄⋊C₈	—	18	2	no	—
126	C₄×C₈	—	18	2	no	—
127	#95	C₃⋊Dic₃	1	2	yes	—
128	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
129	C₂×C₃×C₄	—	2	8	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	8	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	—	2	8	no	—
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	C₂×C₃⋊C₈	1	16	yes	—
137	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
138	C₂×C₃×C₄	#208	1	8	yes	—
139	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
140	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
141	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
142	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
143	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
144	C₂×C₃×C₄	#237	1	4	yes	—
145	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
146	C₂×C₃×C₄	#239	1	4	yes	—
147	C₃⋊C₈	—	6	8	no	—
148	C₃⋊C₈	—	6	8	no	—
149	C₃⋊C₈	—	12	8	no	—
150	C₂×C₃²	#389	1	1	yes	characteristic
151	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₃²	#424	1	1	yes	characteristic
153	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
154	C₂×C₃²	#436	1	1	yes	characteristic
155	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
156	C₂×C₃²	#446	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
158	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
159	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
162	C₂×C₈	—	9	2	no	—
163	C₂×C₈	—	9	2	no	—
164	C₂×C₈	—	18	1	no	—
165	C₂×C₈	—	18	1	no	—
166	C₂×C₈	—	18	1	no	—
167	C₂×C₈	—	18	1	no	—
168	C₂×C₈	—	18	2	no	—
169	C₂×C₈	—	18	2	no	—
170	C₂×C₈	—	18	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	C₃⋊(C₃⋊C₈)	1	4	yes	complemented
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	#634	1	2	yes	—
176	C₄²	C₄⋊(C₃⋊D₃)	1	2	yes	—
177	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
178	C₂²×C₃	C₂×#208	1	4	yes	—
179	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
180	C₂²×C₃	D₄×Dic₃	1	4	yes	—
181	C₂²×C₃	Q₈×Dic₃	1	4	yes	—
182	C₂²×C₃	#948	1	4	yes	—
183	C₂²×C₃	#948	1	4	yes	—
184	C₃×C₄	—	2	8	no	—
185	C₃×C₄	—	2	16	no	—
186	C₃×C₄	—	4	8	no	—
187	C₃×C₄	C₄⋊(C₃⋊C₈)	1	16	yes	complemented
188	C₃×C₄	#888	1	4	yes	—
189	C₃×C₄	#890	1	4	yes	—
190	C₃×C₄	#895	1	4	yes	—
191	C₃×C₄	#897	1	4	yes	—
192	C₃²	#1939	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
202	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
203	C₂×C₄	#3748	1	2	yes	—
204	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
205	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
206	C₂×C₄	C₂×#634	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
208	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
209	C₂×C₄	#3830	1	1	yes	characteristic
210	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
211	C₂×C₄	#3832	1	1	yes	characteristic
212	C₈	—	18	2	no	—
213	C₈	—	18	2	no	—
214	C₈	—	36	2	no	—
215	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
216	C₂×C₃	#4984	1	4	yes	—
217	C₂×C₃	#4999	1	4	yes	—
218	C₂×C₃	D₄×C₃⋊C₈	1	4	yes	—
219	C₂×C₃	#5012	1	4	yes	—
220	C₂×C₃	Q₈×C₃⋊C₈	1	4	yes	—
221	C₂×C₃	#5024	1	4	yes	—
222	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
223	C₂²	C₂×#3748	1	1	yes	characteristic
224	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
225	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
226	C₂²	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
227	C₂²	#64157	1	1	yes	characteristic
228	C₂²	#64157	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	4	yes	complemented
233	C₄	#64097	1	1	yes	characteristic
234	C₄	#64099	1	1	yes	characteristic
235	C₄	#64104	1	1	yes	characteristic
236	C₄	#64106	1	1	yes	characteristic
237	C₃	#69484	1	4	yes	complemented
238	C₂	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
239	C₂	#10592683	1	1	yes	characteristic
240	C₂	#10592698	1	1	yes	characteristic
241	C₂	D₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
242	C₂	#10592711	1	1	yes	characteristic
243	C₂	Q₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
244	C₂	#10592723	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#1939
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
29. C₃²×C₄⋊C₄	C₈	semidirect
88. C₃×C₄⋊C₄	C₃⋊C₈	semidirect
112. C₃²×C₄	C₄⋊C₈	semidirect
172. C₄⋊C₄	C₃⋊(C₃⋊C₈)	semidirect
187. C₃×C₄	C₄⋊(C₃⋊C₈)	semidirect
192. C₃²	#1939	semidirect
232. C₄	C₄⋊(C₃⋊(C₃⋊C₈))	semidirect
237. C₃	#69484	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589390	2
#69484	3