Subgroups — C₂×#10589143 #11797258

Order 1152.

341 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63646	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#10589143	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4506	—	3	1	no	maximal
6	C₂×#4493	—	3	1	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3748	—	2	1	no	—
9	C₂×C₃×#208	—	2	1	no	—
10	#63646	—	2	1	no	—
11	#63646	C₂²	1	2	yes	—
12	C₃×#819	—	2	1	no	—
13	C₃×#819	C₂²	1	2	yes	—
14	C₂×C₃×C₃⋊C₁₆	C₄	1	2	yes	—
15	C₂×C₃×C₃⋊C₁₆	C₄	1	2	yes	—
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂²×C₃×C₁₆	—	3	1	no	—
18	C₂×C₃×#108	—	3	1	no	—
19	C₂×#819	—	3	1	no	—
20	C₂×#819	—	3	1	no	—
21	C₂×#819	—	6	1	no	—
22	C₂×#819	D₃	1	1	yes	characteristic
23	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
24	#4506	—	3	4	no	—
25	#4493	—	3	4	no	—
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	#3748	—	2	2	no	—
34	#3748	—	4	1	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
36	C₃×#208	—	2	2	no	—
37	C₃×#208	—	4	1	no	—
38	C₂×C₃×C₃⋊C₈	—	4	1	no	—
39	C₃×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
40	C₂×#368	—	9	1	no	Sylow
41	C₂×C₃×M₁₆	—	6	1	no	—
42	C₂²×C₃×C₈	—	2	1	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×#208	—	2	1	no	—
46	C₂×#208	—	6	2	no	—
47	C₂×#208	—	6	1	no	—
48	C₂×#208	—	6	1	no	—
49	C₂×C₃×C₁₆	—	3	2	no	—
50	C₂×C₃×C₁₆	—	3	2	no	—
51	C₂×C₃×C₁₆	—	3	2	no	—
52	C₃×#108	—	3	2	no	—
53	C₃×#108	—	6	1	no	—
54	#819	—	2	1	no	—
55	#819	—	3	2	no	—
56	#819	—	3	2	no	—
57	#819	—	6	2	no	—
58	#819	—	6	2	no	—
59	#819	—	6	1	no	—
60	#819	—	6	1	no	—
61	#819	C₂×D₃	1	2	yes	—
62	C₂×C₃⋊C₁₆	Dic₃	1	2	yes	—
63	C₂×C₃⋊C₁₆	Dic₃	1	2	yes	—
64	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
65	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₃⋊(C₃⋊C₈)	—	4	2	no	—
77	C₃×C₃⋊C₈	—	4	2	no	—
78	C₂²×C₁₆	—	3	1	no	—
79	C₂×#108	—	3	1	no	—
80	C₂×#108	—	9	1	no	—
81	#368	—	9	4	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
84	C₂²×C₃×C₄	#67	1	1	yes	characteristic
85	C₃×M₁₆	—	6	2	no	—
86	C₃×M₁₆	—	12	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	12	1	no	—
94	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
95	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
96	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
97	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₃×C₈	#67	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
106	#208	—	2	2	no	—
107	#208	—	4	1	no	—
108	#208	—	6	4	no	—
109	#208	—	6	2	no	—
110	#208	—	6	2	no	—
111	#208	—	12	2	no	—
112	#208	—	12	1	no	—
113	#208	—	12	1	no	—
114	C₂×C₃⋊C₈	—	4	1	no	—
115	C₂×C₃⋊C₈	—	12	2	no	—
116	C₂×C₃⋊C₈	—	12	1	no	—
117	C₂×C₃⋊C₈	—	12	1	no	—
118	C₃×C₁₆	—	3	4	no	—
119	C₃⋊C₁₆	C₂×Dic₃	1	4	yes	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	#107	1	1	yes	characteristic
122	C₂²×C₃²	#107	1	1	yes	characteristic
123	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
124	C₃²×C₄	#107	1	1	yes	characteristic
125	C₃²×C₄	#107	1	1	yes	characteristic
126	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
128	C₂×M₁₆	—	6	1	no	—
129	C₂×M₁₆	—	18	1	no	—
130	C₂²×C₈	D₃²	1	1	yes	characteristic
131	C₂×C₁₆	—	3	2	no	—
132	C₂×C₁₆	—	3	2	no	—
133	C₂×C₁₆	—	3	2	no	—
134	#108	—	3	2	no	—
135	#108	—	6	1	no	—
136	#108	—	9	2	no	—
137	#108	—	18	1	no	—
138	C₂³×C₃	—	2	1	no	—
139	C₂³×C₃	Dic₁₂	1	1	yes	characteristic
140	C₂³×C₃	#216	1	1	yes	characteristic
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
148	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
149	C₂×C₃×C₄	Dic₁₂	1	1	yes	characteristic
150	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
151	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	#213	1	1	yes	characteristic
155	C₂×C₃×C₄	#213	1	1	yes	characteristic
156	C₂×C₃×C₄	#216	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	—	2	1	no	—
163	C₃×C₈	—	12	2	no	—
164	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
165	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
166	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
167	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
168	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
169	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
170	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
171	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
172	C₃⋊C₈	—	4	2	no	—
173	C₃⋊C₈	—	12	4	no	—
174	C₃⋊C₈	—	12	2	no	—
175	C₃⋊C₈	—	12	2	no	—
176	C₂×C₃²	—	2	1	no	—
177	C₂×C₃²	—	2	1	no	—
178	C₂×C₃²	#368	1	2	yes	complemented
179	C₂×C₃²	C₂×#107	1	1	yes	characteristic
180	C₂²×C₄	#626	1	1	yes	characteristic
181	M₁₆	—	6	2	no	—
182	M₁₆	—	12	1	no	—
183	M₁₆	—	18	2	no	—
184	M₁₆	—	36	1	no	—
185	C₂×C₈	—	12	1	no	—
186	C₂×C₈	—	36	1	no	—
187	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
188	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
189	C₂×C₈	#626	1	1	yes	characteristic
190	C₂×C₈	#627	1	1	yes	characteristic
191	C₂×C₈	#627	1	1	yes	characteristic
192	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
193	C₁₆	—	3	4	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	#807	1	1	yes	characteristic
201	C₂²×C₃	#807	1	1	yes	characteristic
202	C₂²×C₃	#818	1	1	yes	characteristic
203	C₂²×C₃	#818	1	1	yes	characteristic
204	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
205	C₂²×C₃	C₂×#216	1	1	yes	characteristic
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	#807	1	1	yes	characteristic
211	C₃×C₄	#807	1	1	yes	characteristic
212	C₃×C₄	#818	1	1	yes	characteristic
213	C₃×C₄	#818	1	1	yes	characteristic
214	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
215	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
216	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
217	C₃×C₄	C₂×#213	1	1	yes	characteristic
218	C₃²	C₂×#368	1	1	yes	characteristic, complemented, Sylow
219	C₂³	#3719	1	1	yes	characteristic
220	C₂×C₄	#3714	1	1	yes	characteristic
221	C₂×C₄	#3714	1	1	yes	characteristic
222	C₂×C₄	#3719	1	1	yes	characteristic
223	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
224	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
225	C₂×C₄	C₂×#626	1	1	yes	characteristic
226	C₈	—	12	2	no	—
227	C₈	—	36	2	no	—
228	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
229	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
230	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
231	C₈	C₂×#627	1	1	yes	characteristic
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	2	no	—
238	C₂×C₃	—	4	1	no	—
239	C₂×C₃	—	4	1	no	—
240	C₂×C₃	#4493	1	2	yes	complemented
241	C₂×C₃	#4506	1	2	yes	complemented
242	C₂×C₃	C₂×#807	1	1	yes	characteristic
243	C₂×C₃	C₂×#818	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	#63575	1	1	yes	characteristic
246	C₂²	#63575	1	1	yes	characteristic
247	C₂²	C₂×#3719	1	1	yes	characteristic
248	C₄	#63575	1	1	yes	characteristic
249	C₄	#63575	1	1	yes	characteristic
250	C₄	C₂×#3714	1	1	yes	characteristic
251	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
252	C₃	—	2	1	no	—
253	C₃	C₂×#4493	1	1	yes	characteristic, complemented
254	C₃	C₂×#4506	1	1	yes	characteristic, complemented
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	#10589143	1	2	yes	direct product factor
258	C₂	C₂×#63575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3714
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	#107
prime radical	C₂³×C₃²	Q₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₂×#368
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #10589143	C₂	direct (N × H)
178. C₂×C₃²	#368	semidirect
218. C₃²	C₂×#368	semidirect
240. C₂×C₃	#4493	semidirect
241. C₂×C₃	#4506	semidirect
253. C₃	C₂×#4493	semidirect
254. C₃	C₂×#4506	semidirect
257. C₂	#10589143	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×#63646	2
C₂×C₃×#819	2
C₂²×C₃×C₃⋊C₁₆	2
#10589143	2
C₂×#4506	3
C₂×#4493	3

Subgroups — C2×#10589143 #11797258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589143 #11797258