Subgroups — C₂×#10589100 #11776789

Order 1152.

481 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	#10589100	C₂	1	4	yes	maximal, direct product factor
4	C₂×#4479	—	3	2	no	maximal
5	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₃⋊C₈	—	2	2	no	—
9	C₄×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
10	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
11	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
12	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₈	—	3	2	no	—
14	C₂×C₄⋊(C₃⋊C₈)	D₃	1	2	yes	—
15	C₂×C₄×C₃⋊C₈	—	3	2	no	—
16	C₂×C₄×C₃⋊C₈	—	6	1	no	—
17	#4479	—	3	8	no	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
21	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₃²×C₄²	D₄	1	1	yes	characteristic
23	C₃²×C₄²	Q₈	1	1	yes	characteristic
24	C₃²×C₄²	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
26	C₂×C₃⋊(C₃⋊C₈)	C₈	1	4	yes	complemented
27	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
28	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₃⋊C₈	—	2	4	no	—
30	C₂×C₃×C₃⋊C₈	—	4	4	no	—
31	C₂×#335	—	9	1	no	Sylow
32	C₂²×C₃×C₈	—	6	2	no	—
33	C₂×C₃×C₄²	—	2	1	no	—
34	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
35	C₂²×C₃⋊C₈	—	2	2	no	—
36	C₂²×C₃⋊C₈	—	3	2	no	—
37	C₂²×C₃⋊C₈	—	3	2	no	—
38	C₂²×C₃⋊C₈	—	6	1	no	—
39	C₂²×C₃⋊C₈	—	6	1	no	—
40	C₃×C₄⋊C₈	—	3	4	no	—
41	C₃×C₄⋊C₈	—	6	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₃⋊C₈)	C₂×D₃	1	4	yes	—
44	C₄×C₃⋊C₈	—	3	4	no	—
45	C₄×C₃⋊C₈	—	3	4	no	—
46	C₄×C₃⋊C₈	—	6	2	no	—
47	C₄×C₃⋊C₈	—	6	2	no	—
48	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
54	C₂×C₃²×C₄	M₁₆	1	2	yes	—
55	C₂×C₃²×C₄	D₈	1	2	yes	—
56	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
57	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₃⋊(C₃⋊C₈)	—	2	2	no	—
61	C₃⋊(C₃⋊C₈)	C₂×C₈	1	4	yes	complemented
62	C₃×C₃⋊C₈	—	4	4	no	—
63	C₂×C₄⋊C₈	—	3	2	no	—
64	C₂×C₄×C₈	—	9	1	no	—
65	#335	—	9	4	no	—
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	Dic₆	1	2	yes	—
70	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
71	C₂²×C₃×C₄	#67	1	2	yes	—
72	C₂×C₃×C₈	—	6	4	no	—
73	C₂×C₃×C₈	—	12	4	no	—
74	C₃×C₄²	—	2	1	no	—
75	C₃×C₄²	—	2	1	no	—
76	C₃×C₄²	—	2	1	no	—
77	C₃×C₄²	—	2	1	no	—
78	C₃×C₄²	Dic₆	1	2	yes	—
79	C₃×C₄²	C₄×D₃	1	2	yes	—
80	C₃×C₄²	#67	1	2	yes	—
81	C₃×C₄²	C₂²×D₃	1	2	yes	—
82	C₂×C₃⋊C₈	—	2	4	no	—
83	C₂×C₃⋊C₈	—	3	8	no	—
84	C₂×C₃⋊C₈	—	3	4	no	—
85	C₂×C₃⋊C₈	—	3	4	no	—
86	C₂×C₃⋊C₈	—	4	4	no	—
87	C₂×C₃⋊C₈	—	6	2	no	—
88	C₂×C₃⋊C₈	—	6	2	no	—
89	C₂×C₃⋊C₈	—	6	4	no	—
90	C₂×C₃⋊C₈	—	6	4	no	—
91	C₂×C₃⋊C₈	—	6	4	no	—
92	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
93	C₂²×C₃²	#107	1	2	yes	—
94	C₂²×C₃²	#108	1	2	yes	—
95	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₃²×C₄	—	2	1	no	—
97	C₃²×C₄	—	2	1	no	—
98	C₃²×C₄	C₄⋊C₈	1	2	yes	—
99	C₃²×C₄	#107	1	2	yes	—
100	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
101	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
102	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
103	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
104	C₂²×C₈	—	6	2	no	—
105	C₂²×C₈	—	9	1	no	—
106	C₂²×C₈	—	9	1	no	—
107	C₂×C₄²	D₃²	1	1	yes	characteristic
108	C₄⋊C₈	—	3	4	no	—
109	C₄⋊C₈	—	6	2	no	—
110	C₄×C₈	—	9	2	no	—
111	C₄×C₈	—	9	2	no	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
114	C₂×C₃×C₄	—	2	4	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
127	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
128	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
129	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
130	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
131	C₂×C₃×C₄	#213	1	4	yes	—
132	C₂×C₃×C₄	#216	1	4	yes	—
133	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
134	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
135	C₂×C₃×C₄	C₂×#67	1	2	yes	—
136	C₃×C₈	—	12	4	no	—
137	C₃⋊C₈	—	3	8	no	—
138	C₃⋊C₈	—	4	4	no	—
139	C₃⋊C₈	—	6	4	no	—
140	C₃⋊C₈	—	6	4	no	—
141	C₃⋊C₈	—	6	4	no	—
142	C₂×C₃²	#335	1	4	yes	complemented
143	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₂×C₃²	C₂×#107	1	1	yes	characteristic
145	C₂×C₃²	C₂×#108	1	1	yes	characteristic
146	C₂²×C₄	#624	1	1	yes	characteristic
147	C₂²×C₄	#625	1	1	yes	characteristic
148	C₂²×C₄	#627	1	1	yes	characteristic
149	C₂×C₈	—	6	4	no	—
150	C₂×C₈	—	9	4	no	—
151	C₂×C₈	—	9	2	no	—
152	C₂×C₈	—	9	2	no	—
153	C₂×C₈	—	12	4	no	—
154	C₂×C₈	—	18	2	no	—
155	C₄²	#624	1	1	yes	characteristic
156	C₄²	#625	1	1	yes	characteristic
157	C₄²	#627	1	1	yes	characteristic
158	C₄²	C₂×D₃²	1	1	yes	characteristic
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	#807	1	4	yes	—
164	C₂²×C₃	Dic₃⋊C₈	1	4	yes	—
165	C₂²×C₃	#822	1	4	yes	—
166	C₂²×C₃	C₂×Dic₃⋊C₄	1	2	yes	—
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	#807	1	4	yes	—
178	C₃×C₄	Dic₃⋊C₈	1	4	yes	—
179	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
180	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
181	C₃×C₄	C₂×#213	1	2	yes	—
182	C₃×C₄	C₂×#216	1	2	yes	—
183	C₃²	C₂×#335	1	1	yes	characteristic, complemented, Sylow
184	C₂³	#3724	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	#3710	1	2	yes	—
187	C₂×C₄	#3712	1	2	yes	—
188	C₂×C₄	#3713	1	2	yes	—
189	C₂×C₄	#3718	1	2	yes	—
190	C₂×C₄	#3724	1	1	yes	characteristic
191	C₂×C₄	#3724	1	1	yes	characteristic
192	C₂×C₄	#3724	1	1	yes	characteristic
193	C₂×C₄	C₂×#624	1	1	yes	characteristic
194	C₂×C₄	C₂×#625	1	1	yes	characteristic
195	C₂×C₄	C₂×#627	1	1	yes	characteristic
196	C₈	—	9	4	no	—
197	C₈	—	12	4	no	—
198	C₈	—	18	2	no	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	#4479	1	8	yes	complemented
204	C₂×C₃	C₂×#807	1	2	yes	—
205	C₂×C₃	C₂×Dic₃⋊C₈	1	2	yes	—
206	C₂×C₃	C₂×#822	1	2	yes	—
207	C₂²	#63572	1	2	yes	—
208	C₂²	#63576	1	2	yes	—
209	C₂²	#63578	1	2	yes	—
210	C₂²	C₂×#3724	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	#63572	1	2	yes	—
214	C₄	#63576	1	2	yes	—
215	C₄	C₂×#3710	1	1	yes	characteristic
216	C₄	C₂×#3712	1	1	yes	characteristic
217	C₄	C₂×#3713	1	1	yes	characteristic
218	C₄	C₂×#3718	1	1	yes	characteristic
219	C₃	—	2	1	no	—
220	C₃	C₂×#4479	1	2	yes	complemented
221	C₂	#10589100	1	4	yes	direct product factor
222	C₂	C₂×#63572	1	1	yes	characteristic
223	C₂	C₂×#63576	1	1	yes	characteristic
224	C₂	C₂×#63578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#625
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	C₂×#335
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #10589100	C₂	direct (N × H)
26. C₂×C₃⋊(C₃⋊C₈)	C₈	semidirect
61. C₃⋊(C₃⋊C₈)	C₂×C₈	semidirect
142. C₂×C₃²	#335	semidirect
183. C₃²	C₂×#335	semidirect
203. C₂×C₃	#4479	semidirect
220. C₃	C₂×#4479	semidirect
221. C₂	#10589100	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×C₃⋊(C₃⋊C₈)	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
#10589100	2
C₂×#4479	3

Subgroups — C2×#10589100 #11776789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589100 #11776789