Subgroups — C₂²×#64133 #11900321

Order 1152.

5520 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#3757	C₂	1	2	yes	maximal
4	C₂²×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64133	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#3750	C₂	1	2	yes	maximal, complemented
7	C₂²×#924	—	3	4	no	maximal
8	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
9	C₂³×C₃²×C₄	—	2	8	no	—
10	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃²×C₄	C₂²	1	12	yes	—
12	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃⋊Dic₃	C₂²	1	2	yes	—
14	C₂³×C₃⋊Dic₃	C₂²	1	2	yes	—
15	C₂²×#634	—	2	8	no	—
16	C₂²×#634	C₂²	1	12	yes	complemented
17	C₂×#3757	C₂²	1	24	yes	—
18	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	12	yes	complemented
19	#64133	C₂²	1	64	yes	direct product factor
20	C₂×#3750	C₂²	1	24	yes	complemented
21	C₂⁴×C₃×C₄	D₃	1	4	yes	—
22	C₂³×Dic₆	—	3	4	no	—
23	C₂²×#217	—	3	8	no	—
24	C₂²×C₄⋊Dic₃	—	3	4	no	—
25	C₂×#924	—	3	96	no	—
26	C₂²×Dic₃⋊C₄	—	3	8	no	—
27	C₂⁴×C₃²	—	2	4	no	—
28	C₂⁴×C₃²	—	2	4	no	—
29	C₂⁴×C₃²	Q₈	1	8	yes	complemented
30	C₂⁴×C₃²	C₂³	1	6	yes	—
31	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
32	C₂²×C₃²×C₄	—	2	4	no	—
33	C₂²×C₃²×C₄	—	2	48	no	—
34	C₂²×C₃²×C₄	D₄	1	8	yes	—
35	C₂²×C₃²×C₄	C₂³	1	6	yes	—
36	C₂²×C₃²×C₄	C₂³	1	16	yes	—
37	C₂²×C₃²×C₄	C₂³	1	6	yes	—
38	C₂²×C₃⋊Dic₃	—	2	8	no	—
39	C₂²×C₃⋊Dic₃	—	2	8	no	—
40	C₂²×C₃⋊Dic₃	C₂³	1	12	yes	—
41	C₂²×C₃⋊Dic₃	C₂³	1	12	yes	—
42	C₂×#634	—	2	48	no	—
43	C₂×#634	C₂³	1	16	yes	complemented
44	#3757	C₂³	1	32	yes	—
45	C₄⋊(C₃⋊Dic₃)	C₂³	1	16	yes	complemented
46	#3750	C₂³	1	32	yes	complemented
47	C₂²×C₂²⋊Q₈	—	9	1	no	Sylow
48	C₂⁵×C₃	C₂×D₃	1	4	yes	—
49	C₂³×C₃×C₄	—	2	32	no	—
50	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
51	C₂³×C₃×C₄	C₂×D₃	1	48	yes	—
52	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
53	C₂³×Dic₃	—	3	8	no	—
54	C₂³×Dic₃	—	3	8	no	—
55	C₂²×Dic₆	—	3	48	no	—
56	C₂²×Dic₆	—	6	32	no	—
57	C₂×#217	—	3	96	no	—
58	C₂×C₄⋊Dic₃	—	3	48	no	—
59	#924	—	3	256	no	—
60	C₂×Dic₃⋊C₄	—	3	96	no	—
61	C₂³×C₃²	—	2	32	no	—
62	C₂³×C₃²	—	2	12	no	—
63	C₂³×C₃²	—	2	12	no	—
64	C₂³×C₃²	C₂×D₄	1	4	yes	—
65	C₂³×C₃²	C₂×Q₈	1	4	yes	—
66	C₂³×C₃²	C₂×Q₈	1	24	yes	complemented
67	C₂³×C₃²	#41	1	4	yes	—
68	C₂³×C₃²	C₂⁴	1	4	yes	—
69	C₂³×C₃²	C₂⁴	1	3	yes	—
70	C₂×C₃²×C₄	—	2	12	no	—
71	C₂×C₃²×C₄	—	2	32	no	—
72	C₂×C₃²×C₄	C₂×D₄	1	24	yes	—
73	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
74	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
75	C₂×C₃⋊Dic₃	—	2	24	no	—
76	C₂×C₃⋊Dic₃	—	2	24	no	—
77	C₂×C₃⋊Dic₃	C₂⁴	1	8	yes	—
78	C₂×C₃⋊Dic₃	C₂⁴	1	8	yes	—
79	#634	—	2	32	no	—
80	C₂³×Q₈	—	9	1	no	—
81	C₂⁴×C₄	C₃⋊D₃	1	1	yes	characteristic
82	C₂×C₂²⋊Q₈	—	9	24	no	—
83	C₂²×C₄⋊C₄	—	9	2	no	—
84	C₂²×C₄⋊C₄	—	9	1	no	—
85	C₂²×C₂²⋊C₄	—	9	2	no	—
86	C₂⁴×C₃	—	2	16	no	—
87	C₂⁴×C₃	—	2	16	no	—
88	C₂⁴×C₃	Dic₆	1	32	yes	complemented
89	C₂⁴×C₃	C₂²×D₃	1	24	yes	—
90	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
91	C₂²×C₃×C₄	—	2	16	no	—
92	C₂²×C₃×C₄	—	2	192	no	—
93	C₂²×C₃×C₄	#67	1	32	yes	—
94	C₂²×C₃×C₄	C₂²×D₃	1	24	yes	—
95	C₂²×C₃×C₄	C₂²×D₃	1	64	yes	—
96	C₂²×C₃×C₄	C₂²×D₃	1	24	yes	—
97	C₂²×Dic₃	—	3	48	no	—
98	C₂²×Dic₃	—	3	48	no	—
99	C₂²×Dic₃	—	6	32	no	—
100	C₂²×Dic₃	—	6	32	no	—
101	C₂×Dic₆	—	3	64	no	—
102	C₂×Dic₆	—	6	192	no	—
103	#217	—	3	128	no	—
104	C₄⋊Dic₃	—	3	64	no	—
105	Dic₃⋊C₄	—	3	128	no	—
106	C₂²×C₃²	—	2	48	no	—
107	C₂²×C₃²	—	2	4	no	—
108	C₂²×C₃²	—	2	4	no	—
109	C₂²×C₃²	C₂²⋊Q₈	1	16	yes	complemented
110	C₂²×C₃²	C₂²×D₄	1	6	yes	—
111	C₂²×C₃²	C₂²×Q₈	1	6	yes	—
112	C₂²×C₃²	C₂²×Q₈	1	8	yes	complemented
113	C₂²×C₃²	C₂×#41	1	6	yes	—
114	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
115	C₃²×C₄	—	2	4	no	—
116	C₃²×C₄	C₂²×D₄	1	8	yes	—
117	C₃⋊Dic₃	—	2	8	no	—
118	C₃⋊Dic₃	—	2	8	no	—
119	C₂⁵	C₂×C₃⋊D₃	1	1	yes	characteristic
120	C₂²×Q₈	—	9	12	no	—
121	C₂²×Q₈	—	18	8	no	—
122	C₂³×C₄	—	2	8	no	—
123	C₂³×C₄	—	9	2	no	—
124	C₂³×C₄	—	9	2	no	—
125	C₂³×C₄	C₂×C₃⋊D₃	1	12	yes	—
126	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
127	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
128	C₂²⋊Q₈	—	9	64	no	—
129	C₂×C₄⋊C₄	—	9	24	no	—
130	C₂×C₄⋊C₄	—	9	12	no	—
131	C₂×C₂²⋊C₄	—	9	24	no	—
132	C₂³×C₃	—	2	128	no	—
133	C₂³×C₃	—	2	48	no	—
134	C₂³×C₃	—	2	48	no	—
135	C₂³×C₃	C₂×Dic₆	1	16	yes	—
136	C₂³×C₃	C₂×Dic₆	1	96	yes	complemented
137	C₂³×C₃	#235	1	16	yes	—
138	C₂³×C₃	C₂×#67	1	16	yes	—
139	C₂³×C₃	C₂³×D₃	1	16	yes	—
140	C₂³×C₃	C₂³×D₃	1	12	yes	—
141	C₂×C₃×C₄	—	2	48	no	—
142	C₂×C₃×C₄	—	2	128	no	—
143	C₂×C₃×C₄	C₂×#67	1	96	yes	—
144	C₂×C₃×C₄	C₂³×D₃	1	16	yes	—
145	C₂×C₃×C₄	C₂³×D₃	1	16	yes	—
146	C₂×Dic₃	—	3	32	no	—
147	C₂×Dic₃	—	3	32	no	—
148	C₂×Dic₃	—	6	96	no	—
149	C₂×Dic₃	—	6	96	no	—
150	Dic₆	—	6	128	no	—
151	C₂×C₃²	—	2	8	no	—
152	C₂×C₃²	C₂×C₂²⋊Q₈	1	12	yes	complemented
153	C₂×C₃²	C₂³×D₄	1	1	yes	characteristic
154	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
155	C₂×C₃²	C₂²×#41	1	1	yes	characteristic
156	C₂⁴	—	2	4	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	#634	1	8	yes	complemented
159	C₂⁴	C₂²×C₃⋊D₃	1	6	yes	—
160	C₂⁴	C₂²×C₃⋊D₃	1	1	yes	characteristic
161	C₂×Q₈	—	9	16	no	—
162	C₂×Q₈	—	18	48	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	48	no	—
165	C₂²×C₄	—	9	12	no	—
166	C₂²×C₄	—	9	12	no	—
167	C₂²×C₄	—	18	8	no	—
168	C₂²×C₄	—	18	8	no	—
169	C₂²×C₄	#638	1	8	yes	—
170	C₂²×C₄	C₂²×C₃⋊D₃	1	6	yes	—
171	C₂²×C₄	C₂²×C₃⋊D₃	1	16	yes	—
172	C₂²×C₄	C₂²×C₃⋊D₃	1	6	yes	—
173	C₄⋊C₄	—	9	32	no	—
174	C₄⋊C₄	—	9	16	no	—
175	C₂²⋊C₄	—	9	32	no	—
176	C₂²×C₃	—	2	16	no	—
177	C₂²×C₃	—	2	16	no	—
178	C₂²×C₃	—	2	192	no	—
179	C₂²×C₃	#924	1	64	yes	complemented
180	C₂²×C₃	C₂²×Dic₆	1	24	yes	—
181	C₂²×C₃	C₂²×Dic₆	1	32	yes	complemented
182	C₂²×C₃	C₂×#235	1	24	yes	—
183	C₂²×C₃	C₂²×#67	1	24	yes	—
184	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
185	C₃×C₄	—	2	16	no	—
186	C₃×C₄	C₂²×#67	1	32	yes	—
187	Dic₃	—	6	32	no	—
188	Dic₃	—	6	32	no	—
189	C₃²	C₂²×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	2	12	no	—
191	C₂³	—	2	12	no	—
192	C₂³	—	2	32	no	—
193	C₂³	C₂×#634	1	4	yes	—
194	C₂³	C₂×#634	1	24	yes	complemented
195	C₂³	#3828	1	4	yes	—
196	C₂³	C₂×#638	1	4	yes	—
197	C₂³	C₂³×C₃⋊D₃	1	4	yes	—
198	C₂³	C₂³×C₃⋊D₃	1	3	yes	—
199	Q₈	—	18	32	no	—
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	—	2	32	no	—
202	C₂×C₄	—	9	8	no	—
203	C₂×C₄	—	9	8	no	—
204	C₂×C₄	—	18	24	no	—
205	C₂×C₄	—	18	24	no	—
206	C₂×C₄	C₂×#638	1	24	yes	—
207	C₂×C₄	C₂³×C₃⋊D₃	1	4	yes	—
208	C₂×C₄	C₂³×C₃⋊D₃	1	4	yes	—
209	C₂×C₃	—	2	32	no	—
210	C₂×C₃	C₂×#924	1	48	yes	complemented
211	C₂×C₃	C₂³×Dic₆	1	4	yes	—
212	C₂×C₃	C₂²×#235	1	4	yes	—
213	C₂×C₃	C₂³×#67	1	4	yes	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	48	no	—
217	C₂²	#64133	1	16	yes	direct product factor
218	C₂²	C₂²×#634	1	6	yes	—
219	C₂²	C₂²×#634	1	8	yes	complemented
220	C₂²	C₂×#3828	1	6	yes	—
221	C₂²	C₂²×#638	1	6	yes	—
222	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	18	8	no	—
225	C₄	—	18	8	no	—
226	C₄	C₂²×#638	1	8	yes	—
227	C₃	C₂²×#924	1	4	yes	complemented
228	C₂	—	2	8	no	—
229	C₂	C₂×#64133	1	12	yes	direct product factor
230	C₂	C₂³×#634	1	1	yes	characteristic
231	C₂	C₂²×#3828	1	1	yes	characteristic
232	C₂	C₂³×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁵
Frattini	C₂²	C₂⁴×C₃⋊D₃
Fitting	C₂⁴×C₃²×C₄	C₂
socle	C₂⁴×C₃²	C₂³
prime radical	C₂⁵×C₃²	C₂²
hypercenter	C₂⁴×C₄	C₃⋊D₃
hypocenter	C₃²	C₂²×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×#634	C₂	semidirect
4. C₂²×C₄⋊(C₃⋊Dic₃)	C₂	semidirect
5. C₂×#64133	C₂	direct (N × H)
6. C₂²×#3750	C₂	semidirect
16. C₂²×#634	C₂²	semidirect
18. C₂×C₄⋊(C₃⋊Dic₃)	C₂²	semidirect
19. #64133	C₂²	direct (N × H)
20. C₂×#3750	C₂²	semidirect
29. C₂⁴×C₃²	Q₈	semidirect
43. C₂×#634	C₂³	semidirect
45. C₄⋊(C₃⋊Dic₃)	C₂³	semidirect
46. #3750	C₂³	semidirect
66. C₂³×C₃²	C₂×Q₈	semidirect
88. C₂⁴×C₃	Dic₆	semidirect
109. C₂²×C₃²	C₂²⋊Q₈	semidirect
112. C₂²×C₃²	C₂²×Q₈	semidirect
136. C₂³×C₃	C₂×Dic₆	semidirect
152. C₂×C₃²	C₂×C₂²⋊Q₈	semidirect
158. C₂⁴	#634	semidirect
179. C₂²×C₃	#924	semidirect
181. C₂²×C₃	C₂²×Dic₆	semidirect
189. C₃²	C₂²×C₂²⋊Q₈	semidirect
194. C₂³	C₂×#634	semidirect
210. C₂×C₃	C₂×#924	semidirect
217. C₂²	#64133	direct (N × H)
219. C₂²	C₂²×#634	semidirect
227. C₃	C₂²×#924	semidirect
229. C₂	C₂×#64133	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²×C₄	2
C₂³×#634	2
C₂²×#3757	2
C₂²×C₄⋊(C₃⋊Dic₃)	2
C₂×#64133	2
C₂²×#3750	2
C₂²×#924	3

Subgroups — C22×#64133 #11900321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64133 #11900321