Subgroups — C₂²×#64097 #11900715

Order 1152.

4872 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#634	C₂	1	2	yes	maximal
2	C₂²×C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3750	C₂	1	2	yes	maximal
5	C₂²×C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#64097	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#888	—	3	4	no	maximal
8	C₂³×C₃²×C₄	C₂²	1	2	yes	—
9	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃⋊Dic₃	C₂²	1	2	yes	—
11	C₂³×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂³×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×#634	—	2	16	no	—
14	C₂²×#634	C₂²	1	24	yes	—
15	C₂×C₃²×C₄⋊C₄	C₂²	1	12	yes	—
16	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	12	yes	—
17	C₂×#3750	C₂²	1	24	yes	—
18	C₂×C₄×C₃⋊Dic₃	C₂²	1	12	yes	—
19	#64097	C₂²	1	64	yes	direct product factor
20	C₂³×Dic₆	—	3	8	no	—
21	C₂²×C₃×C₄⋊C₄	D₃	1	4	yes	—
22	C₂²×C₄⋊Dic₃	—	3	4	no	—
23	C₂²×Dic₃⋊C₄	—	3	8	no	—
24	C₂²×C₄×Dic₃	—	3	4	no	—
25	C₂×#888	—	3	96	no	—
26	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
27	C₂²×C₃²×C₄	—	2	8	no	—
28	C₂²×C₃²×C₄	Q₈	1	8	yes	complemented
29	C₂²×C₃²×C₄	C₂³	1	12	yes	—
30	C₂²×C₃²×C₄	C₂³	1	6	yes	—
31	C₂²×C₃⋊Dic₃	—	2	8	no	—
32	C₂²×C₃⋊Dic₃	D₄	1	8	yes	—
33	C₂²×C₃⋊Dic₃	Q₈	1	8	yes	complemented
34	C₂²×C₃⋊Dic₃	C₂³	1	12	yes	—
35	C₂²×C₃⋊Dic₃	C₂³	1	6	yes	—
36	C₂²×C₃⋊Dic₃	C₂³	1	6	yes	—
37	C₂×#634	—	2	96	no	—
38	C₂×#634	C₂³	1	32	yes	—
39	C₃²×C₄⋊C₄	C₂³	1	16	yes	—
40	C₄⋊(C₃⋊Dic₃)	C₂³	1	16	yes	—
41	#3750	C₂³	1	32	yes	—
42	C₄×C₃⋊Dic₃	C₂³	1	16	yes	—
43	C₂²×C₄⋊Q₈	—	9	1	no	Sylow
44	C₂³×C₃×C₄	C₂×D₃	1	8	yes	—
45	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
46	C₂³×Dic₃	—	3	4	no	—
47	C₂³×Dic₃	—	3	4	no	—
48	C₂³×Dic₃	—	3	8	no	—
49	C₂²×Dic₆	—	3	96	no	—
50	C₂²×Dic₆	—	6	64	no	—
51	C₂×C₃×C₄⋊C₄	C₂×D₃	1	48	yes	—
52	C₂×C₄⋊Dic₃	—	3	48	no	—
53	C₂×Dic₃⋊C₄	—	3	96	no	—
54	C₂×C₄×Dic₃	—	3	48	no	—
55	#888	—	3	256	no	—
56	C₂³×C₃²	C₂×D₄	1	4	yes	—
57	C₂³×C₃²	C₂×Q₈	1	4	yes	—
58	C₂³×C₃²	C₂×Q₈	1	4	yes	—
59	C₂³×C₃²	C₂⁴	1	3	yes	—
60	C₂×C₃²×C₄	—	2	24	no	—
61	C₂×C₃²×C₄	C₂×Q₈	1	24	yes	complemented
62	C₂×C₃²×C₄	C₂⁴	1	8	yes	—
63	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
64	C₂×C₃⋊Dic₃	—	2	24	no	—
65	C₂×C₃⋊Dic₃	C₂×D₄	1	24	yes	—
66	C₂×C₃⋊Dic₃	C₂×Q₈	1	24	yes	complemented
67	C₂×C₃⋊Dic₃	C₂⁴	1	8	yes	—
68	C₂×C₃⋊Dic₃	C₂⁴	1	4	yes	—
69	C₂×C₃⋊Dic₃	C₂⁴	1	4	yes	—
70	#634	—	2	64	no	—
71	C₂³×Q₈	—	9	2	no	—
72	C₂×C₄⋊Q₈	—	9	24	no	—
73	C₂²×C₄⋊C₄	—	9	2	no	—
74	C₂²×C₄⋊C₄	—	9	1	no	—
75	C₂²×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
76	C₂²×C₄²	—	9	1	no	—
77	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
78	C₂²×C₃×C₄	—	2	32	no	—
79	C₂²×C₃×C₄	Dic₆	1	32	yes	complemented
80	C₂²×C₃×C₄	C₂²×D₃	1	48	yes	—
81	C₂²×C₃×C₄	C₂²×D₃	1	24	yes	—
82	C₂²×Dic₃	—	3	32	no	—
83	C₂²×Dic₃	—	3	32	no	—
84	C₂²×Dic₃	—	3	24	no	—
85	C₂²×Dic₃	—	3	24	no	—
86	C₂²×Dic₃	—	3	48	no	—
87	C₂²×Dic₃	—	6	32	no	—
88	C₂×Dic₆	—	3	128	no	—
89	C₂×Dic₆	—	6	384	no	—
90	C₃×C₄⋊C₄	C₂²×D₃	1	64	yes	—
91	C₄⋊Dic₃	—	3	64	no	—
92	Dic₃⋊C₄	—	3	128	no	—
93	C₄×Dic₃	—	3	64	no	—
94	C₂²×C₃²	C₄⋊Q₈	1	16	yes	complemented
95	C₂²×C₃²	C₂²×D₄	1	6	yes	—
96	C₂²×C₃²	C₂²×Q₈	1	6	yes	—
97	C₂²×C₃²	C₂²×Q₈	1	6	yes	—
98	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
99	C₃²×C₄	—	2	8	no	—
100	C₃²×C₄	C₂²×Q₈	1	8	yes	complemented
101	C₃⋊Dic₃	—	2	8	no	—
102	C₃⋊Dic₃	C₂²×D₄	1	8	yes	—
103	C₃⋊Dic₃	C₂²×Q₈	1	8	yes	complemented
104	C₂²×Q₈	—	9	24	no	—
105	C₂²×Q₈	—	18	16	no	—
106	C₂³×C₄	—	9	1	no	—
107	C₂³×C₄	—	9	1	no	—
108	C₂³×C₄	—	9	2	no	—
109	C₂³×C₄	C₂×C₃⋊D₃	1	2	yes	—
110	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₄⋊Q₈	—	9	64	no	—
112	C₂×C₄⋊C₄	—	9	24	no	—
113	C₂×C₄⋊C₄	—	9	12	no	—
114	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	12	yes	—
115	C₂×C₄²	—	9	12	no	—
116	C₂³×C₃	C₂×Dic₆	1	16	yes	—
117	C₂³×C₃	D₃×D₄	1	16	yes	—
118	C₂³×C₃	D₃×Q₈	1	16	yes	—
119	C₂³×C₃	C₂³×D₃	1	12	yes	—
120	C₂×C₃×C₄	—	2	96	no	—
121	C₂×C₃×C₄	C₂×Dic₆	1	96	yes	complemented
122	C₂×C₃×C₄	C₂³×D₃	1	32	yes	—
123	C₂×C₃×C₄	C₂³×D₃	1	16	yes	—
124	C₂×Dic₃	—	3	96	no	—
125	C₂×Dic₃	—	3	96	no	—
126	C₂×Dic₃	—	3	16	no	—
127	C₂×Dic₃	—	3	16	no	—
128	C₂×Dic₃	—	3	32	no	—
129	C₂×Dic₃	—	6	96	no	—
130	Dic₆	—	6	256	no	—
131	C₂×C₃²	C₂×C₄⋊Q₈	1	12	yes	complemented
132	C₂×C₃²	C₂³×D₄	1	1	yes	characteristic
133	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
134	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
135	C₂⁴	C₂²×C₃⋊D₃	1	1	yes	characteristic
136	C₂×Q₈	—	9	32	no	—
137	C₂×Q₈	—	18	96	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	9	6	no	—
140	C₂²×C₄	—	9	6	no	—
141	C₂²×C₄	—	9	8	no	—
142	C₂²×C₄	—	9	8	no	—
143	C₂²×C₄	—	9	12	no	—
144	C₂²×C₄	—	18	8	no	—
145	C₂²×C₄	#634	1	8	yes	complemented
146	C₂²×C₄	C₂²×C₃⋊D₃	1	12	yes	—
147	C₂²×C₄	C₂²×C₃⋊D₃	1	6	yes	—
148	C₄⋊C₄	—	9	32	no	—
149	C₄⋊C₄	—	9	16	no	—
150	C₄⋊C₄	C₂²×C₃⋊D₃	1	16	yes	—
151	C₄²	—	9	16	no	—
152	C₂²×C₃	#888	1	64	yes	complemented
153	C₂²×C₃	C₂²×Dic₆	1	24	yes	—
154	C₂²×C₃	C₂×D₃×D₄	1	24	yes	—
155	C₂²×C₃	C₂×D₃×Q₈	1	24	yes	—
156	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
157	C₃×C₄	—	2	32	no	—
158	C₃×C₄	C₂²×Dic₆	1	32	yes	complemented
159	Dic₃	—	3	32	no	—
160	Dic₃	—	3	32	no	—
161	Dic₃	—	6	32	no	—
162	C₃²	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×#634	1	4	yes	—
164	C₂³	D₄×C₃⋊D₃	1	4	yes	—
165	C₂³	Q₈×C₃⋊D₃	1	4	yes	—
166	C₂³	C₂³×C₃⋊D₃	1	3	yes	—
167	Q₈	—	18	64	no	—
168	C₂×C₄	—	2	24	no	—
169	C₂×C₄	—	9	24	no	—
170	C₂×C₄	—	9	24	no	—
171	C₂×C₄	—	9	4	no	—
172	C₂×C₄	—	9	4	no	—
173	C₂×C₄	—	9	8	no	—
174	C₂×C₄	—	18	24	no	—
175	C₂×C₄	C₂×#634	1	24	yes	complemented
176	C₂×C₄	C₂³×C₃⋊D₃	1	8	yes	—
177	C₂×C₄	C₂³×C₃⋊D₃	1	4	yes	—
178	C₂×C₃	C₂×#888	1	48	yes	complemented
179	C₂×C₃	C₂³×Dic₆	1	4	yes	—
180	C₂×C₃	C₂²×D₃×D₄	1	4	yes	—
181	C₂×C₃	C₂²×D₃×Q₈	1	4	yes	—
182	C₂²	#64097	1	16	yes	direct product factor
183	C₂²	C₂²×#634	1	6	yes	—
184	C₂²	C₂×D₄×C₃⋊D₃	1	6	yes	—
185	C₂²	C₂×Q₈×C₃⋊D₃	1	6	yes	—
186	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
187	C₄	—	2	8	no	—
188	C₄	—	9	8	no	—
189	C₄	—	9	8	no	—
190	C₄	—	18	8	no	—
191	C₄	C₂²×#634	1	8	yes	complemented
192	C₃	C₂²×#888	1	4	yes	complemented
193	C₂	C₂×#64097	1	12	yes	direct product factor
194	C₂	C₂³×#634	1	1	yes	characteristic
195	C₂	C₂²×D₄×C₃⋊D₃	1	1	yes	characteristic
196	C₂	C₂²×Q₈×C₃⋊D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. C₂×#64097	C₂	direct (N × H)
19. #64097	C₂²	direct (N × H)
28. C₂²×C₃²×C₄	Q₈	semidirect
33. C₂²×C₃⋊Dic₃	Q₈	semidirect
61. C₂×C₃²×C₄	C₂×Q₈	semidirect
66. C₂×C₃⋊Dic₃	C₂×Q₈	semidirect
79. C₂²×C₃×C₄	Dic₆	semidirect
94. C₂²×C₃²	C₄⋊Q₈	semidirect
100. C₃²×C₄	C₂²×Q₈	semidirect
103. C₃⋊Dic₃	C₂²×Q₈	semidirect
121. C₂×C₃×C₄	C₂×Dic₆	semidirect
131. C₂×C₃²	C₂×C₄⋊Q₈	semidirect
145. C₂²×C₄	#634	semidirect
152. C₂²×C₃	#888	semidirect
158. C₃×C₄	C₂²×Dic₆	semidirect
162. C₃²	C₂²×C₄⋊Q₈	semidirect
175. C₂×C₄	C₂×#634	semidirect
178. C₂×C₃	C₂×#888	semidirect
182. C₂²	#64097	direct (N × H)
191. C₄	C₂²×#634	semidirect
192. C₃	C₂²×#888	semidirect
193. C₂	C₂×#64097	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#634	2
C₂²×C₃²×C₄⋊C₄	2
C₂²×C₄⋊(C₃⋊Dic₃)	2
C₂²×#3750	2
C₂²×C₄×C₃⋊Dic₃	2
C₂×#64097	2
C₂²×#888	3

Subgroups — C22×#64097 #11900715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64097 #11900715