Subgroups — C₂×Q₈×#634 #11903719

Order 1152.

3336 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	C₂×C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	Q₈×#634	C₂	1	16	yes	maximal, direct product factor
4	C₂×#64097	C₂	1	6	yes	maximal
5	C₂×#64078	C₂	1	3	yes	maximal
6	C₂×C₄×#634	C₂	1	3	yes	maximal
7	C₂×Q₈×Dic₆	—	3	4	no	maximal
8	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×#634	C₂²	1	6	yes	—
10	C₂²×#634	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Q₈	C₂²	1	8	yes	—
12	C₂×C₃²×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂×C₃²×C₄²	C₂²	1	3	yes	—
14	Q₈×C₃⋊Dic₃	C₂²	1	16	yes	—
15	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	3	yes	—
16	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
17	C₂×#3750	C₂²	1	6	yes	—
18	C₂×C₄×C₃⋊Dic₃	C₂²	1	6	yes	—
19	#64097	C₂²	1	48	yes	—
20	#64078	C₂²	1	24	yes	—
21	C₄×#634	C₂²	1	24	yes	—
22	C₂×C₃×C₄×Q₈	D₃	1	4	yes	—
23	C₂×Q₈×Dic₃	—	3	8	no	—
24	Q₈×Dic₆	—	3	64	no	—
25	C₂×#888	—	3	24	no	—
26	C₂×#869	—	3	12	no	—
27	C₂×C₄×Dic₆	—	3	12	no	—
28	C₂×C₃²×Q₈	Q₈	1	8	yes	complemented
29	C₂×C₃²×Q₈	C₂³	1	4	yes	—
30	C₂²×C₃²×C₄	C₂³	1	3	yes	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂³	1	3	yes	—
33	C₂²×C₃⋊Dic₃	C₂³	1	6	yes	—
34	C₂²×C₃⋊Dic₃	C₂³	1	2	yes	—
35	C₂×#634	—	4	12	no	—
36	C₂×#634	Q₈	1	8	yes	direct product factor
37	C₂×#634	C₂³	1	24	yes	—
38	C₂×#634	C₂³	1	4	yes	—
39	C₃²×C₄⋊C₄	C₂³	1	12	yes	—
40	C₃²×C₄²	C₂³	1	12	yes	—
41	C₄⋊(C₃⋊Dic₃)	C₂³	1	12	yes	—
42	C₄⋊(C₃⋊Dic₃)	C₂³	1	24	yes	—
43	#3750	C₂³	1	24	yes	—
44	C₄×C₃⋊Dic₃	C₂³	1	24	yes	—
45	C₂×Q₈²	—	9	1	no	Sylow
46	C₂²×C₃×Q₈	C₂×D₃	1	4	yes	—
47	C₂²×Dic₆	—	3	4	no	—
48	C₂²×Dic₆	—	3	24	no	—
49	C₃×C₄×Q₈	C₂×D₃	1	32	yes	—
50	C₂×C₃×C₄⋊C₄	C₂×D₃	1	12	yes	—
51	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
52	Q₈×Dic₃	—	3	64	no	—
53	C₂×C₄⋊Dic₃	—	3	12	no	—
54	C₂×C₄⋊Dic₃	—	3	24	no	—
55	C₂×Dic₃⋊C₄	—	3	24	no	—
56	C₂×C₄×Dic₃	—	3	24	no	—
57	#888	—	3	192	no	—
58	#869	—	3	96	no	—
59	C₄×Dic₆	—	3	96	no	—
60	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
61	C₃²×Q₈	C₂×Q₈	1	16	yes	complemented
62	C₂×C₃²×C₄	—	2	6	no	—
63	C₂×C₃²×C₄	C₂×Q₈	1	12	yes	—
64	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
65	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
66	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
67	C₂×C₃²×C₄	C₂⁴	1	2	yes	—
68	C₂×C₃⋊Dic₃	—	2	12	no	—
69	C₂×C₃⋊Dic₃	C₂×Q₈	1	8	yes	—
70	C₂×C₃⋊Dic₃	C₂⁴	1	12	yes	—
71	C₂×C₃⋊Dic₃	C₂⁴	1	4	yes	—
72	#634	—	4	24	no	—
73	#634	C₂×Q₈	1	16	yes	direct product factor
74	Q₈²	—	9	16	no	—
75	C₂×C₄⋊Q₈	—	9	6	no	—
76	C₂×C₄⋊Q₈	—	9	3	no	—
77	C₂×C₄×Q₈	—	9	2	no	—
78	C₂×C₄×Q₈	—	9	3	no	—
79	C₂×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
80	C₂×C₃×Q₈	Dic₆	1	32	yes	complemented
81	C₂×C₃×Q₈	C₂²×D₃	1	16	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
83	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
85	C₂²×Dic₃	—	3	8	no	—
86	C₂²×Dic₃	—	3	24	no	—
87	C₂×Dic₆	—	3	32	no	—
88	C₂×Dic₆	—	3	16	no	—
89	C₂×Dic₆	—	3	96	no	—
90	C₂×Dic₆	—	12	48	no	—
91	C₃×C₄⋊C₄	C₂²×D₃	1	48	yes	—
92	C₃×C₄²	C₂²×D₃	1	48	yes	—
93	C₄⋊Dic₃	—	3	48	no	—
94	C₄⋊Dic₃	—	3	96	no	—
95	Dic₃⋊C₄	—	3	96	no	—
96	C₄×Dic₃	—	3	96	no	—
97	C₂²×C₃²	C₂²×Q₈	1	2	yes	—
98	C₂²×C₃²	C₂²×Q₈	1	2	yes	—
99	C₂²×C₃²	H(5,𝔽₂)	1	2	yes	—
100	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
101	C₃²×C₄	—	2	6	no	—
102	C₃²×C₄	C₂²×Q₈	1	12	yes	—
103	C₃²×C₄	C₂²×Q₈	1	4	yes	—
104	C₃⋊Dic₃	—	2	12	no	—
105	C₃⋊Dic₃	C₂²×Q₈	1	8	yes	—
106	C₂²×Q₈	—	9	1	no	—
107	C₂²×Q₈	—	9	6	no	—
108	C₂²×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₄⋊Q₈	—	9	48	no	—
110	C₄⋊Q₈	—	9	24	no	—
111	C₄×Q₈	—	9	16	no	—
112	C₄×Q₈	—	9	24	no	—
113	C₄×Q₈	C₂×C₃⋊D₃	1	8	yes	—
114	C₂×C₄⋊C₄	—	9	6	no	—
115	C₂×C₄⋊C₄	—	9	6	no	—
116	C₂×C₄⋊C₄	—	9	3	no	—
117	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	3	yes	—
118	C₂×C₄²	—	9	6	no	—
119	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
120	C₂³×C₃	C₂³×D₃	1	4	yes	—
121	C₃×Q₈	C₂×Dic₆	1	64	yes	complemented
122	C₂×C₃×C₄	—	2	24	no	—
123	C₂×C₃×C₄	C₂×Dic₆	1	48	yes	—
124	C₂×C₃×C₄	D₃×Q₈	1	16	yes	—
125	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
126	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
127	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
128	C₂×Dic₃	—	3	32	no	—
129	C₂×Dic₃	—	3	16	no	—
130	C₂×Dic₃	—	3	48	no	—
131	C₂×Dic₃	—	6	48	no	—
132	Dic₆	—	3	64	no	—
133	Dic₆	—	12	96	no	—
134	C₂×C₃²	Q₈²	1	4	yes	complemented
135	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
136	C₂×C₃²	C₂³×Q₈	1	1	yes	characteristic
137	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×Q₈	—	9	4	no	—
139	C₂×Q₈	—	9	24	no	—
140	C₂×Q₈	—	9	8	no	—
141	C₂×Q₈	—	36	12	no	—
142	C₂×Q₈	#634	1	8	yes	direct product factor
143	C₂×Q₈	C₂²×C₃⋊D₃	1	4	yes	—
144	C₂²×C₄	—	9	2	no	—
145	C₂²×C₄	—	9	6	no	—
146	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
147	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
148	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
149	C₄⋊C₄	—	9	24	no	—
150	C₄⋊C₄	—	9	24	no	—
151	C₄⋊C₄	—	9	12	no	—
152	C₄⋊C₄	C₂²×C₃⋊D₃	1	12	yes	—
153	C₄²	—	9	24	no	—
154	C₄²	C₂²×C₃⋊D₃	1	12	yes	—
155	C₂²×C₃	C₂²×Dic₆	1	8	yes	—
156	C₂²×C₃	C₂×D₃×Q₈	1	8	yes	—
157	C₂²×C₃	#1009	1	8	yes	—
158	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
159	C₃×C₄	—	2	24	no	—
160	C₃×C₄	C₂²×Dic₆	1	48	yes	—
161	C₃×C₄	C₂×D₃×Q₈	1	16	yes	—
162	Dic₃	—	3	32	no	—
163	Dic₃	—	6	48	no	—
164	C₃²	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂³×C₃⋊D₃	1	1	yes	characteristic
166	Q₈	—	9	16	no	—
167	Q₈	—	36	24	no	—
168	Q₈	C₂×#634	1	16	yes	direct product factor
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	9	8	no	—
171	C₂×C₄	—	9	4	no	—
172	C₂×C₄	—	9	12	no	—
173	C₂×C₄	—	18	12	no	—
174	C₂×C₄	C₂×#634	1	12	yes	—
175	C₂×C₄	Q₈×C₃⋊D₃	1	4	yes	—
176	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
177	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
178	C₂×C₄	C₂³×C₃⋊D₃	1	2	yes	—
179	C₂×C₃	Q₈×Dic₆	1	16	yes	complemented
180	C₂×C₃	C₂³×Dic₆	1	4	yes	—
181	C₂×C₃	C₂²×D₃×Q₈	1	4	yes	—
182	C₂×C₃	C₂×#1009	1	4	yes	—
183	C₂²	C₂²×#634	1	2	yes	—
184	C₂²	C₂×Q₈×C₃⋊D₃	1	2	yes	—
185	C₂²	#64366	1	2	yes	—
186	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
187	C₄	—	2	6	no	—
188	C₄	—	9	8	no	—
189	C₄	—	18	12	no	—
190	C₄	C₂²×#634	1	12	yes	—
191	C₄	C₂×Q₈×C₃⋊D₃	1	4	yes	—
192	C₃	C₂×Q₈×Dic₆	1	4	yes	complemented
193	C₂	Q₈×#634	1	4	yes	direct product factor
194	C₂	C₂³×#634	1	1	yes	characteristic
195	C₂	C₂²×Q₈×C₃⋊D₃	1	1	yes	characteristic
196	C₂	C₂×#64366	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁵
Frattini	C₂²	C₂⁴×C₃⋊D₃
Fitting	C₂×C₃²×C₄×Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂⁴
hypercenter	C₂×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×Q₈²
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈×#634	C₂	direct (N × H)
28. C₂×C₃²×Q₈	Q₈	semidirect
36. C₂×#634	Q₈	direct (N × H)
61. C₃²×Q₈	C₂×Q₈	semidirect
73. #634	C₂×Q₈	direct (N × H)
80. C₂×C₃×Q₈	Dic₆	semidirect
121. C₃×Q₈	C₂×Dic₆	semidirect
134. C₂×C₃²	Q₈²	semidirect
142. C₂×Q₈	#634	direct (N × H)
164. C₃²	C₂×Q₈²	semidirect
168. Q₈	C₂×#634	direct (N × H)
179. C₂×C₃	Q₈×Dic₆	semidirect
192. C₃	C₂×Q₈×Dic₆	semidirect
193. C₂	Q₈×#634	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×C₃²×C₄×Q₈	2
C₂×Q₈×C₃⋊Dic₃	2
Q₈×#634	2
C₂×#64097	2
C₂×#64078	2
C₂×C₄×#634	2
C₂×Q₈×Dic₆	3

Subgroups — C2×Q8×#634 #11903719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#634 #11903719