Subgroups — C₃×C₄×C₄⋊Q₈ #84171

Order 384.

588 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	2	yes	maximal
3	C₃×#391	C₂	1	2	yes	maximal
4	C₃×#389	C₂	1	4	yes	maximal
5	C₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₃×C₄×C₄⋊C₄	C₂	1	4	yes	maximal
7	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	C₂²	1	2	yes	—
10	C₃×C₄⋊Q₈	C₄	1	8	yes	direct product factor
11	C₃×C₄×Q₈	—	2	8	no	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄²	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄²	C₂²	1	4	yes	—
19	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₃×#95	C₂²	1	8	yes	—
23	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
24	C₂×C₄×Q₈	C₂×C₃	1	2	yes	—
25	#391	C₂×C₃	1	2	yes	—
26	#389	C₂×C₃	1	4	yes	—
27	#384	C₂×C₃	1	1	yes	characteristic
28	C₄×C₄⋊C₄	C₂×C₃	1	4	yes	—
29	C₄³	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×Q₈	—	2	4	no	—
31	C₂×C₃×Q₈	—	2	4	no	—
32	C₂×C₃×Q₈	C₂×C₄	1	8	yes	—
33	C₂²×C₃×C₄	C₂³	1	4	yes	—
34	C₂²×C₃×C₄	C₂³	1	4	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₃×C₄⋊C₄	—	2	8	no	—
41	C₃×C₄⋊C₄	—	2	4	no	—
42	C₃×C₄⋊C₄	—	2	8	no	—
43	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
44	C₃×C₄²	—	2	8	no	—
45	C₃×C₄²	—	2	2	no	—
46	C₃×C₄²	—	2	4	no	—
47	C₃×C₄²	C₂×C₄	1	4	yes	—
48	C₃×C₄²	D₄	1	4	yes	—
49	C₃×C₄²	Q₈	1	8	yes	complemented
50	C₂²×Q₈	C₂²×C₃	1	2	yes	—
51	C₄⋊Q₈	C₃×C₄	1	8	yes	direct product factor
52	C₄×Q₈	—	2	8	no	—
53	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
56	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₄²	C₂²×C₃	1	4	yes	—
58	C₂×C₄²	C₂²×C₃	1	2	yes	—
59	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄²	C₂²×C₃	1	2	yes	—
63	#95	C₂²×C₃	1	8	yes	—
64	C₂³×C₃	C₂⁴	1	1	yes	characteristic
65	C₃×Q₈	—	2	16	no	—
66	C₂×C₃×C₄	—	2	4	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	—	2	2	no	—
69	C₂×C₃×C₄	—	2	2	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	—	2	4	no	—
72	C₂×C₃×C₄	—	2	8	no	—
73	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
74	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
80	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
81	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
82	C₂×C₃×C₄	#41	1	2	yes	—
83	C₂×C₃×C₄	#41	1	4	yes	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	2	4	no	—
86	C₂×Q₈	C₂×C₃×C₄	1	8	yes	—
87	C₂²×C₄	C₂³×C₃	1	4	yes	—
88	C₂²×C₄	C₂³×C₃	1	4	yes	—
89	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂²×C₄	C₂³×C₃	1	2	yes	—
91	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂²×C₄	C₂³×C₃	1	2	yes	—
93	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₄⋊C₄	—	2	8	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	8	no	—
97	C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	4	no	—
101	C₄²	C₂×C₃×C₄	1	4	yes	—
102	C₄²	C₃×D₄	1	4	yes	—
103	C₄²	C₃×Q₈	1	8	yes	complemented
104	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₃	C₂²×Q₈	1	2	yes	—
107	C₂²×C₃	C₂×#41	1	1	yes	characteristic
108	C₂²×C₃	C₂×#41	1	2	yes	—
109	C₃×C₄	—	2	8	no	—
110	C₃×C₄	—	2	2	no	—
111	C₃×C₄	—	2	4	no	—
112	C₃×C₄	—	4	4	no	—
113	C₃×C₄	C₄×D₄	1	4	yes	—
114	C₃×C₄	C₄×Q₈	1	8	yes	complemented
115	C₃×C₄	C₄⋊Q₈	1	4	yes	direct product factor
116	C₂³	C₂⁴×C₃	1	1	yes	characteristic
117	Q₈	—	2	16	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
126	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
127	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
128	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
129	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
130	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
131	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
132	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
133	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
134	C₂×C₄	C₃×#41	1	2	yes	—
135	C₂×C₄	C₃×#41	1	4	yes	—
136	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₃	C₂×C₄×Q₈	1	2	yes	—
138	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₃	#532	1	1	yes	characteristic
140	C₂×C₃	#533	1	2	yes	—
141	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
142	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
143	C₂²	C₂²×C₃×Q₈	1	2	yes	—
144	C₂²	C₂×C₃×#41	1	1	yes	characteristic
145	C₂²	C₂×C₃×#41	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	C₃×C₄×D₄	1	4	yes	—
151	C₄	C₃×C₄×Q₈	1	8	yes	complemented
152	C₄	C₃×C₄⋊Q₈	1	4	yes	direct product factor
153	C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
154	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
155	C₂	C₂×C₃×C₄×Q₈	1	2	yes	—
156	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
157	C₂	C₃×#532	1	1	yes	characteristic
158	C₂	C₃×#533	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₄×C₄⋊Q₈	C₃	direct (N × H)
10. C₃×C₄⋊Q₈	C₄	direct (N × H)
49. C₃×C₄²	Q₈	semidirect
51. C₄⋊Q₈	C₃×C₄	direct (N × H)
103. C₄²	C₃×Q₈	semidirect
114. C₃×C₄	C₄×Q₈	semidirect
115. C₃×C₄	C₄⋊Q₈	direct (N × H)
151. C₄	C₃×C₄×Q₈	semidirect
152. C₄	C₃×C₄⋊Q₈	direct (N × H)
153. C₃	C₄×C₄⋊Q₈	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×C₃×C₄⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₃×#391	2
C₃×#389	2
C₃×#384	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄³	2
C₄×C₄⋊Q₈	3

Subgroups — C3×C4×C4⋊Q8 #84171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₄⋊Q₈ #84171