Finite Group Databasebeta

Subgroups — #69127

Order 384.

210 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5753	C₂	1	1	yes	maximal, characteristic, complemented
2	#5142	C₂	1	1	yes	maximal, characteristic, complemented
3	#4908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#800	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
6	#4517	C₂	1	1	yes	maximal, characteristic, complemented
7	#4510	C₂	1	1	yes	maximal, characteristic, complemented
8	#2193	—	3	1	no	maximal, Sylow
9	#1010	—	2	1	no	—
10	C₃×#131	C₂²	1	1	yes	characteristic
11	#949	—	2	1	no	—
12	#909	—	2	1	no	—
13	#909	—	2	2	no	—
14	C₂×Dic₁₂	—	2	1	no	—
15	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
16	#904	—	2	1	no	—
17	#904	C₂²	1	1	yes	characteristic
18	C₂×D₂₄	C₂²	1	1	yes	characteristic
19	C₃×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
21	#823	—	2	1	no	—
22	#819	C₂²	1	1	yes	characteristic
23	#800	—	2	2	no	—
24	#800	—	2	1	no	—
25	D₄₈	—	2	1	no	—
26	#578	—	3	1	no	—
27	#509	—	3	1	no	—
28	C₂×SD₃₂	—	3	1	no	—
29	#472	—	3	1	no	—
30	#362	—	3	1	no	—
31	#359	—	3	1	no	—
32	#350	D₃	1	1	yes	characteristic, complemented
33	C₃×#41	D₄	1	1	yes	characteristic
34	D₃×Q₈	—	4	1	no	—
35	#237	—	4	2	no	—
36	#237	—	4	1	no	—
37	#235	—	2	2	no	—
38	#235	—	2	1	no	—
39	C₂×D₁₂	—	2	1	no	—
40	C₂×Dic₆	—	2	2	no	—
41	C₂×Dic₆	—	2	1	no	—
42	C₃×M₁₆	—	2	1	no	—
43	C₃×M₁₆	D₄	1	1	yes	characteristic
44	C₂×C₃×C₈	—	2	1	no	—
45	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
46	#215	—	4	1	no	—
47	#213	—	4	1	no	—
48	#208	—	2	1	no	—
49	Dic₁₂	—	2	2	no	—
50	Dic₁₂	—	2	1	no	—
51	Dic₁₂	D₄	1	1	yes	characteristic
52	Dic₁₂	D₄	1	2	yes	—
53	D₂₄	—	2	1	no	—
54	D₂₄	—	2	1	no	—
55	D₂₄	D₄	1	1	yes	characteristic
56	#204	—	2	2	no	—
57	#204	—	2	1	no	—
58	C₃×C₁₆	—	2	1	no	—
59	C₃×C₁₆	—	2	1	no	—
60	2₋¹⁺⁴	—	6	1	no	—
61	#137	—	6	2	no	—
62	#137	—	6	1	no	—
63	AGL(1,ℤ/8)	—	6	1	no	—
64	#135	—	3	1	no	—
65	#135	—	6	1	no	—
66	C₂×Q₁₆	—	3	1	no	—
67	C₂×Q₁₆	—	6	1	no	—
68	C₂×D₈	—	3	1	no	—
69	#131	C₂×D₃	1	1	yes	characteristic
70	SD₃₂	—	6	2	no	—
71	SD₃₂	—	6	1	no	—
72	D₁₆	—	6	1	no	—
73	M₃₂	C₂×D₃	1	1	yes	characteristic
74	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
75	#108	—	3	1	no	—
76	#100	—	6	1	no	—
77	C₂²×D₃	—	4	1	no	—
78	C₃×Q₈	D₈	1	1	yes	characteristic
79	C₃×D₄	—	2	1	no	—
80	C₃×D₄	D₈	1	1	yes	characteristic
81	C₂×C₃×C₄	—	2	1	no	—
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	#67	—	4	2	no	—
84	#67	—	4	1	no	—
85	C₂×Dic₃	—	4	2	no	—
86	C₂×Dic₃	—	4	1	no	—
87	D₁₂	—	2	1	no	—
88	D₁₂	—	4	1	no	—
89	D₁₂	—	4	1	no	—
90	C₄×D₃	—	4	2	no	—
91	C₄×D₃	—	4	1	no	—
92	Dic₆	—	2	2	no	—
93	Dic₆	—	2	1	no	—
94	Dic₆	—	4	2	no	—
95	Dic₆	—	4	1	no	—
96	C₃×C₈	—	2	1	no	—
97	C₃×C₈	C₂×D₄	1	1	yes	characteristic
98	C₃×C₈	C₂×D₄	1	1	yes	characteristic
99	C₃⋊C₈	—	4	1	no	—
100	#41	—	6	2	no	—
101	#41	—	6	1	no	—
102	#41	—	12	2	no	—
103	#41	—	12	1	no	—
104	#41	D₁₂	1	1	yes	characteristic
105	C₂×Q₈	—	6	2	no	—
106	C₂×Q₈	—	6	1	no	—
107	C₂×Q₈	—	12	1	no	—
108	C₂×D₄	—	6	1	no	—
109	Q₁₆	—	3	2	no	—
110	Q₁₆	—	3	1	no	—
111	Q₁₆	—	6	2	no	—
112	Q₁₆	—	6	1	no	—
113	SD₁₆	—	6	2	no	—
114	SD₁₆	—	6	1	no	—
115	SD₁₆	—	12	1	no	—
116	D₈	—	3	1	no	—
117	D₈	—	6	1	no	—
118	D₈	—	6	1	no	—
119	D₈	—	12	1	no	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	6	1	no	—
122	M₁₆	D₁₂	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	1	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×D₃	—	4	1	no	—
130	C₂×D₃	—	8	1	no	—
131	C₂×D₃	—	16	1	no	—
132	C₃×C₄	—	2	1	no	—
133	C₃×C₄	#120	1	1	yes	characteristic
134	C₃×C₄	C₂×D₈	1	1	yes	characteristic
135	Dic₃	—	4	1	no	—
136	Dic₃	—	4	2	no	—
137	C₂³	—	12	1	no	—
138	Q₈	—	6	2	no	—
139	Q₈	—	6	1	no	—
140	Q₈	—	12	2	no	—
141	Q₈	—	12	1	no	—
142	Q₈	D₂₄	1	1	yes	characteristic
143	D₄	—	2	1	no	—
144	D₄	—	6	1	no	—
145	D₄	—	12	2	no	—
146	D₄	—	12	1	no	—
147	D₄	—	12	1	no	—
148	D₄	—	12	1	no	—
149	D₄	D₂₄	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	12	2	no	—
152	C₂×C₄	—	12	2	no	—
153	C₂×C₄	—	12	1	no	—
154	C₂×C₄	—	12	1	no	—
155	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	12	1	no	—
158	C₈	D₃×D₄	1	1	yes	characteristic
159	C₈	D₃×D₄	1	1	yes	characteristic
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	4	1	no	—
162	C₂×C₃	#447	1	1	yes	characteristic
163	D₃	—	8	1	no	—
164	D₃	—	16	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	12	1	no	—
167	C₂²	—	24	1	no	—
168	C₂²	—	48	1	no	—
169	C₂²	#908	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	12	2	no	—
172	C₄	—	12	1	no	—
173	C₄	#882	1	1	yes	characteristic
174	C₄	C₂×D₂₄	1	1	yes	characteristic
175	C₃	#2193	1	1	yes	characteristic, complemented, Sylow
176	C₂	—	2	1	no	—
177	C₂	—	4	1	no	—
178	C₂	—	24	1	no	—
179	C₂	—	48	1	no	—
180	C₂	#4732	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4732
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#350	C₂
socle	C₂×C₃	#447
hypercenter	#350	D₃
hypocenter	C₃	#2193
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
1. #5753	C₂	semidirect
2. #5142	C₂	semidirect
3. #4908	C₂	semidirect
4. C₂×#800	C₂	semidirect
5. C₃×#350	C₂	semidirect
6. #4517	C₂	semidirect
7. #4510	C₂	semidirect
32. #350	D₃	semidirect
175. C₃	#2193	semidirect

Maximal subgroups

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

subgroup	index
#5753	2
#5142	2
#4908	2
C₂×#800	2
C₃×#350	2
#4517	2
#4510	2
#2193	3