Finite Group Databasebeta

Subgroups — #72124

Order 384.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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	C₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	#5193	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#828	C₂	1	1	yes	maximal, characteristic, complemented
4	#5142	C₂	1	1	yes	maximal, characteristic, complemented
5	#4567	C₂	1	1	yes	maximal, characteristic, complemented
6	#4565	C₂	1	1	yes	maximal, characteristic, complemented
7	#4561	C₂	1	1	yes	maximal, characteristic, complemented
8	#2193	—	3	1	no	maximal, Sylow
9	C₃×2₋¹⁺⁴	—	2	1	no	—
10	C₃×#137	—	2	2	no	—
11	C₃×#137	—	2	1	no	—
12	C₃×#135	—	2	1	no	—
13	C₃×#135	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₁₆	—	2	1	no	—
15	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
16	C₃×#131	C₂²	1	1	yes	characteristic
17	#949	—	2	1	no	—
18	C₂×D₂₄	C₂²	1	1	yes	characteristic
19	#828	—	2	2	no	—
20	#828	—	2	1	no	—
21	#826	—	2	1	no	—
22	#823	—	2	1	no	—
23	#819	C₂²	1	1	yes	characteristic
24	#812	C₂²	1	1	yes	characteristic
25	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
26	#578	D₃	1	1	yes	characteristic, complemented
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	—	3	1	no	—
33	C₃×#41	—	2	2	no	—
34	C₃×#41	—	2	1	no	—
35	C₃×#41	—	4	2	no	—
36	C₃×#41	—	4	1	no	—
37	C₃×#41	D₄	1	1	yes	characteristic
38	C₂×C₃×Q₈	—	2	2	no	—
39	C₂×C₃×Q₈	—	2	1	no	—
40	C₂×C₃×Q₈	—	4	1	no	—
41	C₂×D₁₂	—	2	1	no	—
42	C₃×Q₁₆	—	2	2	no	—
43	C₃×Q₁₆	—	2	1	no	—
44	C₃×Q₁₆	D₄	1	1	yes	characteristic
45	C₃×Q₁₆	D₄	1	2	yes	—
46	C₃×SD₁₆	—	2	2	no	—
47	C₃×SD₁₆	—	2	1	no	—
48	C₃×D₈	D₄	1	1	yes	characteristic
49	C₃×M₁₆	—	2	1	no	—
50	C₃×M₁₆	D₄	1	1	yes	characteristic
51	C₂×C₃×C₈	—	2	1	no	—
52	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
53	#215	—	4	1	no	—
54	#213	—	4	1	no	—
55	#208	—	2	1	no	—
56	D₂₄	—	2	1	no	—
57	D₂₄	—	2	1	no	—
58	C₃⋊C₁₆	—	2	1	no	—
59	C₃⋊C₁₆	—	2	1	no	—
60	2₋¹⁺⁴	—	2	1	no	—
61	#137	—	2	2	no	—
62	#137	—	2	1	no	—
63	AGL(1,ℤ/8)	—	6	1	no	—
64	#135	—	2	1	no	—
65	#135	C₂×D₃	1	1	yes	characteristic
66	C₂×Q₁₆	—	2	1	no	—
67	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
74	C₂×C₁₆	—	3	1	no	—
75	#108	—	3	1	no	—
76	#100	—	6	1	no	—
77	C₂²×D₃	—	4	1	no	—
78	C₃×Q₈	—	2	2	no	—
79	C₃×Q₈	—	2	1	no	—
80	C₃×Q₈	—	4	2	no	—
81	C₃×Q₈	—	4	1	no	—
82	C₃×Q₈	D₈	1	1	yes	characteristic
83	C₃×D₄	—	2	1	no	—
84	C₃×D₄	—	2	1	no	—
85	C₃×D₄	—	4	2	no	—
86	C₃×D₄	—	4	1	no	—
87	C₃×D₄	D₈	1	1	yes	characteristic
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	4	2	no	—
90	C₂×C₃×C₄	—	4	2	no	—
91	C₂×C₃×C₄	—	4	1	no	—
92	C₂×C₃×C₄	—	4	1	no	—
93	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	D₁₂	—	4	1	no	—
95	D₁₂	—	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	—	2	2	no	—
101	#41	—	2	1	no	—
102	#41	—	4	2	no	—
103	#41	—	4	1	no	—
104	#41	#67	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×D₄	—	6	1	no	—
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	2	1	no	—
111	Q₁₆	#67	1	1	yes	characteristic
112	Q₁₆	#67	1	2	yes	—
113	SD₁₆	—	2	2	no	—
114	SD₁₆	—	2	1	no	—
115	SD₁₆	—	12	1	no	—
116	D₈	—	6	1	no	—
117	D₈	—	6	1	no	—
118	D₈	—	12	1	no	—
119	D₈	#67	1	1	yes	characteristic
120	M₁₆	—	2	1	no	—
121	M₁₆	—	6	1	no	—
122	M₁₆	#67	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₁₆	—	6	1	no	—
126	C₁₆	—	6	1	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	4	1	no	—
129	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×D₃	—	8	1	no	—
131	C₂×D₃	—	16	1	no	—
132	C₃×C₄	—	2	1	no	—
133	C₃×C₄	—	4	2	no	—
134	C₃×C₄	—	4	1	no	—
135	C₃×C₄	#120	1	1	yes	characteristic
136	C₃×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂³	—	12	1	no	—
138	Q₈	—	2	2	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	4	1	no	—
142	Q₈	#213	1	1	yes	characteristic
143	D₄	—	2	1	no	—
144	D₄	—	2	1	no	—
145	D₄	—	4	2	no	—
146	D₄	—	4	1	no	—
147	D₄	—	12	1	no	—
148	D₄	—	12	1	no	—
149	D₄	#213	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	C₂×#67	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	12	1	no	—
158	C₈	C₂×#67	1	1	yes	characteristic
159	C₈	C₂×#67	1	1	yes	characteristic
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	4	1	no	—
162	C₂×C₃	—	8	1	no	—
163	C₂×C₃	#447	1	1	yes	characteristic
164	D₃	—	16	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	24	1	no	—
168	C₂²	—	48	1	no	—
169	C₂²	#932	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	1	no	—
173	C₄	C₂×#213	1	1	yes	characteristic
174	C₄	#953	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	—	8	1	no	—
179	C₂	—	48	1	no	—
180	C₂	#5217	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#5217
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#578	C₂
socle	C₂×C₃	#447
hypercenter	#578	D₃
hypocenter	C₃	#2193
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #578

Complements

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

N	H ≅ G/N	type
1. C₃×#578	C₂	semidirect
2. #5193	C₂	semidirect
3. C₂×#828	C₂	semidirect
4. #5142	C₂	semidirect
5. #4567	C₂	semidirect
6. #4565	C₂	semidirect
7. #4561	C₂	semidirect
26. #578	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
C₃×#578	2
#5193	2
C₂×#828	2
#5142	2
#4567	2
#4565	2
#4561	2
#2193	3