Finite Group Databasebeta

Subgroups — #69193

Order 384.

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

normal subgroups only

This group has 256 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	C₂×D₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#827	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#800	C₂	1	1	yes	maximal, characteristic, complemented
4	#4887	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#4491	C₂	1	1	yes	maximal, characteristic, complemented
8	#2187	—	3	1	no	maximal, Sylow
9	C₂×D₃×D₄	—	2	1	no	—
10	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×#213	—	2	1	no	—
12	D₃×D₈	—	2	2	no	—
13	D₃×D₈	—	2	2	no	—
14	C₂×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₂₄	C₂²	1	1	yes	characteristic
16	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
17	#897	—	2	1	no	—
18	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₃×#107	C₂²	1	1	yes	characteristic, complemented
20	#827	—	2	2	no	—
21	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
22	#810	—	2	1	no	—
23	#800	—	2	2	no	—
24	C₂²×D₈	—	3	1	no	—
25	C₂×SD₃₂	—	3	1	no	—
26	C₂×SD₃₂	—	3	1	no	—
27	#467	—	3	1	no	—
28	#357	—	3	1	no	—
29	#357	D₃	1	1	yes	characteristic, complemented
30	C₂²⋊C₁₆	—	3	1	no	—
31	C₂³×D₃	—	4	1	no	—
32	C₂×C₃×D₄	—	2	1	no	—
33	C₂×#67	—	4	1	no	—
34	D₃×D₄	—	2	4	no	—
35	D₃×D₄	—	4	2	no	—
36	C₂×D₁₂	—	2	1	no	—
37	C₂×C₄×D₃	D₄	1	1	yes	characteristic
38	C₂×Dic₆	—	2	1	no	—
39	C₃×D₈	—	2	1	no	—
40	C₃×D₈	D₄	1	2	yes	—
41	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₄⋊C₄	—	2	1	no	—
43	#213	—	2	4	no	—
44	(C₂×D₃)⋊C₄	—	4	1	no	—
45	C₄⋊Dic₃	—	4	1	no	—
46	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
47	Dic₁₂	—	2	1	no	—
48	Dic₁₂	—	2	1	no	—
49	D₂₄	—	2	1	no	—
50	D₂₄	D₄	1	2	yes	—
51	D₃×C₈	—	2	1	no	—
52	D₃×C₈	—	2	1	no	—
53	C₃×C₁₆	—	2	1	no	—
54	C₃⋊C₁₆	—	2	1	no	—
55	C₂²×D₄	—	6	1	no	—
56	C₂×Q₁₆	—	3	1	no	—
57	C₂×D₈	—	3	1	no	—
58	C₂×D₈	—	6	1	no	—
59	C₂×D₈	—	6	2	no	—
60	C₂×D₈	—	6	2	no	—
61	C₂×D₈	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₈	—	3	1	no	—
63	C₂²⋊Q₈	—	6	1	no	—
64	SD₃₂	—	6	2	no	—
65	SD₃₂	—	6	2	no	—
66	C₂×C₁₆	—	3	1	no	—
67	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
68	#107	C₂×D₃	1	1	yes	characteristic, complemented
69	#103	—	6	1	no	—
70	C₂³×C₃	—	4	1	no	—
71	C₂²×D₃	—	4	4	no	—
72	C₂²×D₃	—	4	1	no	—
73	C₂²×D₃	—	8	2	no	—
74	C₂²×D₃	—	8	2	no	—
75	C₂²×D₃	D₈	1	1	yes	characteristic
76	C₃×D₄	—	2	2	no	—
77	C₃×D₄	—	4	1	no	—
78	C₂×C₃×C₄	—	4	1	no	—
79	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	#67	—	4	4	no	—
81	C₂×Dic₃	—	4	1	no	—
82	C₂×Dic₃	D₈	1	1	yes	characteristic
83	D₁₂	—	2	2	no	—
84	D₁₂	—	4	1	no	—
85	C₄×D₃	—	2	1	no	—
86	C₄×D₃	—	2	1	no	—
87	Dic₆	—	4	1	no	—
88	Dic₆	—	4	1	no	—
89	C₃×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃×C₈	C₂×D₄	1	1	yes	characteristic
91	C₃⋊C₈	—	2	1	no	—
92	C₂⁴	—	12	1	no	—
93	C₂×Q₈	—	6	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	6	4	no	—
96	C₂×D₄	—	6	1	no	—
97	C₂×D₄	—	12	1	no	—
98	C₂×D₄	—	12	2	no	—
99	C₂²×C₄	—	3	1	no	—
100	Q₁₆	—	6	1	no	—
101	Q₁₆	—	6	1	no	—
102	D₈	—	2	1	no	—
103	D₈	—	3	2	no	—
104	D₈	—	6	4	no	—
105	D₈	—	6	1	no	—
106	D₈	D₁₂	1	2	yes	—
107	C₂×C₈	—	3	1	no	—
108	C₂×C₈	—	6	1	no	—
109	C₂×C₈	—	6	1	no	—
110	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	12	1	no	—
113	C₂²⋊C₄	—	12	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	6	1	no	—
116	C₂²×C₃	—	4	2	no	—
117	C₂²×C₃	—	8	1	no	—
118	C₂²×C₃	—	8	1	no	—
119	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
120	C₂×D₃	—	2	1	no	—
121	C₂×D₃	—	2	1	no	—
122	C₂×D₃	—	4	2	no	—
123	C₂×D₃	—	8	8	no	—
124	C₂×D₃	—	8	1	no	—
125	C₂×D₃	—	8	1	no	—
126	C₂×D₃	SD₃₂	1	2	yes	complemented
127	C₃×C₄	—	8	1	no	—
128	C₃×C₄	#120	1	1	yes	characteristic
129	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Dic₃	—	2	1	no	—
131	Dic₃	—	8	1	no	—
132	C₂³	—	3	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	12	4	no	—
135	C₂³	—	12	1	no	—
136	C₂³	—	24	2	no	—
137	C₂³	—	24	2	no	—
138	Q₈	—	12	1	no	—
139	Q₈	—	12	1	no	—
140	D₄	—	2	2	no	—
141	D₄	—	4	1	no	—
142	D₄	—	6	2	no	—
143	D₄	—	12	4	no	—
144	D₄	—	12	1	no	—
145	C₂×C₄	—	3	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	6	1	no	—
148	C₂×C₄	—	6	1	no	—
149	C₂×C₄	—	12	1	no	—
150	C₂×C₄	D₃×D₄	1	1	yes	characteristic
151	C₈	—	6	1	no	—
152	C₈	C₂×D₁₂	1	1	yes	characteristic
153	C₈	D₃×D₄	1	1	yes	characteristic
154	C₂×C₃	—	8	2	no	—
155	C₂×C₃	#447	1	1	yes	characteristic
156	C₂×C₃	C₂×SD₃₂	1	1	yes	characteristic
157	C₂×C₃	#509	1	1	yes	characteristic
158	D₃	—	2	2	no	—
159	D₃	—	8	2	no	—
160	C₂²	—	3	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	6	1	no	—
163	C₂²	—	6	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	12	2	no	—
167	C₂²	—	24	8	no	—
168	C₂²	—	24	1	no	—
169	C₂²	—	24	1	no	—
170	C₂²	D₃×D₈	1	1	yes	characteristic
171	C₄	—	6	1	no	—
172	C₄	—	8	1	no	—
173	C₄	—	24	1	no	—
174	C₄	#882	1	1	yes	characteristic
175	C₄	#914	1	1	yes	characteristic
176	C₃	#2187	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	6	2	no	—
178	C₂	—	8	2	no	—
179	C₂	—	24	2	no	—
180	C₂	#4773	1	1	yes	characteristic
181	C₂	#4911	1	1	yes	characteristic
182	C₂	D₃×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#357	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×D₃×D₈	C₂
hypercenter	#357	D₃
hypocenter	C₃	#2187
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
2. C₂×#827	C₂	semidirect
3. C₂×#800	C₂	semidirect
4. #4887	C₂	semidirect
5. C₃×#357	C₂	semidirect
6. (C₂×D₃)⋊C₁₆	C₂	semidirect
7. #4491	C₂	semidirect
14. C₂×Dic₁₂	C₂²	semidirect
18. C₂×C₃×C₁₆	C₂²	semidirect
19. C₃×#107	C₂²	semidirect
21. C₂×C₃⋊C₁₆	C₂²	semidirect
29. #357	D₃	semidirect
67. C₂×C₁₆	C₂×D₃	semidirect
68. #107	C₂×D₃	semidirect
126. C₂×D₃	SD₃₂	semidirect
176. C₃	#2187	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₈	2
C₂×#827	2
C₂×#800	2
#4887	2
C₃×#357	2
(C₂×D₃)⋊C₁₆	2
#4491	2
#2187	3