Finite Group Databasebeta

Subgroups — #71870

Order 384.

234 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 73 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×#536	C₂	1	1	yes	maximal, characteristic
2	#5240	C₂	1	1	yes	maximal, characteristic
3	#5237	C₂	1	1	yes	maximal, characteristic
4	#4539	C₂	1	4	yes	maximal, complemented
5	#2123	—	3	1	no	maximal, Sylow
6	C₂×C₃×#41	C₂²	1	1	yes	characteristic
7	C₂²×C₃×Q₈	C₄	1	1	yes	characteristic, complemented
8	C₃×C₄⋊Q₈	—	2	1	no	—
9	C₃×#124	—	2	1	no	—
10	C₃×#123	C₄	1	2	yes	complemented
11	C₃×#123	C₂²	1	2	yes	—
12	C₃×C₂²⋊Q₈	—	2	1	no	—
13	C₃×C₂²⋊Q₈	—	2	1	no	—
14	C₃×#117	C₄	1	1	yes	characteristic, complemented
15	#926	—	2	1	no	—
16	C₂×#208	—	2	1	no	—
17	#836	—	2	1	no	—
18	#834	—	2	1	no	—
19	#834	C₂²	1	2	yes	—
20	#833	C₂²	1	2	yes	—
21	#536	D₃	1	1	yes	characteristic
22	#413	—	3	1	no	—
23	#410	—	3	1	no	—
24	#352	—	3	4	no	—
25	C₃×#41	—	4	1	no	—
26	C₃×#41	—	4	1	no	—
27	C₂×C₃×Q₈	—	2	1	no	—
28	C₂×C₃×Q₈	—	2	1	no	—
29	C₂×C₃×Q₈	—	4	1	no	—
30	C₂×C₃×Q₈	—	4	1	no	—
31	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
32	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×D₄	D₄	1	1	yes	characteristic
34	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
39	C₃×C₄⋊C₄	—	2	1	no	—
40	C₃×C₄⋊C₄	—	2	2	no	—
41	C₃×C₄⋊C₄	—	2	2	no	—
42	C₃×C₂²⋊C₄	—	2	2	no	—
43	C₃×C₂²⋊C₄	—	2	2	no	—
44	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
45	C₃×C₄²	—	2	1	no	—
46	#217	—	2	2	no	—
47	C₄⋊Dic₃	—	4	1	no	—
48	C₄×Dic₃	—	4	1	no	—
49	#208	—	2	2	no	—
50	#208	—	4	1	no	—
51	C₂×C₃⋊C₈	—	4	1	no	—
52	C₂×#41	C₂×D₃	1	1	yes	characteristic
53	C₂²×Q₈	Dic₃	1	1	yes	characteristic, complemented
54	C₂×M₁₆	—	6	1	no	—
55	C₄⋊Q₈	—	2	1	no	—
56	#124	—	2	1	no	—
57	#123	Dic₃	1	2	yes	complemented
58	#123	C₂×D₃	1	2	yes	—
59	C₂²⋊Q₈	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	#117	—	6	1	no	—
62	#117	Dic₃	1	1	yes	characteristic, complemented
63	#101	—	6	1	no	—
64	#100	—	3	2	no	—
65	#99	—	3	2	no	—
66	#99	—	6	1	no	—
67	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
69	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
70	C₃×Q₈	—	4	1	no	—
71	C₃×Q₈	—	4	2	no	—
72	C₃×Q₈	—	4	1	no	—
73	C₃×Q₈	—	4	1	no	—
74	C₃×D₄	—	4	1	no	—
75	C₃×D₄	—	4	1	no	—
76	C₃×D₄	—	4	1	no	—
77	C₂×C₃×C₄	—	2	1	no	—
78	C₂×C₃×C₄	—	2	1	no	—
79	C₂×C₃×C₄	—	2	1	no	—
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	1	no	—
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	4	2	no	—
86	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₃	—	4	2	no	—
91	C₃⋊C₈	—	4	2	no	—
92	#41	—	4	1	no	—
93	#41	—	4	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	4	1	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×Q₈	#67	1	1	yes	characteristic
99	C₂×D₄	C₂×Dic₃	1	1	yes	characteristic
100	C₂×D₄	#67	1	1	yes	characteristic
101	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₄	C₂×Dic₃	1	2	yes	—
104	C₂²×C₄	#67	1	1	yes	characteristic
105	C₂²×C₄	#67	1	1	yes	characteristic
106	M₁₆	—	6	2	no	—
107	M₁₆	—	12	1	no	—
108	C₂×C₈	—	12	1	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	12	1	no	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	6	2	no	—
116	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
117	C₄²	—	2	1	no	—
118	C₄²	—	12	1	no	—
119	C₂²×C₃	—	2	1	no	—
120	C₂²×C₃	—	2	1	no	—
121	C₂²×C₃	—	2	1	no	—
122	C₂²×C₃	—	4	1	no	—
123	C₂²×C₃	—	4	1	no	—
124	C₂²×C₃	—	4	1	no	—
125	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₃×C₄	—	2	1	no	—
127	C₃×C₄	—	2	1	no	—
128	C₃×C₄	—	2	1	no	—
129	C₃×C₄	—	4	2	no	—
130	C₃×C₄	—	4	2	no	—
131	C₃×C₄	#99	1	1	yes	characteristic
132	C₃×C₄	#99	1	1	yes	characteristic
133	Dic₃	—	8	2	no	—
134	C₂³	#217	1	1	yes	characteristic
135	C₂³	C₂²×Dic₃	1	1	yes	characteristic
136	C₂³	C₂×#67	1	1	yes	characteristic
137	Q₈	—	4	1	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	12	2	no	—
154	C₂×C₄	#217	1	1	yes	characteristic
155	C₂×C₄	#217	1	1	yes	characteristic
156	C₂×C₄	#217	1	1	yes	characteristic
157	C₂×C₄	C₂×#67	1	1	yes	characteristic
158	C₈	—	12	2	no	—
159	C₂×C₃	—	2	1	no	—
160	C₂×C₃	—	4	1	no	—
161	C₂×C₃	—	4	1	no	—
162	C₂×C₃	—	4	1	no	—
163	C₂×C₃	C₂×#99	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	C₂×#217	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	24	2	no	—
177	C₄	#834	1	1	yes	characteristic
178	C₄	#834	1	1	yes	characteristic
179	C₃	#2123	1	1	yes	characteristic, complemented, Sylow
180	C₂	—	2	1	no	—
181	C₂	—	4	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	4	1	no	—
184	C₂	C₂×#834	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#834
commutator	C₂³×C₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₃
Fitting	C₃×#536	C₂
socle	C₂×C₃	C₂×#99
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂×C₃×D₄	C₂³
hypercenter	#536	D₃
hypocenter	C₃	#2123
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

Complements

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

N	H ≅ G/N	type
4. #4539	C₂	semidirect
7. C₂²×C₃×Q₈	C₄	semidirect
10. C₃×#123	C₄	semidirect
14. C₃×#117	C₄	semidirect
53. C₂²×Q₈	Dic₃	semidirect
57. #123	Dic₃	semidirect
62. #117	Dic₃	semidirect
179. C₃	#2123	semidirect

Maximal subgroups

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

subgroup	index
C₃×#536	2
#5240	2
#5237	2
#4539	2
#2123	3