Finite Group Databasebeta

Subgroups — #68419

Order 384.

202 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 67 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	#5728	C₂	1	1	yes	maximal, characteristic
2	#4833	C₂	1	1	yes	maximal, characteristic
3	#4833	C₂	1	1	yes	maximal, characteristic
4	C₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
5	#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	#4486	C₂	1	1	yes	maximal, characteristic, complemented
7	#4482	C₂	1	1	yes	maximal, characteristic, complemented
8	#1544	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#946	—	2	1	no	—
11	Q₈×Dic₃	—	2	1	no	—
12	#897	—	2	1	no	—
13	#892	C₄	1	2	yes	—
14	#890	C₄	1	2	yes	—
15	#887	—	2	1	no	—
16	C₄²×D₃	C₂²	1	1	yes	characteristic
17	#869	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	(C₂×D₃)⋊C₈	—	2	1	no	—
21	(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₈×Dic₃	—	2	1	no	—
23	C₈×Dic₃	—	2	1	no	—
24	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#533	—	3	1	no	—
27	#431	—	3	1	no	—
28	#431	—	3	1	no	—
29	#332	—	3	1	no	—
30	#332	—	3	1	no	—
31	#332	—	3	1	no	—
32	#332	D₃	1	1	yes	characteristic, complemented
33	C₂×C₃×Q₈	—	2	1	no	—
34	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃	D₄	1	1	yes	characteristic
36	C₂×C₄×D₃	D₄	1	1	yes	characteristic
37	C₂×Dic₆	—	2	1	no	—
38	C₂×C₃×C₈	—	2	1	no	—
39	C₂×C₃×C₈	—	2	1	no	—
40	C₃×C₄⋊C₄	—	2	1	no	—
41	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₃×C₄²	C₂³	1	1	yes	characteristic
43	(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₄⋊Dic₃	—	2	1	no	—
45	C₄⋊Dic₃	—	2	2	no	—
46	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	Dic₃⋊C₄	—	2	2	no	—
48	C₄×Dic₃	—	2	2	no	—
49	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₃	D₄	1	1	yes	characteristic
51	C₄×Dic₃	D₄	1	1	yes	characteristic
52	C₂×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃⋊C₈	—	2	1	no	—
54	C₄⋊Q₈	—	3	1	no	—
55	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
56	#125	—	3	2	no	—
57	C₂²⋊Q₈	—	6	1	no	—
58	C₂²⋊Q₈	—	6	1	no	—
59	C₄×Q₈	—	6	1	no	—
60	C₄×Q₈	—	6	1	no	—
61	#117	—	3	2	no	—
62	C₂×C₄²	—	3	1	no	—
63	C₄⋊C₈	—	3	1	no	—
64	C₄⋊C₈	—	3	1	no	—
65	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
67	C₂²⋊C₈	—	6	1	no	—
68	C₂²⋊C₈	—	6	1	no	—
69	C₄×C₈	—	6	1	no	—
70	C₄×C₈	—	6	1	no	—
71	C₂²×D₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₃×Q₈	—	4	1	no	—
73	C₃×Q₈	—	4	1	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₃	—	2	2	no	—
79	C₂×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄×D₃	—	2	1	no	—
83	C₄×D₃	—	2	1	no	—
84	C₄×D₃	—	2	1	no	—
85	C₄×D₃	—	2	1	no	—
86	C₄×D₃	—	2	2	no	—
87	Dic₆	—	4	1	no	—
88	Dic₆	—	4	1	no	—
89	C₃×C₈	—	4	1	no	—
90	C₃×C₈	—	4	1	no	—
91	C₃⋊C₈	—	4	1	no	—
92	C₃⋊C₈	—	4	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	6	1	no	—
95	C₂²×C₄	—	3	1	no	—
96	C₂²×C₄	—	3	1	no	—
97	C₂²×C₄	—	3	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	6	1	no	—
101	C₂×C₈	—	6	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	3	2	no	—
104	C₄⋊C₄	—	6	2	no	—
105	C₄⋊C₄	—	6	2	no	—
106	C₄⋊C₄	—	6	1	no	—
107	C₄⋊C₄	C₄×D₃	1	2	yes	—
108	C₂²⋊C₄	—	6	2	no	—
109	C₄²	—	3	1	no	—
110	C₄²	—	3	1	no	—
111	C₄²	—	3	1	no	—
112	C₄²	—	6	2	no	—
113	C₄²	C₂²×D₃	1	1	yes	characteristic
114	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×D₃	—	2	1	no	—
116	C₂×D₃	—	2	1	no	—
117	C₂×D₃	—	2	1	no	—
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	4	2	no	—
120	C₃×C₄	#135	1	1	yes	characteristic
121	C₃×C₄	#135	1	1	yes	characteristic
122	C₃×C₄	#135	1	1	yes	characteristic
123	C₃×C₄	#135	1	1	yes	characteristic
124	Dic₃	—	2	1	no	—
125	Dic₃	—	2	1	no	—
126	Dic₃	—	4	2	no	—
127	Dic₃	#101	1	2	yes	—
128	C₂³	—	3	1	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	12	1	no	—
132	Q₈	—	12	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	3	1	no	—
135	C₂×C₄	—	3	1	no	—
136	C₂×C₄	—	3	1	no	—
137	C₂×C₄	—	6	2	no	—
138	C₂×C₄	—	6	1	no	—
139	C₂×C₄	—	6	1	no	—
140	C₂×C₄	—	6	1	no	—
141	C₂×C₄	—	6	1	no	—
142	C₂×C₄	—	6	2	no	—
143	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄	D₃×D₄	1	1	yes	characteristic
145	C₂×C₄	D₃×D₄	1	1	yes	characteristic
146	C₈	—	4	1	no	—
147	C₈	—	4	1	no	—
148	C₈	—	12	1	no	—
149	C₈	—	12	1	no	—
150	C₂×C₃	C₂×#101	1	1	yes	characteristic
151	C₂×C₃	#416	1	1	yes	characteristic
152	C₂×C₃	#416	1	1	yes	characteristic
153	D₃	—	4	1	no	—
154	C₂²	—	6	1	no	—
155	C₂²	—	6	1	no	—
156	C₂²	—	6	1	no	—
157	C₂²	D₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	3	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	6	1	no	—
162	C₄	—	6	1	no	—
163	C₄	—	12	2	no	—
164	C₄	#912	1	1	yes	characteristic
165	C₄	#916	1	1	yes	characteristic
166	C₄	#916	1	1	yes	characteristic
167	C₄	#919	1	1	yes	characteristic
168	C₃	#1544	1	1	yes	characteristic, complemented, Sylow
169	C₂	—	12	1	no	—
170	C₂	D₃×#101	1	1	yes	characteristic
171	C₂	#4772	1	1	yes	characteristic
172	C₂	#4804	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#332	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#332	D₃
hypocenter	C₃	#1544
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

Complements

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

N	H ≅ G/N	type
4. C₃×#332	C₂	semidirect
5. #4547	C₂	semidirect
6. #4486	C₂	semidirect
7. #4482	C₂	semidirect
32. #332	D₃	semidirect
168. C₃	#1544	semidirect

Maximal subgroups

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

subgroup	index
#5728	2
#4833	2
C₃×#332	2
#4547	2
#4486	2
#4482	2
#1544	3