Finite Group Databasebeta

Subgroups — #71045

Order 384.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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	#5553	C₂	1	1	yes	maximal, characteristic
2	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	#5094	C₂	1	1	yes	maximal, characteristic, complemented
4	#5088	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#4485	C₂	1	1	yes	maximal, characteristic, complemented
7	#4484	C₂	1	1	yes	maximal, characteristic, complemented
8	#1642	—	3	1	no	maximal, Sylow
9	C₂×#235	—	2	1	no	—
10	C₂×C₃×SD₁₆	—	2	1	no	—
11	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
13	#940	—	2	1	no	—
14	#930	—	2	1	no	—
15	#893	—	2	1	no	—
16	#884	—	2	2	no	—
17	#874	C₂²	1	1	yes	characteristic
18	C₄×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×#103	—	2	1	no	—
21	#810	—	2	1	no	—
22	#809	—	2	2	no	—
23	#808	—	2	1	no	—
24	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#550	—	3	1	no	—
27	#486	—	3	1	no	—
28	#475	—	3	1	no	—
29	#461	D₃	1	1	yes	characteristic, complemented
30	#331	—	3	1	no	—
31	Q₈⋊C₈	—	3	1	no	—
32	D₄⋊C₈	—	3	1	no	—
33	C₂×C₃×Q₈	—	2	1	no	—
34	C₂×C₃×D₄	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	—	2	1	no	—
36	C₂×#67	—	2	2	no	—
37	C₂×#67	—	2	2	no	—
38	#235	—	4	2	no	—
39	#235	—	4	2	no	—
40	C₂×D₁₂	—	2	1	no	—
41	C₂×D₁₂	—	2	1	no	—
42	C₂×D₁₂	D₄	1	2	yes	—
43	C₂×C₄×D₃	—	2	2	no	—
44	C₂×Dic₆	D₄	1	1	yes	characteristic
45	C₃×SD₁₆	—	4	2	no	—
46	C₂×C₃×C₈	—	2	1	no	—
47	C₃×C₄⋊C₄	—	2	1	no	—
48	C₃×C₄⋊C₄	—	2	1	no	—
49	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₃×C₂²⋊C₄	—	2	1	no	—
51	C₃×C₄²	C₂³	1	1	yes	characteristic
52	(C₂×D₃)⋊C₄	—	2	2	no	—
53	C₄⋊Dic₃	D₄	1	1	yes	characteristic
54	Dic₃⋊C₄	—	2	1	no	—
55	C₄×Dic₃	—	2	1	no	—
56	C₂×C₃⋊C₈	—	2	1	no	—
57	C₂×C₃⋊C₈	—	2	1	no	—
58	C₂×#41	—	6	1	no	—
59	C₂×SD₁₆	—	2	1	no	—
60	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
61	#127	—	3	1	no	—
62	#127	—	6	1	no	—
63	#121	—	6	2	no	—
64	#121	—	6	1	no	—
65	C₄×Q₈	—	3	1	no	—
66	C₄×D₄	—	6	1	no	—
67	C₄×D₄	C₂×D₃	1	1	yes	characteristic
68	#106	—	6	1	no	—
69	C₄⋊C₈	—	3	1	no	—
70	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
71	#103	—	2	1	no	—
72	#103	—	6	1	no	—
73	#102	—	6	2	no	—
74	C₄×C₈	—	3	1	no	—
75	C₂³×C₃	—	2	1	no	—
76	C₂²×D₃	—	2	2	no	—
77	C₃×Q₈	—	4	1	no	—
78	C₃×Q₈	—	4	1	no	—
79	C₃×D₄	—	2	1	no	—
80	C₃×D₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	1	no	—
82	C₂×C₃×C₄	—	4	1	no	—
83	C₂×C₃×C₄	—	4	1	no	—
84	C₂×C₃×C₄	—	4	1	no	—
85	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
88	#67	—	4	4	no	—
89	#67	—	8	2	no	—
90	C₂×Dic₃	—	2	1	no	—
91	C₂×Dic₃	—	2	1	no	—
92	D₁₂	—	4	2	no	—
93	D₁₂	—	4	1	no	—
94	D₁₂	—	4	1	no	—
95	D₁₂	—	4	1	no	—
96	D₁₂	—	4	1	no	—
97	C₄×D₃	—	4	4	no	—
98	Dic₆	—	2	1	no	—
99	Dic₆	SD₁₆	1	2	yes	complemented
100	C₃×C₈	—	4	1	no	—
101	C₃⋊C₈	—	4	1	no	—
102	C₃⋊C₈	—	4	1	no	—
103	#41	—	12	2	no	—
104	#41	—	12	2	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	3	1	no	—
107	C₂×D₄	—	3	2	no	—
108	C₂×D₄	—	6	2	no	—
109	C₂×D₄	—	6	2	no	—
110	C₂×D₄	—	6	1	no	—
111	C₂×D₄	—	6	1	no	—
112	C₂×D₄	#67	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	6	2	no	—
115	SD₁₆	—	4	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	6	1	no	—
118	C₂×C₈	—	6	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	3	1	no	—
122	C₄⋊C₄	—	6	1	no	—
123	C₄⋊C₄	#67	1	1	yes	characteristic
124	C₂²⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	6	2	no	—
126	C₄²	—	6	1	no	—
127	C₄²	C₂²×D₃	1	1	yes	characteristic
128	C₂²×C₃	—	4	1	no	—
129	C₂²×C₃	—	4	1	no	—
130	C₂²×C₃	—	4	1	no	—
131	C₂²×C₃	#120	1	1	yes	characteristic
132	C₂×D₃	—	4	2	no	—
133	C₂×D₃	—	4	2	no	—
134	C₂×D₃	—	4	2	no	—
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	4	1	no	—
137	C₃×C₄	—	8	1	no	—
138	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
139	C₃×C₄	#135	1	1	yes	characteristic
140	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
142	Dic₃	—	2	2	no	—
143	Dic₃	—	4	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	6	2	no	—
146	Q₈	—	3	2	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	6	1	no	—
150	D₄	—	2	1	no	—
151	D₄	—	2	1	no	—
152	D₄	—	12	2	no	—
153	D₄	—	12	4	no	—
154	D₄	—	12	1	no	—
155	D₄	—	12	1	no	—
156	D₄	—	12	1	no	—
157	D₄	—	12	1	no	—
158	D₄	—	24	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	6	1	no	—
164	C₂×C₄	—	6	1	no	—
165	C₂×C₄	—	12	4	no	—
166	C₂×C₄	D₃×D₄	1	1	yes	characteristic
167	C₂×C₄	D₃×D₄	1	1	yes	characteristic
168	C₂×C₄	C₂×#67	1	1	yes	characteristic
169	C₈	—	4	1	no	—
170	C₈	—	12	1	no	—
171	C₈	—	12	1	no	—
172	C₂×C₃	—	8	1	no	—
173	C₂×C₃	#449	1	1	yes	characteristic
174	C₂×C₃	#450	1	1	yes	characteristic
175	C₂×C₃	#455	1	1	yes	characteristic
176	D₃	—	8	2	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	12	2	no	—
181	C₂²	—	12	2	no	—
182	C₂²	—	12	2	no	—
183	C₂²	#937	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	6	2	no	—
187	C₄	—	8	1	no	—
188	C₄	—	12	1	no	—
189	C₄	D₃×SD₁₆	1	1	yes	characteristic
190	C₄	#914	1	1	yes	characteristic
191	C₄	#932	1	1	yes	characteristic
192	C₄	#950	1	1	yes	characteristic
193	C₃	#1642	1	1	yes	characteristic, complemented, Sylow
194	C₂	—	8	1	no	—
195	C₂	—	24	2	no	—
196	C₂	#5037	1	1	yes	characteristic
197	C₂	#5169	1	1	yes	characteristic
198	C₂	#5202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#937
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#461	C₂
socle	C₂²×C₃	#120
prime radical	#5553	C₂
hypercenter	#461	D₃
hypocenter	C₃	#1642
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
2. C₃×#461	C₂	semidirect
3. #5094	C₂	semidirect
4. #5088	C₂	semidirect
5. D₄⋊(C₃⋊C₈)	C₂	semidirect
6. #4485	C₂	semidirect
7. #4484	C₂	semidirect
29. #461	D₃	semidirect
99. Dic₆	SD₁₆	semidirect
193. C₃	#1642	semidirect

Maximal subgroups

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

subgroup	index
#5553	2
C₃×#461	2
#5094	2
#5088	2
D₄⋊(C₃⋊C₈)	2
#4485	2
#4484	2
#1642	3