Finite Group Databasebeta

Subgroups — #70401

Order 384.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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×#421	C₂	1	1	yes	maximal, characteristic
2	#5237	C₂	1	1	yes	maximal, characteristic
3	C₂×#837	C₂	1	1	yes	maximal, characteristic
4	#5124	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#4533	C₂	1	1	yes	maximal, characteristic, complemented
7	#4532	C₂	1	1	yes	maximal, characteristic, complemented
8	#1932	—	3	1	no	maximal, Sylow
9	C₂×C₃×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×#208	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	1	yes	characteristic
15	C₂×#208	C₂²	1	1	yes	characteristic
16	C₃×#104	—	2	2	no	—
17	#837	—	2	1	no	—
18	#837	—	2	1	no	—
19	#836	C₄	1	2	yes	complemented
20	#833	C₄	1	2	yes	complemented
21	#822	—	2	2	no	—
22	#815	—	2	1	no	—
23	C₈×Dic₃	—	2	1	no	—
24	#430	—	3	1	no	—
25	#421	D₃	1	1	yes	characteristic
26	C₂×#104	—	3	1	no	—
27	#413	—	3	1	no	—
28	C₄×M₁₆	—	3	1	no	—
29	#344	—	3	1	no	—
30	#339	—	3	1	no	—
31	C₃×#41	—	2	1	no	—
32	C₃×#41	—	2	1	no	—
33	C₃×#41	—	2	1	no	—
34	C₃×#41	—	2	1	no	—
35	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×D₄	—	2	1	no	—
37	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	—	2	1	no	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	—	2	1	no	—
41	C₃×M₁₆	—	2	1	no	—
42	C₃×M₁₆	D₄	1	2	yes	—
43	C₂×C₃×C₈	—	2	1	no	—
44	C₃×C₄⋊C₄	—	2	1	no	—
45	C₃×C₂²⋊C₄	—	2	1	no	—
46	C₃×C₄²	—	2	1	no	—
47	C₄×Dic₃	—	2	1	no	—
48	C₄×Dic₃	D₄	1	1	yes	characteristic
49	C₄×Dic₃	D₄	1	1	yes	characteristic
50	#208	—	2	1	no	—
51	#208	—	2	1	no	—
52	#208	—	2	1	no	—
53	#208	—	2	1	no	—
54	#208	C₂×C₄	1	2	yes	—
55	#208	C₂×C₄	1	2	yes	—
56	C₂×C₃⋊C₈	—	2	1	no	—
57	C₂×C₃⋊C₈	—	2	1	no	—
58	C₂×C₃⋊C₈	—	2	1	no	—
59	C₂×#41	C₂×D₃	1	1	yes	characteristic
60	C₂×M₁₆	—	3	1	no	—
61	C₂×M₁₆	—	3	1	no	—
62	C₂×M₁₆	—	3	1	no	—
63	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
64	#117	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄²	—	3	1	no	—
66	#108	—	6	2	no	—
67	#104	—	2	2	no	—
68	#104	—	6	1	no	—
69	#104	—	6	1	no	—
70	#101	—	3	2	no	—
71	#100	—	3	2	no	—
72	#97	—	6	1	no	—
73	C₄×C₈	—	6	1	no	—
74	C₂³×C₃	—	2	1	no	—
75	C₂³×C₃	#41	1	1	yes	characteristic
76	C₃×Q₈	—	2	1	no	—
77	C₃×Q₈	—	2	1	no	—
78	C₃×D₄	—	2	1	no	—
79	C₃×D₄	—	2	1	no	—
80	C₃×D₄	—	4	1	no	—
81	C₃×D₄	—	4	1	no	—
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	4	1	no	—
84	C₂×C₃×C₄	—	4	1	no	—
85	C₂×C₃×C₄	—	4	1	no	—
86	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄	#41	1	1	yes	characteristic
90	C₂×C₃×C₄	#41	1	1	yes	characteristic
91	C₂×C₃×C₄	#41	1	1	yes	characteristic
92	C₂×Dic₃	—	2	1	no	—
93	C₂×Dic₃	—	2	1	no	—
94	C₂×Dic₃	—	4	2	no	—
95	C₃×C₈	—	2	2	no	—
96	C₃⋊C₈	—	2	2	no	—
97	C₃⋊C₈	—	2	2	no	—
98	C₃⋊C₈	—	4	1	no	—
99	#41	—	2	1	no	—
100	#41	—	2	1	no	—
101	#41	—	2	1	no	—
102	#41	—	2	1	no	—
103	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	6	1	no	—
108	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
109	M₁₆	—	2	1	no	—
110	M₁₆	—	3	2	no	—
111	M₁₆	—	3	2	no	—
112	M₁₆	—	6	1	no	—
113	M₁₆	—	6	1	no	—
114	M₁₆	—	6	1	no	—
115	M₁₆	—	6	1	no	—
116	M₁₆	#67	1	2	yes	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	6	1	no	—
119	C₂×C₈	—	6	1	no	—
120	C₂×C₈	—	6	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	3	1	no	—
125	C₄²	—	3	1	no	—
126	C₄²	—	6	1	no	—
127	C₂²×C₃	—	4	1	no	—
128	C₂²×C₃	—	4	1	no	—
129	C₂²×C₃	—	8	1	no	—
130	C₂²×C₃	#117	1	1	yes	characteristic
131	C₂²×C₃	#123	1	1	yes	characteristic
132	C₂²×C₃	#124	1	1	yes	characteristic
133	C₃×C₄	—	4	1	no	—
134	C₃×C₄	—	8	1	no	—
135	C₃×C₄	C₄×D₄	1	1	yes	characteristic
136	C₃×C₄	C₄×D₄	1	1	yes	characteristic
137	C₃×C₄	#121	1	1	yes	characteristic
138	C₃×C₄	#126	1	1	yes	characteristic
139	Dic₃	—	2	2	no	—
140	Dic₃	—	4	1	no	—
141	C₂³	—	2	1	no	—
142	C₂³	#237	1	1	yes	characteristic
143	Q₈	—	2	1	no	—
144	Q₈	—	2	1	no	—
145	D₄	—	2	1	no	—
146	D₄	—	2	1	no	—
147	D₄	—	4	1	no	—
148	D₄	—	4	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	6	1	no	—
154	C₂×C₄	—	6	1	no	—
155	C₂×C₄	—	12	2	no	—
156	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₄	#235	1	1	yes	characteristic
158	C₂×C₄	D₃×D₄	1	1	yes	characteristic
159	C₂×C₄	#237	1	1	yes	characteristic
160	C₂×C₄	#237	1	1	yes	characteristic
161	C₂×C₄	C₂×#67	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	6	2	no	—
164	C₈	—	6	2	no	—
165	C₈	—	12	1	no	—
166	C₂×C₃	—	2	1	no	—
167	C₂×C₃	—	2	1	no	—
168	C₂×C₃	—	2	1	no	—
169	C₂×C₃	—	8	1	no	—
170	C₂×C₃	#388	1	1	yes	characteristic
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	8	1	no	—
174	C₂²	#877	1	1	yes	characteristic
175	C₂²	#883	1	1	yes	characteristic
176	C₂²	#936	1	1	yes	characteristic
177	C₄	—	4	1	no	—
178	C₄	—	6	2	no	—
179	C₄	—	8	1	no	—
180	C₄	—	12	1	no	—
181	C₄	#879	1	1	yes	characteristic
182	C₄	#67⋊C₄	1	1	yes	characteristic
183	C₄	C₄×#67	1	1	yes	characteristic
184	C₄	#939	1	1	yes	characteristic
185	C₃	#1932	1	1	yes	characteristic, complemented, Sylow
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	—	8	1	no	—
190	C₂	#4945	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#939
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#421	C₂
socle	C₂×C₃	#388
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#421	D₃
hypocenter	C₃	#1932
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
4. #5124	C₂	semidirect
5. Dic₃×M₁₆	C₂	semidirect
6. #4533	C₂	semidirect
7. #4532	C₂	semidirect
19. #836	C₄	semidirect
20. #833	C₄	semidirect
185. C₃	#1932	semidirect

Maximal subgroups

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

subgroup	index
C₃×#421	2
#5237	2
C₂×#837	2
#5124	2
Dic₃×M₁₆	2
#4533	2
#4532	2
#1932	3