Finite Group Databasebeta

Subgroups — #70301

Order 384.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#417	C₂	1	1	yes	maximal, characteristic
2	#5234	C₂	1	1	yes	maximal, characteristic
3	#5232	C₂	1	1	yes	maximal, characteristic
4	C₂×#807	C₂	1	1	yes	maximal, characteristic, complemented
5	#4558	C₂	1	1	yes	maximal, characteristic, complemented
6	#4554	C₂	1	1	yes	maximal, characteristic, complemented
7	#4530	C₂	1	1	yes	maximal, characteristic, complemented
8	#2078	—	3	1	no	maximal, Sylow
9	C₂×C₃×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₄	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	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#103	—	2	1	no	—
17	C₃×#102	—	2	1	no	—
18	#835	—	2	1	no	—
19	#832	—	2	1	no	—
20	#830	—	2	2	no	—
21	#807	—	2	2	no	—
22	C₂×#107	—	3	1	no	—
23	#417	—	3	1	no	—
24	#417	D₃	1	1	yes	characteristic
25	#408	—	3	1	no	—
26	#343	—	3	1	no	—
27	#343	—	3	1	no	—
28	#341	—	3	1	no	—
29	C₃×#41	—	2	1	no	—
30	C₃×#41	—	2	1	no	—
31	C₃×#41	—	2	1	no	—
32	C₃×#41	—	2	1	no	—
33	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
34	C₂×C₃×D₄	—	2	1	no	—
35	C₂×C₃×D₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	—	2	1	no	—
37	C₂²×C₃×C₄	—	2	1	no	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	—	2	1	no	—
40	C₃×M₁₆	—	2	1	no	—
41	C₃×M₁₆	—	2	1	no	—
42	C₂×C₃×C₈	—	2	1	no	—
43	C₃×C₄⋊C₄	—	2	1	no	—
44	C₃×C₄⋊C₄	—	2	1	no	—
45	C₄⋊Dic₃	—	2	1	no	—
46	C₄⋊Dic₃	—	2	1	no	—
47	#208	—	2	1	no	—
48	#208	—	2	1	no	—
49	#208	—	2	1	no	—
50	#208	—	2	1	no	—
51	C₂×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃⋊C₈	D₄	1	2	yes	—
56	C₂×#41	C₂×D₃	1	1	yes	characteristic
57	C₂×M₁₆	—	3	1	no	—
58	C₂×M₁₆	—	3	1	no	—
59	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₈	—	3	1	no	—
61	C₂×C₄⋊C₄	—	3	1	no	—
62	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
63	#107	—	6	2	no	—
64	#103	—	2	1	no	—
65	#103	—	6	1	no	—
66	#102	—	2	1	no	—
67	#102	—	6	1	no	—
68	C₂²⋊C₈	—	6	2	no	—
69	C₂³×C₃	—	2	1	no	—
70	C₂³×C₃	#41	1	1	yes	characteristic
71	C₃×Q₈	—	2	1	no	—
72	C₃×Q₈	—	2	1	no	—
73	C₃×D₄	—	2	1	no	—
74	C₃×D₄	—	2	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	—	4	1	no	—
79	C₂×C₃×C₄	—	4	1	no	—
80	C₂×C₃×C₄	—	4	1	no	—
81	C₂×C₃×C₄	—	8	1	no	—
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄	#41	1	1	yes	characteristic
85	C₂×C₃×C₄	#41	1	1	yes	characteristic
86	C₂×C₃×C₄	#41	1	1	yes	characteristic
87	C₂×C₃×C₄	#41	1	1	yes	characteristic
88	C₂×Dic₃	—	4	1	no	—
89	C₂×Dic₃	—	8	1	no	—
90	C₃×C₈	—	4	1	no	—
91	C₃⋊C₈	—	2	2	no	—
92	C₃⋊C₈	—	4	1	no	—
93	C₃⋊C₈	—	4	1	no	—
94	#41	—	2	1	no	—
95	#41	—	2	1	no	—
96	#41	—	2	1	no	—
97	#41	—	2	1	no	—
98	C₂×Q₈	#67	1	1	yes	characteristic
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	#67	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	6	1	no	—
104	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	6	1	no	—
108	M₁₆	—	6	1	no	—
109	M₁₆	—	6	1	no	—
110	M₁₆	—	6	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	3	2	no	—
113	C₂×C₈	—	6	1	no	—
114	C₂×C₈	—	6	1	no	—
115	C₂×C₈	—	6	1	no	—
116	C₂×C₈	—	6	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	6	1	no	—
120	C₄⋊C₄	—	6	1	no	—
121	C₂²×C₃	—	2	1	no	—
122	C₂²×C₃	—	2	1	no	—
123	C₂²×C₃	—	4	1	no	—
124	C₂²×C₃	—	4	1	no	—
125	C₂²×C₃	—	4	1	no	—
126	C₂²×C₃	#121	1	1	yes	characteristic
127	C₂²×C₃	#126	1	1	yes	characteristic
128	C₂²×C₃	#126	1	1	yes	characteristic
129	C₃×C₄	—	4	1	no	—
130	C₃×C₄	—	8	1	no	—
131	C₃×C₄	#123	1	1	yes	characteristic
132	C₃×C₄	#123	1	1	yes	characteristic
133	C₃×C₄	#123	1	1	yes	characteristic
134	C₃×C₄	#124	1	1	yes	characteristic
135	Dic₃	—	8	1	no	—
136	C₂³	—	2	1	no	—
137	C₂³	#237	1	1	yes	characteristic
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	12	1	no	—
150	C₂×C₄	—	24	1	no	—
151	C₂×C₄	#235	1	1	yes	characteristic
152	C₂×C₄	#235	1	1	yes	characteristic
153	C₂×C₄	D₃×D₄	1	1	yes	characteristic
154	C₂×C₄	#237	1	1	yes	characteristic
155	C₂×C₄	#237	1	1	yes	characteristic
156	C₂×C₄	C₂×#67	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	6	2	no	—
159	C₈	—	12	1	no	—
160	C₈	—	12	1	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	—	8	1	no	—
164	C₂×C₃	#397	1	1	yes	characteristic
165	C₂×C₃	#472	1	1	yes	characteristic
166	C₂×C₃	#473	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	#879	1	1	yes	characteristic
173	C₂²	#879	1	1	yes	characteristic
174	C₂²	#939	1	1	yes	characteristic
175	C₄	—	4	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	24	1	no	—
178	C₄	#883	1	1	yes	characteristic
179	C₄	#885	1	1	yes	characteristic
180	C₄	#929	1	1	yes	characteristic
181	C₄	#935	1	1	yes	characteristic
182	C₃	#2078	1	1	yes	characteristic, complemented, Sylow
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	8	1	no	—
186	C₂	#4952	1	1	yes	characteristic
187	C₂	#5199	1	1	yes	characteristic
188	C₂	#5204	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#939
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#417	C₂
socle	C₂²×C₃	#121
prime radical	C₂×C₃×#41	C₂²
hypercenter	#417	D₃
hypocenter	C₃	#2078
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
4. C₂×#807	C₂	semidirect
5. #4558	C₂	semidirect
6. #4554	C₂	semidirect
7. #4530	C₂	semidirect
182. C₃	#2078	semidirect

Maximal subgroups

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

subgroup	index
C₃×#417	2
#5234	2
#5232	2
C₂×#807	2
#4558	2
#4554	2
#4530	2
#2078	3