Finite Group Databasebeta

Subgroups — #68945

Order 384.

252 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 79 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	#5745	C₂	1	1	yes	maximal, characteristic
2	#5132	C₂	1	1	yes	maximal, characteristic
3	#5125	C₂	1	1	yes	maximal, characteristic, complemented
4	#5124	C₂	1	1	yes	maximal, characteristic, complemented
5	#5110	C₂	1	1	yes	maximal, characteristic
6	C₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	#4552	C₂	1	1	yes	maximal, characteristic, complemented
8	#1858	—	3	1	no	maximal, Sylow
9	C₂×#235	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₈	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	#907	—	2	2	no	—
17	D₃×M₁₆	—	2	2	no	—
18	#901	—	2	2	no	—
19	#824	C₄	1	2	yes	—
20	#823	C₄	1	2	yes	—
21	#822	—	2	2	no	—
22	(C₂×D₃)⋊C₈	—	2	2	no	—
23	#819	—	2	2	no	—
24	#568	—	3	1	no	—
25	#430	—	3	1	no	—
26	#430	—	3	1	no	—
27	#413	—	3	1	no	—
28	#408	—	3	1	no	—
29	#341	—	3	1	no	—
30	#341	D₃	1	1	yes	characteristic, complemented
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×#67	Q₈	1	2	yes	—
33	#235	—	2	1	no	—
34	#235	—	2	1	no	—
35	#235	—	2	1	no	—
36	#235	—	2	1	no	—
37	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₃	D₄	1	2	yes	—
39	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
40	C₃×M₁₆	—	2	1	no	—
41	C₃×M₁₆	—	2	1	no	—
42	C₃×M₁₆	C₂×C₄	1	2	yes	—
43	C₃×M₁₆	D₄	1	2	yes	—
44	C₂×C₃×C₈	—	2	1	no	—
45	C₂×C₃×C₈	—	2	1	no	—
46	C₂×C₃×C₈	—	2	1	no	—
47	C₂×C₃×C₈	—	2	1	no	—
48	C₂×C₃×C₈	—	2	1	no	—
49	#208	—	2	1	no	—
50	#208	—	2	1	no	—
51	#208	—	2	1	no	—
52	#208	—	2	1	no	—
53	#208	C₂×C₄	1	2	yes	—
54	#208	D₄	1	2	yes	—
55	C₂×C₃⋊C₈	—	2	1	no	—
56	C₂×C₃⋊C₈	—	2	1	no	—
57	C₂×C₃⋊C₈	—	2	1	no	—
58	ΓL(1,𝔽₂₅)	—	2	4	no	—
59	D₃×C₈	—	2	4	no	—
60	C₂×#41	—	3	1	no	—
61	#131	—	6	2	no	—
62	#131	—	6	2	no	—
63	C₂×M₁₆	—	3	1	no	—
64	C₂×M₁₆	—	3	1	no	—
65	C₂×M₁₆	—	3	1	no	—
66	C₂×M₁₆	—	6	2	no	—
67	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
68	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
70	#108	—	6	2	no	—
71	#108	—	6	2	no	—
72	#101	—	3	2	no	—
73	#100	—	3	2	no	—
74	C₂²⋊C₈	—	6	2	no	—
75	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×D₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	#41	1	1	yes	characteristic
82	C₂×C₃×C₄	#41	1	1	yes	characteristic
83	#67	—	2	2	no	—
84	#67	—	2	2	no	—
85	C₂×Dic₃	C₄⋊C₄	1	2	yes	—
86	D₁₂	—	2	1	no	—
87	D₁₂	—	2	1	no	—
88	C₄×D₃	—	2	2	no	—
89	C₄×D₃	—	2	2	no	—
90	Dic₆	—	2	1	no	—
91	Dic₆	—	2	1	no	—
92	C₃×C₈	—	2	2	no	—
93	C₃×C₈	—	2	2	no	—
94	C₃×C₈	—	4	1	no	—
95	C₃⋊C₈	—	2	2	no	—
96	C₃⋊C₈	—	2	2	no	—
97	C₃⋊C₈	—	4	1	no	—
98	#41	—	6	1	no	—
99	#41	—	6	1	no	—
100	#41	—	6	1	no	—
101	#41	—	6	1	no	—
102	C₂×Q₈	—	3	1	no	—
103	C₂×D₄	—	3	2	no	—
104	C₂×D₄	—	3	1	no	—
105	C₂²×C₄	—	3	2	no	—
106	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
107	M₁₆	—	2	1	no	—
108	M₁₆	—	2	1	no	—
109	M₁₆	—	3	2	no	—
110	M₁₆	—	3	2	no	—
111	M₁₆	—	6	4	no	—
112	M₁₆	—	6	1	no	—
113	M₁₆	—	6	1	no	—
114	M₁₆	—	6	1	no	—
115	M₁₆	—	6	1	no	—
116	M₁₆	C₄×D₃	1	2	yes	—
117	M₁₆	D₁₂	1	2	yes	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	6	4	no	—
124	C₂×C₈	—	6	1	no	—
125	C₂×C₈	—	6	1	no	—
126	C₂×C₈	—	6	1	no	—
127	C₂²×C₃	—	4	1	no	—
128	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×D₃	—	2	2	no	—
132	C₂×D₃	—	4	2	no	—
133	C₃×C₄	C₄×D₄	1	1	yes	characteristic
134	C₃×C₄	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄	#120	1	1	yes	characteristic
136	C₃×C₄	#123	1	1	yes	characteristic
137	Dic₃	—	2	2	no	—
138	C₂³	—	3	2	no	—
139	C₂³	D₃×Q₈	1	1	yes	characteristic
140	Q₈	—	6	1	no	—
141	Q₈	—	6	1	no	—
142	D₄	—	6	2	no	—
143	D₄	—	6	2	no	—
144	D₄	—	6	1	no	—
145	D₄	—	6	1	no	—
146	C₂×C₄	—	3	2	no	—
147	C₂×C₄	—	6	2	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
151	C₂×C₄	#235	1	1	yes	characteristic
152	C₂×C₄	D₃×D₄	1	1	yes	characteristic
153	C₂×C₄	D₃×D₄	1	1	yes	characteristic
154	C₂×C₄	#237	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	2	2	no	—
157	C₈	—	4	1	no	—
158	C₈	—	6	2	no	—
159	C₈	—	6	2	no	—
160	C₈	—	12	1	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	—	2	1	no	—
164	C₂×C₃	#385	1	1	yes	characteristic
165	D₃	—	4	2	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	6	2	no	—
168	C₂²	—	12	2	no	—
169	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂²	#896	1	1	yes	characteristic
171	C₂²	#897	1	1	yes	characteristic
172	C₄	—	6	2	no	—
173	C₄	C₄×D₁₂	1	1	yes	characteristic
174	C₄	#67⋊C₄	1	1	yes	characteristic
175	C₄	#882	1	1	yes	characteristic
176	C₄	#883	1	1	yes	characteristic
177	C₃	#1858	1	1	yes	characteristic, complemented, Sylow
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	12	2	no	—
182	C₂	#4668	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#882
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#341	C₂
socle	C₂×C₃	#385
prime radical	C₂×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#341	D₃
hypocenter	C₃	#1858
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
3. #5125	C₂	semidirect
4. #5124	C₂	semidirect
6. C₃×#341	C₂	semidirect
7. #4552	C₂	semidirect
30. #341	D₃	semidirect
177. C₃	#1858	semidirect

Maximal subgroups

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

subgroup	index
#5745	2
#5132	2
#5125	2
#5124	2
#5110	2
C₃×#341	2
#4552	2
#1858	3