Finite Group Databasebeta

Subgroups — #78674

Order 384.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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	#5830	C₂	1	2	yes	maximal, complemented
2	C₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
3	#5135	C₂	1	2	yes	maximal, complemented
4	#5115	C₂	1	2	yes	maximal, complemented
5	#4737	C₂	1	4	yes	maximal, complemented
6	#4732	C₂	1	4	yes	maximal, complemented
7	#3199	—	3	1	no	maximal, Sylow
8	C₂×C₃×#41	C₂²	1	1	yes	characteristic
9	C₂²×D₁₂	C₂²	1	2	yes	complemented
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
12	#947	—	2	2	no	—
13	#940	—	2	2	no	—
14	#937	—	2	2	no	—
15	#937	—	2	2	no	—
16	#930	C₂²	1	4	yes	—
17	#930	C₂²	1	4	yes	complemented
18	#926	C₂²	1	2	yes	complemented
19	C₂×D₂₄	C₂²	1	2	yes	—
20	C₂×D₂₄	C₂²	1	2	yes	—
21	C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
22	#821	C₂²	1	4	yes	complemented
23	#821	C₂²	1	4	yes	complemented
24	#818	C₂²	1	2	yes	complemented
25	#817	C₂²	1	2	yes	complemented
26	#535	—	3	2	no	—
27	#482	—	3	4	no	—
28	#469	—	3	2	no	—
29	#466	—	3	2	no	—
30	#447	—	3	4	no	—
31	#408	D₃	1	1	yes	characteristic, complemented
32	C₂³×D₃	—	2	2	no	—
33	C₃×#41	—	4	1	no	—
34	C₃×#41	—	4	1	no	—
35	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
36	C₂×C₃×D₄	D₄	1	1	yes	characteristic
37	C₂×C₃×D₄	D₄	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂×#67	—	2	4	no	—
41	C₂×#67	—	2	4	no	—
42	C₂×D₁₂	—	2	2	no	—
43	C₂×D₁₂	—	4	2	no	—
44	C₂×D₁₂	—	4	2	no	—
45	C₂×D₁₂	C₂³	1	4	yes	—
46	C₃×M₁₆	—	4	1	no	—
47	C₂×C₃×C₈	—	4	1	no	—
48	C₂×C₃×C₈	C₂³	1	2	yes	—
49	C₂×C₃×C₈	C₂³	1	2	yes	—
50	#217	—	2	2	no	—
51	(C₂×D₃)⋊C₄	—	2	4	no	—
52	(C₂×D₃)⋊C₄	—	2	4	no	—
53	C₄⋊Dic₃	C₂³	1	4	yes	—
54	C₄×Dic₃	—	2	2	no	—
55	D₂₄	—	4	2	no	—
56	D₂₄	—	4	2	no	—
57	C₂×#41	C₂×D₃	1	1	yes	characteristic
58	C₂²×D₄	—	3	2	no	—
59	C₂×D₈	—	3	2	no	—
60	C₂×D₈	—	3	2	no	—
61	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
62	C₂²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
63	#127	—	6	2	no	—
64	#124	—	6	2	no	—
65	#121	—	3	4	no	—
66	#121	—	3	4	no	—
67	#120	—	6	2	no	—
68	#120	—	6	2	no	—
69	#117	—	3	2	no	—
70	#107	—	3	2	no	—
71	#106	—	3	2	no	—
72	#102	—	3	4	no	—
73	#102	—	3	4	no	—
74	C₂²⋊C₈	C₂×D₃	1	4	yes	complemented
75	C₂³×C₃	C₂×D₄	1	2	yes	—
76	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×D₃	—	2	4	no	—
78	C₂²×D₃	—	4	4	no	—
79	C₂²×D₃	—	8	4	no	—
80	C₃×Q₈	—	4	1	no	—
81	C₃×D₄	—	4	2	no	—
82	C₃×D₄	—	4	1	no	—
83	C₂×C₃×C₄	—	2	1	no	—
84	C₂×C₃×C₄	—	2	1	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	#67	—	8	4	no	—
91	#67	—	8	4	no	—
92	C₂×Dic₃	—	2	4	no	—
93	D₁₂	—	2	4	no	—
94	D₁₂	—	2	4	no	—
95	D₁₂	—	4	4	no	—
96	C₃×C₈	—	2	2	no	—
97	C₃×C₈	—	2	2	no	—
98	C₂⁴	—	6	2	no	—
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	C₂×Q₈	D₁₂	1	1	yes	characteristic
102	C₂×D₄	—	3	4	no	—
103	C₂×D₄	—	6	4	no	—
104	C₂×D₄	—	6	4	no	—
105	C₂×D₄	—	6	2	no	—
106	C₂×D₄	—	12	2	no	—
107	C₂×D₄	—	12	2	no	—
108	C₂×D₄	D₁₂	1	1	yes	characteristic
109	C₂×D₄	D₁₂	1	2	yes	—
110	C₂²×C₄	C₂²×D₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
112	D₈	—	12	2	no	—
113	D₈	—	12	2	no	—
114	M₁₆	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	C₂²×D₃	1	2	yes	—
117	C₂×C₈	C₂²×D₃	1	2	yes	—
118	C₄⋊C₄	—	3	4	no	—
119	C₂²⋊C₄	—	6	2	no	—
120	C₂²⋊C₄	—	6	4	no	—
121	C₂²⋊C₄	—	6	4	no	—
122	C₄²	—	6	2	no	—
123	C₂²×C₃	—	2	2	no	—
124	C₂²×C₃	—	2	1	no	—
125	C₂²×C₃	—	2	2	no	—
126	C₂²×C₃	—	2	4	no	—
127	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₃	—	4	4	no	—
129	C₂×D₃	—	4	4	no	—
130	C₂×D₃	—	4	4	no	—
131	C₂×D₃	—	8	8	no	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
135	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
136	Dic₃	—	4	4	no	—
137	C₂³	—	6	4	no	—
138	C₂³	—	12	4	no	—
139	C₂³	—	24	4	no	—
140	C₂³	C₂×D₁₂	1	2	yes	—
141	C₂³	C₂×D₁₂	1	1	yes	characteristic
142	Q₈	—	4	1	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	1	no	—
145	D₄	—	6	4	no	—
146	D₄	—	6	4	no	—
147	D₄	—	12	4	no	—
148	D₄	—	24	4	no	—
149	D₄	—	24	4	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	6	4	no	—
155	C₂×C₄	C₂×D₁₂	1	2	yes	—
156	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₂×C₃	—	4	2	no	—
161	C₂×C₃	—	4	1	no	—
162	C₂×C₃	#534	1	1	yes	characteristic
163	C₂×C₃	#576	1	2	yes	—
164	D₃	—	8	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	4	no	—
169	C₂²	—	12	4	no	—
170	C₂²	—	12	4	no	—
171	C₂²	—	12	4	no	—
172	C₂²	—	24	8	no	—
173	C₂²	C₂²×D₁₂	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	12	4	no	—
177	C₄	#1004	1	1	yes	characteristic
178	C₄	#1004	1	1	yes	characteristic
179	C₃	#3199	1	1	yes	characteristic, complemented, Sylow
180	C₂	—	4	2	no	—
181	C₂	—	4	1	no	—
182	C₂	—	24	4	no	—
183	C₂	#5493	1	1	yes	characteristic
184	C₂	#5752	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#408	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	#408	D₃
hypocenter	C₃	#3199
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

Complements

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

N	H ≅ G/N	type
1. #5830	C₂	semidirect
2. C₃×#408	C₂	semidirect
3. #5135	C₂	semidirect
4. #5115	C₂	semidirect
5. #4737	C₂	semidirect
6. #4732	C₂	semidirect
9. C₂²×D₁₂	C₂²	semidirect
10. C₂×C₃×M₁₆	C₂²	semidirect
11. C₂²×C₃×C₈	C₂²	semidirect
17. #930	C₂²	semidirect
18. #926	C₂²	semidirect
21. C₃×C₂²⋊C₈	C₂²	semidirect
22. #821	C₂²	semidirect
23. #821	C₂²	semidirect
24. #818	C₂²	semidirect
25. #817	C₂²	semidirect
31. #408	D₃	semidirect
61. C₂×M₁₆	C₂×D₃	semidirect
62. C₂²×C₈	C₂×D₃	semidirect
74. C₂²⋊C₈	C₂×D₃	semidirect
179. C₃	#3199	semidirect

Maximal subgroups

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

subgroup	index
#5830	2
C₃×#408	2
#5135	2
#5115	2
#4737	2
#4732	2
#3199	3