Finite Group Databasebeta

Subgroups — #68645

Order 384.

186 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 75 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	#5106	C₂	1	1	yes	maximal, characteristic
2	#5102	C₂	1	1	yes	maximal, characteristic
3	#5101	C₂	1	1	yes	maximal, characteristic
4	C₃×#336	C₂	1	1	yes	maximal, characteristic
5	#4552	C₂	1	1	yes	maximal, characteristic
6	#4523	C₂	1	1	yes	maximal, characteristic
7	#4523	C₂	1	1	yes	maximal, characteristic
8	#1922	—	3	1	no	maximal, Sylow
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	#926	C₂²	1	1	yes	characteristic
13	#926	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	1	yes	characteristic
15	C₂×#208	C₂²	1	1	yes	characteristic
16	#818	C₄	1	1	yes	characteristic, complemented
17	#818	C₄	1	1	yes	characteristic, complemented
18	#817	C₄	1	1	yes	characteristic, complemented
19	#817	C₄	1	1	yes	characteristic, complemented
20	#815	—	2	1	no	—
21	Dic₃⋊C₈	—	2	1	no	—
22	Dic₃⋊C₈	—	2	1	no	—
23	C₈×Dic₃	—	2	1	no	—
24	#427	—	3	1	no	—
25	#424	—	3	1	no	—
26	#405	—	3	1	no	—
27	#341	—	3	1	no	—
28	#339	—	3	1	no	—
29	#339	—	3	1	no	—
30	#336	D₃	1	1	yes	characteristic
31	C₂²×C₃×C₄	—	2	1	no	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₈	—	2	1	no	—
34	C₂×C₃×C₈	—	2	1	no	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₂×C₃×C₈	—	2	1	no	—
37	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄²	—	2	1	no	—
42	C₃×C₄²	—	2	1	no	—
43	#217	—	2	1	no	—
44	#217	—	2	1	no	—
45	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₃	—	2	1	no	—
50	C₄×Dic₃	—	2	1	no	—
51	#208	—	2	1	no	—
52	#208	—	2	1	no	—
53	#208	—	2	1	no	—
54	#208	Q₈	1	2	yes	—
55	C₂×C₃⋊C₈	—	2	1	no	—
56	C₂×C₃⋊C₈	—	2	1	no	—
57	C₂×M₁₆	—	3	1	no	—
58	C₂×M₁₆	—	3	1	no	—
59	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
61	#117	—	3	1	no	—
62	#117	—	3	1	no	—
63	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
64	#107	—	3	1	no	—
65	#107	—	3	1	no	—
66	#106	—	3	1	no	—
67	#106	—	3	1	no	—
68	C₄⋊C₈	—	6	1	no	—
69	C₄⋊C₈	—	6	1	no	—
70	#97	—	6	1	no	—
71	C₄×C₈	—	6	1	no	—
72	C₂³×C₃	#41	1	1	yes	characteristic
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	4	1	no	—
75	C₂×C₃×C₄	—	4	1	no	—
76	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃×C₄	#41	1	1	yes	characteristic
80	C₂×C₃×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×C₄	#41	1	1	yes	characteristic
82	C₂×Dic₃	—	2	1	no	—
83	C₂×Dic₃	—	2	1	no	—
84	C₂×Dic₃	—	2	1	no	—
85	C₂×Dic₃	—	2	1	no	—
86	C₃×C₈	—	2	1	no	—
87	C₃×C₈	—	2	1	no	—
88	C₃×C₈	—	2	1	no	—
89	C₃×C₈	—	2	1	no	—
90	C₃⋊C₈	—	2	2	no	—
91	C₃⋊C₈	—	4	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
94	M₁₆	—	3	2	no	—
95	M₁₆	—	6	1	no	—
96	M₁₆	—	6	1	no	—
97	M₁₆	—	6	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	6	1	no	—
103	C₂×C₈	—	6	1	no	—
104	C₂×C₈	C₄×D₃	1	1	yes	characteristic
105	C₂×C₈	C₄×D₃	1	1	yes	characteristic
106	C₂×C₈	D₁₂	1	1	yes	characteristic
107	C₂×C₈	D₁₂	1	1	yes	characteristic
108	C₄⋊C₄	—	3	1	no	—
109	C₄⋊C₄	—	3	1	no	—
110	C₄⋊C₄	—	3	1	no	—
111	C₄⋊C₄	—	3	1	no	—
112	C₂²⋊C₄	—	6	1	no	—
113	C₂²⋊C₄	—	6	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	6	1	no	—
117	C₄²	—	6	1	no	—
118	C₂²×C₃	—	2	1	no	—
119	C₂²×C₃	—	2	1	no	—
120	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₃	#125	1	1	yes	characteristic
122	C₂²×C₃	#126	1	1	yes	characteristic
123	C₃×C₄	—	4	2	no	—
124	C₃×C₄	#117	1	1	yes	characteristic
125	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
126	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
127	C₃×C₄	#123	1	1	yes	characteristic
128	Dic₃	—	4	1	no	—
129	Dic₃	—	4	1	no	—
130	Dic₃	—	4	1	no	—
131	Dic₃	—	4	1	no	—
132	C₂³	#235	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	6	1	no	—
137	C₂×C₄	—	6	1	no	—
138	C₂×C₄	—	6	1	no	—
139	C₂×C₄	—	6	1	no	—
140	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
142	C₂×C₄	#237	1	1	yes	characteristic
143	C₂×C₄	#237	1	1	yes	characteristic
144	C₂×C₄	#237	1	1	yes	characteristic
145	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	6	2	no	—
151	C₈	—	12	1	no	—
152	C₂×C₃	—	2	1	no	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	#387	1	1	yes	characteristic
155	C₂×C₃	#443	1	1	yes	characteristic
156	C₂×C₃	#444	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	C₄×D₁₂	1	1	yes	characteristic
160	C₂²	#879	1	1	yes	characteristic
161	C₂²	#889	1	1	yes	characteristic
162	C₄	—	4	2	no	—
163	C₄	—	12	1	no	—
164	C₄	—	12	1	no	—
165	C₄	—	12	1	no	—
166	C₄	—	12	1	no	—
167	C₄	#877	1	1	yes	characteristic
168	C₄	#886	1	1	yes	characteristic
169	C₄	#887	1	1	yes	characteristic
170	C₄	#897	1	1	yes	characteristic
171	C₃	#1922	1	1	yes	characteristic, complemented, Sylow
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#4653	1	1	yes	characteristic
175	C₂	#4700	1	1	yes	characteristic
176	C₂	#4717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#336	C₂
socle	C₂²×C₃	C₄×D₄
prime radical	C₂³×C₃	#41
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#336	D₃
hypocenter	C₃	#1922
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #336

Complements

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

N	H ≅ G/N	type
16. #818	C₄	semidirect
17. #818	C₄	semidirect
18. #817	C₄	semidirect
19. #817	C₄	semidirect
171. C₃	#1922	semidirect

Maximal subgroups

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

subgroup	index
#5106	2
#5102	2
#5101	2
C₃×#336	2
#4552	2
#4523	2
#1922	3