Finite Group Databasebeta

Subgroups — #71925

Order 384.

208 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 61 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×#554	C₂	1	1	yes	maximal, characteristic
2	#5079	C₂	1	1	yes	maximal, characteristic, complemented
3	#5020	C₂	1	1	yes	maximal, characteristic, complemented
4	#5019	C₂	1	1	yes	maximal, characteristic
5	#4546	C₂	1	1	yes	maximal, characteristic, complemented
6	#4535	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	#1643	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×#125	—	2	1	no	—
11	C₃×C₂²⋊Q₈	—	2	1	no	—
12	C₃×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₄	—	2	1	no	—
15	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	—	2	1	no	—
17	C₂×#214	—	2	1	no	—
18	#869	C₂²	1	1	yes	characteristic
19	#835	—	2	1	no	—
20	#835	—	2	2	no	—
21	#810	—	2	1	no	—
22	#808	—	2	1	no	—
23	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#554	D₃	1	1	yes	characteristic
27	#487	—	3	1	no	—
28	#475	—	3	1	no	—
29	#461	—	3	1	no	—
30	#333	—	3	1	no	—
31	Q₈⋊C₈	—	3	1	no	—
32	D₄⋊C₈	—	3	1	no	—
33	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
34	C₂×C₃×Q₈	D₄	1	2	yes	—
35	C₂×C₃×D₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	—	2	2	no	—
37	C₂²×C₃×C₄	—	2	1	no	—
38	C₂×Dic₆	—	2	1	no	—
39	C₃×C₄⋊C₄	—	2	2	no	—
40	C₃×C₄⋊C₄	—	2	2	no	—
41	C₃×C₄⋊C₄	—	2	1	no	—
42	C₃×C₄⋊C₄	—	2	1	no	—
43	C₃×C₄⋊C₄	—	2	1	no	—
44	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₃×C₂²⋊C₄	—	2	2	no	—
47	C₃×C₂²⋊C₄	—	2	1	no	—
48	C₃×C₄²	—	2	1	no	—
49	C₃×C₄²	C₂³	1	1	yes	characteristic
50	#214	—	4	2	no	—
51	C₄⋊Dic₃	—	2	1	no	—
52	C₄⋊Dic₃	—	2	1	no	—
53	C₂×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃⋊C₈	—	2	1	no	—
56	C₂×SD₁₆	—	6	1	no	—
57	C₄⋊Q₈	—	3	1	no	—
58	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
59	#125	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₂²⋊Q₈	—	2	2	no	—
62	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
63	C₄×D₄	—	2	1	no	—
64	C₄×D₄	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	2	1	no	—
66	#106	—	6	1	no	—
67	C₄⋊C₈	—	3	1	no	—
68	C₄⋊C₈	—	3	1	no	—
69	#103	—	6	1	no	—
70	#103	—	6	2	no	—
71	#103	—	6	1	no	—
72	C₄×C₈	—	3	1	no	—
73	C₂³×C₃	—	2	1	no	—
74	C₃×Q₈	—	2	1	no	—
75	C₃×Q₈	—	2	1	no	—
76	C₃×Q₈	—	4	2	no	—
77	C₃×D₄	—	2	1	no	—
78	C₃×D₄	SD₁₆	1	2	yes	—
79	C₂×C₃×C₄	—	2	1	no	—
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	1	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	4	1	no	—
84	C₂×C₃×C₄	—	4	1	no	—
85	C₂×C₃×C₄	—	8	2	no	—
86	C₂×C₃×C₄	C₂×D₄	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×Dic₃	—	4	1	no	—
90	Dic₆	—	4	1	no	—
91	Dic₆	—	4	1	no	—
92	C₃⋊C₈	—	4	1	no	—
93	C₃⋊C₈	—	4	1	no	—
94	C₃⋊C₈	—	4	1	no	—
95	C₂×Q₈	—	6	1	no	—
96	C₂×Q₈	#67	1	1	yes	characteristic
97	C₂×Q₈	#67	1	2	yes	—
98	C₂×D₄	#67	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	1	no	—
101	SD₁₆	—	12	2	no	—
102	C₂×C₈	—	6	1	no	—
103	C₂×C₈	—	6	1	no	—
104	C₂×C₈	—	6	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	6	1	no	—
111	C₄⋊C₄	—	6	1	no	—
112	C₄⋊C₄	#67	1	1	yes	characteristic
113	C₄⋊C₄	#67	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	C₂²×D₃	1	1	yes	characteristic
118	C₂²×C₃	—	2	2	no	—
119	C₂²×C₃	—	4	1	no	—
120	C₂²×C₃	—	4	1	no	—
121	C₂²×C₃	#120	1	1	yes	characteristic
122	C₃×C₄	—	2	1	no	—
123	C₃×C₄	—	4	1	no	—
124	C₃×C₄	—	4	1	no	—
125	C₃×C₄	—	4	1	no	—
126	C₃×C₄	—	4	2	no	—
127	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
128	C₃×C₄	#135	1	1	yes	characteristic
129	C₃×C₄	#137	1	1	yes	characteristic
130	C₃×C₄	#137	1	1	yes	characteristic
131	Dic₃	—	8	1	no	—
132	C₂³	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	2	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	12	1	no	—
137	Q₈	—	12	1	no	—
138	D₄	—	2	1	no	—
139	D₄	#215	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	12	1	no	—
148	C₂×C₄	C₂×#67	1	1	yes	characteristic
149	C₂×C₄	C₂×#67	1	1	yes	characteristic
150	C₂×C₄	C₂×#67	1	1	yes	characteristic
151	C₈	—	12	1	no	—
152	C₈	—	12	1	no	—
153	C₈	—	12	1	no	—
154	C₂×C₃	—	4	2	no	—
155	C₂×C₃	#448	1	1	yes	characteristic
156	C₂×C₃	#452	1	1	yes	characteristic
157	C₂×C₃	#455	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	#953	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	2	no	—
167	C₄	—	24	1	no	—
168	C₄	C₂×#215	1	1	yes	characteristic
169	C₄	#942	1	1	yes	characteristic
170	C₄	#950	1	1	yes	characteristic
171	C₄	#951	1	1	yes	characteristic
172	C₃	#1643	1	1	yes	characteristic, complemented, Sylow
173	C₂	—	4	2	no	—
174	C₂	#5227	1	1	yes	characteristic
175	C₂	#5239	1	1	yes	characteristic
176	C₂	#5242	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#953
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#554	C₂
socle	C₂²×C₃	#120
prime radical	C₂×C₃×D₄	D₄
hypercenter	#554	D₃
hypocenter	C₃	#1643
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #554

Complements

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

N	H ≅ G/N	type
2. #5079	C₂	semidirect
3. #5020	C₂	semidirect
5. #4546	C₂	semidirect
6. #4535	C₂	semidirect
172. C₃	#1643	semidirect

Maximal subgroups

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

subgroup	index
C₃×#554	2
#5079	2
#5020	2
#5019	2
#4546	2
#4535	2
D₄⋊(C₃⋊C₈)	2
#1643	3