Finite Group Databasebeta

Subgroups — #70404

Order 384.

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

normal subgroups only

This group has 190 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×#421	C₂	1	1	yes	maximal, characteristic
2	#5237	C₂	1	1	yes	maximal, characteristic
3	#5233	C₂	1	1	yes	maximal, characteristic
4	#4999	C₂	1	1	yes	maximal, characteristic, complemented
5	#4558	C₂	1	1	yes	maximal, characteristic, complemented
6	#4556	C₂	1	1	yes	maximal, characteristic, complemented
7	#4531	C₂	1	1	yes	maximal, characteristic, complemented
8	#2080	—	3	1	no	maximal, Sylow
9	C₂×C₃×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₃×#117	C₂²	1	1	yes	characteristic
12	#926	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×#104	—	2	2	no	—
17	#836	—	2	1	no	—
18	#835	—	2	1	no	—
19	#833	—	2	1	no	—
20	#832	—	2	1	no	—
21	C₄⋊(C₃⋊C₈)	—	2	2	no	—
22	#424	—	3	1	no	—
23	#421	D₃	1	1	yes	characteristic
24	#416	—	3	1	no	—
25	#413	—	3	1	no	—
26	#344	—	3	1	no	—
27	#341	—	3	1	no	—
28	H(3,ℤ/4)	—	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	C₂³	1	1	yes	characteristic
38	C₃×M₁₆	—	2	1	no	—
39	C₃×M₁₆	—	2	1	no	—
40	C₂×C₃×C₈	—	2	1	no	—
41	C₃×C₄⋊C₄	—	2	1	no	—
42	C₃×C₂²⋊C₄	—	2	1	no	—
43	C₃×C₄²	D₄	1	2	yes	—
44	#217	—	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	—	2	1	no	—
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	#117	—	3	1	no	—
62	#117	C₂×D₃	1	1	yes	characteristic
63	C₄⋊C₈	—	6	2	no	—
64	#104	—	2	2	no	—
65	#103	—	6	1	no	—
66	#102	—	6	1	no	—
67	#101	—	6	1	no	—
68	#100	—	6	1	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	2	no	—
78	C₂×C₃×C₄	—	2	1	no	—
79	C₂×C₃×C₄	—	4	1	no	—
80	C₂×C₃×C₄	—	4	1	no	—
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄	#41	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×Dic₃	—	4	1	no	—
88	C₃×C₈	—	4	1	no	—
89	C₃⋊C₈	—	4	1	no	—
90	C₃⋊C₈	—	4	1	no	—
91	C₃⋊C₈	—	4	1	no	—
92	#41	—	2	1	no	—
93	#41	—	2	1	no	—
94	#41	—	2	1	no	—
95	#41	—	2	1	no	—
96	C₂×Q₈	D₁₂	1	1	yes	characteristic
97	C₂×D₄	—	2	1	no	—
98	C₂×D₄	D₁₂	1	1	yes	characteristic
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	6	1	no	—
104	M₁₆	—	6	1	no	—
105	M₁₆	—	6	1	no	—
106	M₁₆	—	6	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	6	1	no	—
109	C₂×C₈	—	6	1	no	—
110	C₂×C₈	—	6	1	no	—
111	C₂×C₈	—	6	1	no	—
112	C₂×C₈	—	6	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	6	1	no	—
115	C₂²⋊C₄	—	2	1	no	—
116	C₂²⋊C₄	—	6	1	no	—
117	C₄²	—	6	1	no	—
118	C₄²	#67	1	2	yes	—
119	C₂²×C₃	—	4	1	no	—
120	C₂²×C₃	—	4	1	no	—
121	C₂²×C₃	—	8	1	no	—
122	C₂²×C₃	#123	1	1	yes	characteristic
123	C₂²×C₃	#123	1	1	yes	characteristic
124	C₂²×C₃	#126	1	1	yes	characteristic
125	C₃×C₄	—	4	2	no	—
126	C₃×C₄	—	4	1	no	—
127	C₃×C₄	#121	1	1	yes	characteristic
128	C₃×C₄	#123	1	1	yes	characteristic
129	C₃×C₄	#124	1	1	yes	characteristic
130	C₃×C₄	#126	1	1	yes	characteristic
131	Dic₃	—	8	1	no	—
132	C₂³	—	2	1	no	—
133	C₂³	#237	1	1	yes	characteristic
134	Q₈	—	2	1	no	—
135	Q₈	—	2	1	no	—
136	D₄	—	2	1	no	—
137	D₄	—	2	1	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	12	1	no	—
145	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
146	C₂×C₄	#235	1	1	yes	characteristic
147	C₂×C₄	#237	1	1	yes	characteristic
148	C₂×C₄	#237	1	1	yes	characteristic
149	C₂×C₄	#237	1	1	yes	characteristic
150	C₂×C₄	C₂×#67	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	—	12	1	no	—
153	C₈	—	12	1	no	—
154	C₈	—	12	1	no	—
155	C₂×C₃	—	2	1	no	—
156	C₂×C₃	—	2	1	no	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	—	8	1	no	—
159	C₂×C₃	#397	1	1	yes	characteristic
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	8	1	no	—
163	C₂²	#879	1	1	yes	characteristic
164	C₂²	#886	1	1	yes	characteristic
165	C₂²	#935	1	1	yes	characteristic
166	C₄	—	4	2	no	—
167	C₄	—	4	1	no	—
168	C₄	—	24	1	no	—
169	C₄	#879	1	1	yes	characteristic
170	C₄	#886	1	1	yes	characteristic
171	C₄	#930	1	1	yes	characteristic
172	C₄	#936	1	1	yes	characteristic
173	C₃	#2080	1	1	yes	characteristic, complemented, Sylow
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#4953	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#936
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#421	C₂
socle	C₂×C₃	#397
prime radical	C₂×C₃×#41	C₂²
hypercenter	#421	D₃
hypocenter	C₃	#2080
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
4. #4999	C₂	semidirect
5. #4558	C₂	semidirect
6. #4556	C₂	semidirect
7. #4531	C₂	semidirect
173. C₃	#2080	semidirect

Maximal subgroups

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

subgroup	index
C₃×#421	2
#5237	2
#5233	2
#4999	2
#4558	2
#4556	2
#4531	2
#2080	3