Finite Group Databasebeta

Subgroups — #70338

Order 384.

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

normal subgroups only

This group has 212 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	C₃×#419	C₂	1	1	yes	maximal, characteristic
2	C₂×#836	C₂	1	1	yes	maximal, characteristic
3	C₂×#835	C₂	1	1	yes	maximal, characteristic
4	#4997	C₂	1	1	yes	maximal, characteristic
5	#4995	C₂	1	1	yes	maximal, characteristic
6	#4558	C₂	1	1	yes	maximal, characteristic
7	#4554	C₂	1	1	yes	maximal, characteristic
8	#1930	—	3	1	no	maximal, Sylow
9	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	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×#103	—	2	1	no	—
17	C₃×#103	—	2	1	no	—
18	#836	C₄	1	4	yes	complemented
19	#835	—	2	1	no	—
20	#835	—	2	1	no	—
21	#808	—	2	1	no	—
22	#807	—	2	1	no	—
23	#803	—	2	1	no	—
24	C₄×C₃⋊C₈	—	2	1	no	—
25	#428	—	3	1	no	—
26	#419	D₃	1	1	yes	characteristic
27	C₂×#103	—	3	1	no	—
28	C₂×#101	—	3	1	no	—
29	#405	—	3	1	no	—
30	#343	—	3	1	no	—
31	#341	—	3	1	no	—
32	C₂×C₃×Q₈	—	2	1	no	—
33	C₂×C₃×Q₈	—	2	1	no	—
34	C₂×C₃×Q₈	—	2	1	no	—
35	C₂×C₃×Q₈	—	2	1	no	—
36	C₂×C₃×Q₈	—	2	1	no	—
37	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄	—	2	1	no	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	—	2	1	no	—
41	C₃×M₁₆	—	2	1	no	—
42	C₃×M₁₆	—	2	1	no	—
43	C₂×C₃×C₈	—	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	1	no	—
47	C₃×C₄²	—	2	1	no	—
48	C₄⋊Dic₃	—	2	1	no	—
49	C₄⋊Dic₃	—	2	1	no	—
50	#208	—	2	1	no	—
51	#208	—	2	1	no	—
52	#208	C₂×C₄	1	2	yes	—
53	#208	C₂×C₄	1	2	yes	—
54	C₂×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃⋊C₈	—	2	1	no	—
56	C₂×C₃⋊C₈	—	2	1	no	—
57	C₂×C₃⋊C₈	—	2	1	no	—
58	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
59	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
60	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
61	C₂×M₁₆	—	3	1	no	—
62	C₂×M₁₆	—	3	1	no	—
63	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₈	—	3	1	no	—
65	#117	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	—	3	1	no	—
67	#107	—	6	1	no	—
68	#106	—	6	1	no	—
69	#103	—	2	1	no	—
70	#103	—	2	1	no	—
71	#103	—	6	1	no	—
72	#103	—	6	1	no	—
73	#101	—	3	4	no	—
74	#97	—	6	1	no	—
75	C₄×C₈	—	6	1	no	—
76	C₂³×C₃	#41	1	1	yes	characteristic
77	C₃×Q₈	—	2	2	no	—
78	C₃×Q₈	—	2	2	no	—
79	C₃×Q₈	—	4	1	no	—
80	C₃×Q₈	—	4	1	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	2	1	no	—
84	C₂×C₃×C₄	—	4	1	no	—
85	C₂×C₃×C₄	—	4	1	no	—
86	C₂×C₃×C₄	C₂²×C₄	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×C₃×C₄	#41	1	1	yes	characteristic
90	C₂×C₃×C₄	#41	1	1	yes	characteristic
91	C₂×C₃×C₄	#41	1	1	yes	characteristic
92	C₂×Dic₃	—	4	1	no	—
93	C₂×Dic₃	—	8	1	no	—
94	C₃×C₈	—	4	1	no	—
95	C₃⋊C₈	—	2	2	no	—
96	C₃⋊C₈	—	2	2	no	—
97	C₃⋊C₈	—	2	1	no	—
98	C₃⋊C₈	—	2	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	C₄×D₃	1	2	yes	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	6	1	no	—
107	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	3	2	no	—
111	M₁₆	—	3	2	no	—
112	M₁₆	—	6	1	no	—
113	M₁₆	—	6	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	3	1	no	—
116	C₂×C₈	—	3	1	no	—
117	C₂×C₈	—	6	1	no	—
118	C₂×C₈	—	6	1	no	—
119	C₂×C₈	—	6	1	no	—
120	C₂×C₈	—	6	1	no	—
121	C₄⋊C₄	—	6	1	no	—
122	C₄⋊C₄	—	6	1	no	—
123	C₄⋊C₄	#67	1	1	yes	characteristic
124	C₄⋊C₄	#67	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	#117	1	1	yes	characteristic
130	C₂²×C₃	#121	1	1	yes	characteristic
131	C₂²×C₃	#126	1	1	yes	characteristic
132	C₃×C₄	—	4	2	no	—
133	C₃×C₄	—	4	1	no	—
134	C₃×C₄	—	4	1	no	—
135	C₃×C₄	C₄×D₄	1	1	yes	characteristic
136	C₃×C₄	C₄×D₄	1	1	yes	characteristic
137	C₃×C₄	#123	1	1	yes	characteristic
138	C₃×C₄	#124	1	1	yes	characteristic
139	Dic₃	—	8	1	no	—
140	C₂³	#237	1	1	yes	characteristic
141	Q₈	—	2	2	no	—
142	Q₈	—	2	2	no	—
143	Q₈	—	4	1	no	—
144	Q₈	—	4	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	12	1	no	—
151	C₂×C₄	—	24	1	no	—
152	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄	#235	1	1	yes	characteristic
154	C₂×C₄	D₃×D₄	1	1	yes	characteristic
155	C₂×C₄	#237	1	1	yes	characteristic
156	C₂×C₄	#237	1	1	yes	characteristic
157	C₂×C₄	C₂×#67	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	6	2	no	—
160	C₈	—	6	1	no	—
161	C₈	—	6	1	no	—
162	C₈	—	6	2	no	—
163	C₂×C₃	—	2	1	no	—
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	#388	1	1	yes	characteristic
166	C₂×C₃	#471	1	1	yes	characteristic
167	C₂×C₃	#473	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#877	1	1	yes	characteristic
171	C₂²	#879	1	1	yes	characteristic
172	C₂²	#939	1	1	yes	characteristic
173	C₄	—	4	2	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	24	1	no	—
177	C₄	#67⋊C₄	1	1	yes	characteristic
178	C₄	#883	1	1	yes	characteristic
179	C₄	C₄×#67	1	1	yes	characteristic
180	C₄	#936	1	1	yes	characteristic
181	C₃	#1930	1	1	yes	characteristic, complemented, Sylow
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	#4945	1	1	yes	characteristic
185	C₂	#5203	1	1	yes	characteristic
186	C₂	#5204	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#939
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#419	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#419	D₃
hypocenter	C₃	#1930
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
18. #836	C₄	semidirect
181. C₃	#1930	semidirect

Maximal subgroups

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

subgroup	index
C₃×#419	2
C₂×#836	2
C₂×#835	2
#4997	2
#4995	2
#4558	2
#4554	2
#1930	3