Finite Group Databasebeta

Subgroups — #70669

Order 384.

196 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 83 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×#428	C₂	1	1	yes	maximal, characteristic
2	C₂×#822	C₂	1	1	yes	maximal, characteristic
3	#4997	C₂	1	1	yes	maximal, characteristic
4	#4995	C₂	1	1	yes	maximal, characteristic
5	C₂×#807	C₂	1	1	yes	maximal, characteristic
6	#4554	C₂	1	1	yes	maximal, characteristic
7	#4530	C₂	1	1	yes	maximal, characteristic
8	#1952	—	3	1	no	maximal, Sylow
9	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₄	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×#107	—	2	1	no	—
17	C₃×#106	—	2	1	no	—
18	#822	C₄	1	4	yes	complemented
19	#808	—	2	1	no	—
20	#807	—	2	1	no	—
21	#807	—	2	1	no	—
22	#807	—	2	1	no	—
23	#803	—	2	1	no	—
24	C₄×C₃⋊C₈	—	2	1	no	—
25	C₂×#108	—	3	1	no	—
26	#428	—	3	1	no	—
27	#428	D₃	1	1	yes	characteristic
28	C₂×#107	—	3	1	no	—
29	#405	—	3	1	no	—
30	#343	—	3	1	no	—
31	#343	—	3	1	no	—
32	C₂²×C₃×C₄	—	2	1	no	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃	—	2	1	no	—
35	C₃×M₁₆	—	2	1	no	—
36	C₃×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₈	—	2	1	no	—
38	C₃×C₄⋊C₄	—	2	1	no	—
39	C₃×C₄⋊C₄	—	2	1	no	—
40	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₃×C₂²⋊C₄	—	2	1	no	—
43	C₃×C₄²	—	2	1	no	—
44	C₄⋊Dic₃	—	2	1	no	—
45	C₄⋊Dic₃	—	2	1	no	—
46	#208	—	2	1	no	—
47	#208	—	2	1	no	—
48	#208	C₂×C₄	1	2	yes	—
49	#208	Q₈	1	2	yes	—
50	C₂×C₃⋊C₈	—	2	1	no	—
51	C₂×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
53	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
54	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₂×C₃⋊C₈	Q₈	1	2	yes	—
56	C₂×M₁₆	—	3	1	no	—
57	C₂×M₁₆	—	3	1	no	—
58	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₈	—	3	1	no	—
60	#117	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	3	1	no	—
62	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
63	#108	—	3	4	no	—
64	#107	—	2	1	no	—
65	#107	—	6	1	no	—
66	#107	—	6	1	no	—
67	#107	—	6	1	no	—
68	#106	—	2	1	no	—
69	#106	—	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	1	no	—
74	C₂×C₃×C₄	—	2	1	no	—
75	C₂×C₃×C₄	—	4	1	no	—
76	C₂×C₃×C₄	—	8	1	no	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₃×C₄	#41	1	1	yes	characteristic
83	C₂×Dic₃	—	4	1	no	—
84	C₂×Dic₃	—	8	1	no	—
85	C₃×C₈	—	2	2	no	—
86	C₃⋊C₈	—	2	2	no	—
87	C₃⋊C₈	—	2	2	no	—
88	C₃⋊C₈	C₄⋊C₄	1	2	yes	—
89	C₃⋊C₈	C₄⋊C₄	1	2	yes	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	6	1	no	—
92	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
93	M₁₆	—	2	1	no	—
94	M₁₆	—	3	2	no	—
95	M₁₆	—	3	2	no	—
96	M₁₆	—	6	1	no	—
97	M₁₆	—	6	1	no	—
98	M₁₆	C₄×D₃	1	2	yes	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	3	2	no	—
101	C₂×C₈	—	3	1	no	—
102	C₂×C₈	—	3	1	no	—
103	C₂×C₈	—	3	2	no	—
104	C₂×C₈	—	6	1	no	—
105	C₂×C₈	—	6	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	6	1	no	—
109	C₄⋊C₄	—	6	1	no	—
110	C₄⋊C₄	#67	1	1	yes	characteristic
111	C₄⋊C₄	#67	1	1	yes	characteristic
112	C₂²⋊C₄	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	—	2	1	no	—
116	C₂²×C₃	C₄×Q₈	1	1	yes	characteristic
117	C₂²×C₃	#121	1	1	yes	characteristic
118	C₂²×C₃	#125	1	1	yes	characteristic
119	C₃×C₄	—	4	1	no	—
120	C₃×C₄	—	4	1	no	—
121	C₃×C₄	—	8	1	no	—
122	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₃×C₄	C₄×D₄	1	1	yes	characteristic
124	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
125	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
126	Dic₃	—	8	1	no	—
127	C₂³	#237	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	8	1	no	—
132	C₂×C₄	—	12	1	no	—
133	C₂×C₄	—	24	1	no	—
134	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₄	#235	1	1	yes	characteristic
136	C₂×C₄	D₃×D₄	1	1	yes	characteristic
137	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
138	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
139	C₂×C₄	C₂×#67	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	3	2	no	—
142	C₈	—	3	2	no	—
143	C₈	—	6	2	no	—
144	C₈	—	6	2	no	—
145	C₂×C₃	—	2	1	no	—
146	C₂×C₃	—	2	1	no	—
147	C₂×C₃	#389	1	1	yes	characteristic
148	C₂×C₃	#472	1	1	yes	characteristic
149	C₂×C₃	#473	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#887	1	1	yes	characteristic
153	C₂²	#889	1	1	yes	characteristic
154	C₂²	#939	1	1	yes	characteristic
155	C₄	—	4	1	no	—
156	C₄	—	4	1	no	—
157	C₄	—	8	1	no	—
158	C₄	—	24	1	no	—
159	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
160	C₄	#896	1	1	yes	characteristic
161	C₄	C₄×#67	1	1	yes	characteristic
162	C₄	#944	1	1	yes	characteristic
163	C₃	#1952	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#4976	1	1	yes	characteristic
167	C₂	#5199	1	1	yes	characteristic
168	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×#428	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#428	D₃
hypocenter	C₃	#1952
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
18. #822	C₄	semidirect
163. C₃	#1952	semidirect

Maximal subgroups

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

subgroup	index
C₃×#428	2
C₂×#822	2
#4997	2
#4995	2
C₂×#807	2
#4554	2
#4530	2
#1952	3