Finite Group Databasebeta

Subgroups — #70371

Order 384.

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

normal subgroups only

This group has 230 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	C₂×C₃×#104	C₂	1	1	yes	maximal, characteristic
2	#5234	C₂	1	1	yes	maximal, characteristic
3	#5233	C₂	1	1	yes	maximal, characteristic
4	#5119	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#4558	C₂	1	1	yes	maximal, characteristic, complemented
7	#4523	C₂	1	1	yes	maximal, characteristic, complemented
8	#1937	—	3	1	no	maximal, Sylow
9	C₂×C₃×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	#926	C₂²	1	1	yes	characteristic
13	C₂×#208	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#104	—	2	2	no	—
17	#835	C₄	1	2	yes	complemented
18	#832	C₄	1	2	yes	complemented
19	#830	—	2	2	no	—
20	Dic₃⋊C₈	—	2	2	no	—
21	C₄×C₃⋊C₈	—	2	2	no	—
22	#424	—	3	1	no	—
23	C₂×#104	D₃	1	1	yes	characteristic
24	#416	—	3	1	no	—
25	#408	—	3	1	no	—
26	C₂×C₄×C₈	—	3	1	no	—
27	#341	—	3	1	no	—
28	#339	—	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	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×D₄	—	2	1	no	—
35	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	—	2	1	no	—
37	C₂²×C₃×C₄	—	2	1	no	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₃×M₁₆	—	2	1	no	—
40	C₃×M₁₆	—	2	1	no	—
41	C₂×C₃×C₈	—	2	1	no	—
42	C₃×C₄²	—	2	1	no	—
43	C₃×C₄²	D₄	1	2	yes	—
44	#217	—	2	1	no	—
45	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
46	C₄×Dic₃	—	2	1	no	—
47	#208	—	2	1	no	—
48	#208	—	2	1	no	—
49	C₂×C₃⋊C₈	—	2	1	no	—
50	C₂×C₃⋊C₈	—	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	C₂×C₄	1	2	yes	—
55	C₂×C₃⋊C₈	D₄	1	2	yes	—
56	C₂×#41	C₂×D₃	1	1	yes	characteristic
57	C₂×M₁₆	—	3	1	no	—
58	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₈	—	3	1	no	—
60	C₂²×C₈	—	3	1	no	—
61	#117	—	3	1	no	—
62	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₄⋊C₈	—	6	2	no	—
64	#104	—	2	2	no	—
65	#103	—	3	2	no	—
66	#102	—	3	2	no	—
67	C₂²⋊C₈	—	6	2	no	—
68	C₄×C₈	—	6	2	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	4	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	1	no	—
80	C₂×C₃×C₄	—	4	1	no	—
81	C₂×C₃×C₄	—	4	1	no	—
82	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄	#41	1	1	yes	characteristic
86	C₂×C₃×C₄	#41	1	1	yes	characteristic
87	C₂×C₃×C₄	#41	1	1	yes	characteristic
88	C₂×Dic₃	—	2	2	no	—
89	C₃×C₈	—	4	1	no	—
90	C₃⋊C₈	—	2	2	no	—
91	C₃⋊C₈	—	2	2	no	—
92	C₃⋊C₈	—	4	1	no	—
93	#41	—	2	1	no	—
94	#41	—	2	1	no	—
95	#41	—	2	1	no	—
96	#41	—	2	1	no	—
97	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	C₄×D₃	1	1	yes	characteristic
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	6	1	no	—
106	M₁₆	—	6	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	3	2	no	—
109	C₂×C₈	—	3	2	no	—
110	C₂×C₈	—	6	1	no	—
111	C₂×C₈	—	6	1	no	—
112	C₂×C₈	—	6	1	no	—
113	C₂×C₈	—	6	1	no	—
114	C₂×C₈	—	6	1	no	—
115	C₄⋊C₄	—	3	2	no	—
116	C₂²⋊C₄	—	6	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	6	1	no	—
119	C₄²	#67	1	2	yes	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	4	1	no	—
122	C₂²×C₃	—	4	2	no	—
123	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₃	#123	1	1	yes	characteristic
125	C₂²×C₃	#126	1	1	yes	characteristic
126	C₃×C₄	—	2	4	no	—
127	C₃×C₄	—	4	1	no	—
128	C₃×C₄	#117	1	1	yes	characteristic
129	C₃×C₄	C₄×D₄	1	1	yes	characteristic
130	C₃×C₄	#121	1	1	yes	characteristic
131	C₃×C₄	#124	1	1	yes	characteristic
132	Dic₃	—	4	2	no	—
133	C₂³	—	2	1	no	—
134	C₂³	#237	1	1	yes	characteristic
135	Q₈	—	2	1	no	—
136	Q₈	—	2	1	no	—
137	D₄	—	2	1	no	—
138	D₄	—	2	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	6	2	no	—
147	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄	#235	1	1	yes	characteristic
149	C₂×C₄	D₃×D₄	1	1	yes	characteristic
150	C₂×C₄	#237	1	1	yes	characteristic
151	C₂×C₄	#237	1	1	yes	characteristic
152	C₂×C₄	C₂×#67	1	1	yes	characteristic
153	C₈	—	4	1	no	—
154	C₈	—	6	2	no	—
155	C₈	—	6	2	no	—
156	C₈	—	12	1	no	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	—	8	1	no	—
160	C₂×C₃	#388	1	1	yes	characteristic
161	C₂×C₃	#443	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	2	no	—
165	C₂²	#879	1	1	yes	characteristic
166	C₂²	#67⋊C₄	1	1	yes	characteristic
167	C₂²	#935	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	4	1	no	—
170	C₄	—	12	2	no	—
171	C₄	#877	1	1	yes	characteristic
172	C₄	#885	1	1	yes	characteristic
173	C₄	C₄×#67	1	1	yes	characteristic
174	C₄	#938	1	1	yes	characteristic
175	C₃	#1937	1	1	yes	characteristic, complemented, Sylow
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	8	1	no	—
179	C₂	#4824	1	2	yes	—
180	C₂	#4944	1	1	yes	characteristic

Complements

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

N	H ≅ G/N	type
4. #5119	C₂	semidirect
5. C₂×C₄×C₃⋊C₈	C₂	semidirect
6. #4558	C₂	semidirect
7. #4523	C₂	semidirect
17. #835	C₄	semidirect
18. #832	C₄	semidirect
175. C₃	#1937	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#104	2
#5234	2
#5233	2
#5119	2
C₂×C₄×C₃⋊C₈	2
#4558	2
#4523	2
#1937	3