Finite Group Databasebeta

Subgroups — #9338

Order 256.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	#3238	C₂	1	1	yes	maximal, characteristic
2	#2995	C₂	1	1	yes	maximal, characteristic
3	#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	#1556	C₂	1	1	yes	maximal, characteristic, complemented
5	#1529	C₂	1	1	yes	maximal, characteristic
6	#1307	C₂	1	1	yes	maximal, characteristic, complemented
7	#1291	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#529	C₂²	1	1	yes	characteristic
10	#501	—	2	1	no	—
11	#489	—	2	1	no	—
12	#483	C₄	1	2	yes	—
13	#480	C₄	1	2	yes	—
14	#477	—	2	1	no	—
15	#476	—	2	1	no	—
16	#446	—	2	1	no	—
17	#436	—	2	1	no	—
18	#435	—	2	1	no	—
19	#435	—	2	1	no	—
20	#433	—	2	1	no	—
21	#432	C₂²	1	1	yes	characteristic
22	#432	C₂²	1	1	yes	characteristic
23	#432	C₂²	1	1	yes	characteristic
24	#424	—	2	1	no	—
25	#423	C₂²	1	1	yes	characteristic
26	C₄×M₁₆	C₂²	1	1	yes	characteristic
27	#343	—	2	1	no	—
28	#343	—	2	1	no	—
29	#332	—	2	1	no	—
30	#332	—	2	1	no	—
31	#330	—	2	1	no	—
32	#329	—	2	1	no	—
33	C₂²×Q₈	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	D₄	1	1	yes	characteristic
37	C₂×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₄⋊Q₈	—	2	1	no	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	#126	—	2	1	no	—
44	#125	D₄	1	1	yes	characteristic
45	#124	D₄	1	1	yes	characteristic
46	#123	—	2	1	no	—
47	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
48	#121	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈	D₄	1	1	yes	characteristic
50	C₄×D₄	—	2	1	no	—
51	C₄×D₄	D₄	1	1	yes	characteristic
52	#117	—	2	1	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	#107	—	2	2	no	—
57	#106	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₄⋊C₈	D₄	1	1	yes	characteristic
66	#103	—	2	2	no	—
67	#102	—	2	2	no	—
68	C₂²⋊C₈	—	2	1	no	—
69	C₂²⋊C₈	—	2	1	no	—
70	C₂²⋊C₈	—	2	1	no	—
71	C₂²⋊C₈	—	2	1	no	—
72	C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	#97	—	2	1	no	—
74	#97	—	2	1	no	—
75	#97	—	2	1	no	—
76	#97	—	2	1	no	—
77	#97	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	—	4	1	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	#41	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	—	4	1	no	—
93	M₁₆	—	4	1	no	—
94	M₁₆	—	4	1	no	—
95	M₁₆	—	8	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	8	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
113	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	2	1	no	—
116	C₂²⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	#41	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	C₄×D₄	1	1	yes	characteristic
126	Q₈	—	4	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	8	2	no	—
129	D₄	—	4	1	no	—
130	D₄	—	8	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄	#120	1	1	yes	characteristic
145	C₂×C₄	#121	1	1	yes	characteristic
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	#123	1	1	yes	characteristic
148	C₈	—	4	2	no	—
149	C₈	—	4	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	4	1	no	—
154	C₈	—	8	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	#386	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	2	no	—
171	C₄	#455	1	1	yes	characteristic
172	C₄	#456	1	1	yes	characteristic
173	C₂	—	4	1	no	—
174	C₂	—	8	1	no	—
175	C₂	#1882	1	1	yes	characteristic
176	C₂	#1903	1	1	yes	characteristic
177	C₂	#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1997	C₂	semidirect
4. #1556	C₂	semidirect
6. #1307	C₂	semidirect
7. #1291	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3238	2
#2995	2
#1997	2
#1556	2
#1529	2
#1307	2
#1291	2