Finite Group Databasebeta

← #29389

Subgroups — #29389

Order 256.

387 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2083C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic
5#2058C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic
7#1926C212yesmaximal
8#1921C212yesmaximal, complemented
9#1919C212yesmaximal, complemented
10#1862C212yesmaximal
11#1839C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8—21no—
14C2×C4×D4C2211yescharacteristic
15C2×#117—21no—
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#102C2212yes—
19C2×#102C2212yes—
20#391—21no—
21#390C2211yescharacteristic
22#390C2211yescharacteristic
23#389—21no—
24#389C2212yes—
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#388—21no—
28#388—21no—
29#384C2211yescharacteristic
30C4×C4⋊C4—21no—
31C4×C4⋊C4C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33C4×C22⋊C4—21no—
34C43C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—24no—
49C4×Q8—44no—
50C4×D4—22no—
51C4×D4—22no—
52#117—44no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—24no—
79#106—24no—
80#102—24no—
81#102—24no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—21no—
86#95—22no—
87C24—21no—
88C2×Q8—24no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×D4—22no—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4#4112yes—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C4⋊C4—24no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—44no—
128C4⋊C4#4114yes—
129C4⋊C4#4112yes—
130C22⋊C4—24no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C42—22no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—22no—
142C42—22no—
143C42—42no—
144C42—44no—
145C23—22no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23C22×D411yescharacteristic
153Q8—48no—
154D4—24no—
155D4—42no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—48no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4#13512yes—
187C2×C4AGL(1,ℤ/8)12yes—
188C2×C4AGL(1,ℤ/8)12yes—
189C2×C4#13712yes—
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4112yes—
192C2×C4C2×#4111yescharacteristic
193C2×C42−1+412yes—
194C8—42no—
195C8—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22#52511yescharacteristic
204C22#52511yescharacteristic
205C22#53611yescharacteristic
206C22C2×#13511yescharacteristic
207C22C2×AGL(1,ℤ/8)11yescharacteristic
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22C2×#13711yescharacteristic
210C4—22no—
211C4—22no—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—41no—
216C4—42no—
217C4—41no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4#55212yes—
222C4#55612yes—
223C2—82no—
224C2#259911yescharacteristic
225C2#310511yescharacteristic
226C2#310711yescharacteristic
227C2#310911yescharacteristic
228C2#311211yescharacteristic
229C2#322811yescharacteristic
230C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊Q8C2semidirect
3. #2083C2semidirect
5. #2058C2semidirect
8. #1921C2semidirect
9. #1919C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
#22932
#20832
#20712
#20582
#20232
#19262
#19212
#19192
#18622
#18392