Finite Group Databasebeta

← #29260

Subgroups — #29260

Order 256.

400 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 129 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
2C4×#124C211yesmaximal, characteristic
3#2095C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic
7#1983C212yesmaximal
8#1949C212yesmaximal, complemented
9#1865C212yesmaximal, complemented
10#1862C212yesmaximal
11#1831C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24#391—21no—
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#389—21no—
28#389C2212yes—
29#388—21no—
30#388—21no—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2212yes—
35C4×C22⋊C4—21no—
36C4×C22⋊C4—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—24no—
50#124—22no—
51#124Q814yescomplemented
52C4×Q8—42no—
53C4×Q8—44no—
54C4×D4—42no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
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—22no—
70C2×C42—21no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80C4⋊C8—24no—
81#103—24no—
82#103—42no—
83#102—24no—
84#102—42no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91C24—21no—
92C2×Q8—24no—
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8C2×Q812yes—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4C2×Q812yes—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C4⋊C4#4114yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42—41no—
146C42—42no—
147C42—44no—
148C42C2×Q812yes—
149C42#4112yes—
150C23—22no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23C22×D411yescharacteristic
158Q8—24no—
159Q8—48no—
160Q8—42no—
161D4—24no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—84no—
192C2×C4C22⋊Q812yes—
193C2×C4C22⋊Q812yes—
194C2×C4#13512yes—
195C2×C4AGL(1,ℤ/8)12yes—
196C2×C4#13712yes—
197C2×C4C22×Q811yescharacteristic
198C2×C4C2×#4112yes—
199C2×C4C2×#4111yescharacteristic
200C2×C4H(5,𝔽2)12yes—
201C8—42no—
202C8—42no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22C2×C22⋊Q811yescharacteristic
211C22#52511yescharacteristic
212C22#53511yescharacteristic
213C22C2×#13511yescharacteristic
214C22C2×AGL(1,ℤ/8)11yescharacteristic
215C22C2×#13711yescharacteristic
216C22#57511yescharacteristic
217C4—22no—
218C4—22no—
219C4—22no—
220C4—22no—
221C4—22no—
222C4—42no—
223C4—44no—
224C4—41no—
225C4—81no—
226C4—82no—
227C4—81no—
228C4#55412yes—
229C4#55912yes—
230C2—82no—
231C2#258011yescharacteristic
232C2#307611yescharacteristic
233C2#307911yescharacteristic
234C2#310511yescharacteristic
235C2#311711yescharacteristic
236C2#321011yescharacteristic
237C2#321311yescharacteristic
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. #2095C2semidirect
5. #2024C2semidirect
8. #1949C2semidirect
9. #1865C2semidirect
11. #1831C2semidirect
51. #124Q8semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
C4×#1242
#20952
#20942
#20242
#20232
#19832
#19492
#18652
#18622
#18312