Finite Group Databasebeta

← #23817

Subgroups — #23817

Order 256.

405 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 131 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#2101C212yesmaximal, complemented
4#2072C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic
6#2042C212yesmaximal
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9C4×#103C212yesmaximal, complemented
10C4×#102C212yesmaximal
11#1744C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×C4×C8C2212yes—
22#391—21no—
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#389—21no—
26#389C2212yes—
27#388—21no—
28#388—21no—
29#384C2211yescharacteristic
30#382C2211yescharacteristic
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2212yes—
33C4×C22⋊C4—21no—
34C4×C22⋊C4—21no—
35C43C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
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#124—22no—
52C4×Q8—24no—
53C4×Q8—44no—
54C4×D4—24no—
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#103—24no—
79#103—24no—
80#102—24no—
81#102—24no—
82C4×C8—24no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89C24—21no—
90C2×Q8—24no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8#4112yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—21no—
100C2×D4#4112yes—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4#4114yes—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—21no—
143C42—21no—
144C42—22no—
145C42—42no—
146C42—44no—
147C42#4114yes—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23C22×D411yescharacteristic
156Q8—24no—
157Q8—48no—
158Q8—42no—
159D4—24no—
160D4—42no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—84no—
189C2×C4#12412yes—
190C2×C4#12412yes—
191C2×C4#13512yes—
192C2×C4#13512yes—
193C2×C4AGL(1,ℤ/8)12yes—
194C2×C4#13712yes—
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4112yes—
198C2×C4C2×#4112yes—
199C8—42no—
200C8—42no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22#52511yescharacteristic
209C22#52511yescharacteristic
210C22C2×#12411yescharacteristic
211C22C2×#13511yescharacteristic
212C22C2×#13511yescharacteristic
213C22C2×AGL(1,ℤ/8)11yescharacteristic
214C22C2×#13711yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—44no—
219C4—41no—
220C4—41no—
221C4—42no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4#48914yes—
226C4#52912yes—
227C4#52912yes—
228C2—82no—
229C2#239811yescharacteristic
230C2#310511yescharacteristic
231C2#310611yescharacteristic
232C2#310711yescharacteristic
233C2#310811yescharacteristic
234C2#313111yescharacteristic
235C2C2×#48911yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #2101C2semidirect
4. #2072C2semidirect
7. #2024C2semidirect
9. C4×#103C2semidirect
11. #1744C2semidirect

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
#21012
#20722
#20712
#20422
#20242
#20232
C4×#1032
C4×#1022
#17442