Finite Group Databasebeta

← #30064

Subgroups — #30064

Order 256.

359 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#2861C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#2096C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2042C212yesmaximal
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9#1928C212yesmaximal, complemented
10#1926C212yesmaximal
11#1839C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C4×Q8—21no—
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22#391—21no—
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#390C2211yescharacteristic
26#390C2211yescharacteristic
27#389—21no—
28#389C2212yes—
29#388—21no—
30#388—21no—
31#388—21no—
32#388—21no—
33#384C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—24no—
49#125—42no—
50#124—42no—
51C4×Q8—44no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42—22no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—42no—
77#106—42no—
78#103—24no—
79#103—42no—
80#102—24no—
81#102—42no—
82#95—22no—
83#95—22no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88C24—21no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—21no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—44no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4#4114yes—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—21no—
142C42—21no—
143C42—41no—
144C42—44no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23C22×D411yescharacteristic
153Q8—48no—
154Q8—42no—
155Q8—42no—
156D4—42no—
157D4—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
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—21no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4#13512yes—
189C2×C4AGL(1,ℤ/8)12yes—
190C2×C4#13712yes—
191C2×C4C2×#4112yes—
192C2×C4H(5,𝔽2)12yes—
193C2×C42−1+411yescharacteristic
194C2×C42−1+411yescharacteristic
195C8—42no—
196C8—42no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22#52511yescharacteristic
205C22#53511yescharacteristic
206C22#53611yescharacteristic
207C22C2×#13511yescharacteristic
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22C2×#13711yescharacteristic
210C22#57511yescharacteristic
211C4—22no—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—81no—
220C4—82no—
221C4—81no—
222C4#54212yes—
223C4#54212yes—
224C2—161no—
225C2#263211yescharacteristic
226C2#310811yescharacteristic
227C2#311411yescharacteristic
228C2#320711yescharacteristic
229C2#321611yescharacteristic
230C2#323311yescharacteristic
231C2#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
2. C4×C4⋊Q8C2semidirect
3. #2096C2semidirect
5. #2059C2semidirect
7. #2024C2semidirect
9. #1928C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28612
C4×C4⋊Q82
#20962
#20932
#20592
#20422
#20242
#20232
#19282
#19262
#18392