Finite Group Databasebeta

← #30379

Subgroups — #30379

Order 256.

339 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 121 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#2860C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic
8#2065C212yesmaximal
9#1941C212yesmaximal, complemented
10#1934C212yesmaximal
11C4×C4⋊C8C211yesmaximal, characteristic, complemented
12C2×#126—21no—
13C2×#125—21no—
14C2×#106C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#102C2212yes—
18C2×#102C2212yes—
19C2×C4×C8C2212yes—
20#390C2211yescharacteristic
21#390C2211yescharacteristic
22#390C2211yescharacteristic
23#389—21no—
24#389—21no—
25#389C2212yes—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#389C2212yes—
29#389C2211yescharacteristic
30#388—21no—
31#388—21no—
32#388—21no—
33#384C2211yescharacteristic
34#384C2211yescharacteristic
35#382—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×D4C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#126—44no—
47#125—44no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
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×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#106—24no—
76#106—42no—
77C4⋊C8—24no—
78#102—24no—
79#102—42no—
80C4×C8—42no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95—21no—
85#95—21no—
86C24—21no—
87C2×D4—22no—
88C2×D4—21no—
89C2×D4—22no—
90C2×D4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8#4114yes—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C42—22no—
133C42—22no—
134C42—22no—
135C42—21no—
136C42—22no—
137C42—21no—
138C42—41no—
139C42SD1614yes—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23C22×D411yescharacteristic
148D4—42no—
149D4—42no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4C2×SD1612yes—
179C2×C4C2×SD1612yes—
180C2×C4C2×SD1612yes—
181C2×C4#13512yes—
182C2×C4AGL(1,ℤ/8)12yes—
183C2×C4C2×#4112yes—
184C2×C4H(5,𝔽2)11yescharacteristic
185C2×C42−1+412yes—
186C2×C42−1+411yescharacteristic
187C8—24no—
188C8—42no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#53211yescharacteristic
197C22#53611yescharacteristic
198C22#53711yescharacteristic
199C22C22×SD1611yescharacteristic
200C22C2×#13511yescharacteristic
201C22C2×AGL(1,ℤ/8)11yescharacteristic
202C22#57511yescharacteristic
203C4—22no—
204C4—22no—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—41no—
209C4—81no—
210C4—82no—
211C4—81no—
212C4—81no—
213C4—82no—
214C4#53912yes—
215C4#54112yes—
216C2—161no—
217C2#266411yescharacteristic
218C2#316711yescharacteristic
219C2#317111yescharacteristic
220C2#322711yescharacteristic
221C2#323211yescharacteristic
222C2#324611yescharacteristic
223C2#324711yescharacteristic
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. #2859C2semidirect
3. #2100C2semidirect
5. #2088C2semidirect
6. #2084C2semidirect
9. #1941C2semidirect
11. C4×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#28602
#28592
#21002
#20942
#20882
#20842
#20712
#20652
#19412
#19342
C4×C4⋊C82