Finite Group Databasebeta

← #30428

Subgroups — #30428

Order 256.

460 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 107 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#3061C212yesmaximal, complemented
2#3012C212yesmaximal, complemented
3#2093C211yesmaximal, characteristic, complemented
4#2078C212yesmaximal, complemented
5#2039C212yesmaximal, complemented
6#2006C211yesmaximal, characteristic, complemented
7#1907C212yesmaximal, complemented
8#1876C212yesmaximal, complemented
9#1792C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)C2211yescharacteristic
11C2×AGL(1,ℤ/8)C2212yes—
12C2×#135C2212yes—
13C2×#131C2212yescomplemented
14#537C2212yes—
15#537C2211yescharacteristic, complemented
16#468—22no—
17#468—22no—
18#467—22no—
19#466—22no—
20#452—22no—
21#450—22no—
22#449—22no—
23#447—22no—
24C2×#107C2212yescomplemented
25C2×#104C2212yes—
26C2×#104C2212yescomplemented
27#417C2212yes—
28#417C2212yes—
29#417C2212yescomplemented
30C2×#102C2212yes—
31C2×#100C2211yescharacteristic
32#408C2212yes—
33#390C2211yescharacteristic
34#384C2211yescharacteristic, complemented
35#343C2211yescharacteristic, complemented
36#343C2212yescomplemented
37#341C2212yescomplemented
38#336C2211yescharacteristic, complemented
39H(5,𝔽2)—22no—
40H(5,𝔽2)—42no—
41H(5,𝔽2)—41no—
42C2×#41—22no—
43C2×#41C2312yes—
44C2×#41C2312yes—
45C22×D4—22no—
46C22×D4—41no—
47C22×D4C2311yescharacteristic
48AGL(1,ℤ/8)—24no—
49AGL(1,ℤ/8)—42no—
50#135—42no—
51#135—42no—
52C2×Q16D412yes—
53C2×SD16—22no—
54C2×SD16D412yes—
55C2×SD16D412yes—
56C2×D8—22no—
57C2×D8D412yes—
58#131—42no—
59#131—42no—
60C2×M16—22no—
61C2×M16C2312yes—
62C2×M16C2312yes—
63C22×C8—22no—
64C22×C8C2312yes—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#121—22no—
69#121—22no—
70#121—22no—
71#121—42no—
72#121—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—41no—
77C2×C42C2311yescharacteristic
78#107—22no—
79#107—22no—
80#104—42no—
81#104—42no—
82#103—22no—
83#103—22no—
84#103—22no—
85#102—22no—
86#102—22no—
87#102—22no—
88#102—22no—
89#102—22no—
90#100—41no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93C24—21no—
94C24—41no—
95#41—22no—
96#41—22no—
97#41—22no—
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102#41—22no—
103#41—44no—
104#41—44no—
105C2×Q8C2×D412yes—
106C2×Q8C2×D412yes—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—42no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—41no—
118C2×D4—42no—
119C2×D4C2×D412yes—
120C2×D4C2×D412yes—
121C2×D4#4112yes—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—41no—
128C22×C4C2411yescharacteristic
129Q16—42no—
130SD16—42no—
131SD16—42no—
132SD16—44no—
133D8—42no—
134D8—44no—
135M16—22no—
136M16—22no—
137M16—22no—
138M16—22no—
139M16—44no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—44no—
145C2×C8C2×D412yes—
146C2×C8C2×D412yes—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—82no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—82no—
157C42—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—44no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23—161no—
167C23—161no—
168C23H(5,𝔽2)11yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—22no—
176D4—24no—
177D4—22no—
178D4—48no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—161no—
185D4—162no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—44no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4C22×D412yes—
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)11yescharacteristic
203C2×C4H(5,𝔽2)12yes—
204C8—22no—
205C8—22no—
206C8—42no—
207C8—42no—
208C22—21no—
209C22—21no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—82no—
218C22—81no—
219C22—81no—
220C22—161no—
221C22—164no—
222C22#53811yescharacteristic
223C22D4211yescharacteristic
224C22#56011yescharacteristic
225C4—42no—
226C4—42no—
227C4—42no—
228C4—81no—
229C4—82no—
230C4#53412yes—
231C4#54612yes—
232C2—21no—
233C2—21no—
234C2—82no—
235C2—82no—
236C2—82no—
237C2#266811yescharacteristic
238C2#328911yescharacteristic
239C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3061C2semidirect
2. #3012C2semidirect
3. #2093C2semidirect
4. #2078C2semidirect
5. #2039C2semidirect
6. #2006C2semidirect
7. #1907C2semidirect
8. #1876C2semidirect
9. #1792C2semidirect
13. C2×#131C22semidirect
15. #537C22semidirect
24. C2×#107C22semidirect
26. C2×#104C22semidirect
29. #417C22semidirect
34. #384C22semidirect
35. #343C22semidirect
36. #343C22semidirect
37. #341C22semidirect
38. #336C22semidirect

Maximal subgroups

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

subgroupindex
#30612
#30122
#20932
#20782
#20392
#20062
#19072
#18762
#17922