Finite Group Databasebeta

← #29823

Subgroups — #29823

Order 256.

391 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#497C211yesmaximal, characteristic, complemented
2#3040C211yesmaximal, characteristic
3#3039C211yesmaximal, characteristic, complemented
4#2923C211yesmaximal, characteristic, complemented
5#2047C212yesmaximal, complemented
6#2046C212yesmaximal, complemented
7#2020C212yesmaximal, complemented
8#2019C212yesmaximal
9#1967C211yesmaximal, characteristic, complemented
10#1966C211yesmaximal, characteristic, complemented
11#1838C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×#137C2211yescharacteristic
14C2×AGL(1,ℤ/8)C2211yescharacteristic
15C22×Q16C2212yes—
16C22×SD16C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C4×#41C2211yescharacteristic
20#497—22no—
21#497—22no—
22#464—22no—
23#463—22no—
24#462—22no—
25#461—22no—
26#433—22no—
27#432—22no—
28#424C2212yes—
29#423C2211yescharacteristic
30#423C2211yescharacteristic
31#419C2212yes—
32#419C2211yescharacteristic
33#418C2211yescharacteristic
34C2×#101C2212yes—
35C2×#101C2212yes—
36C2×#101C2212yes—
37C2×#100C2212yes—
38#408C2212yes—
39C4×M16C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#341C2212yes—
42#341C2212yes—
43C2×#41C2311yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×Q8C2312yes—
46C22×D4C2311yescharacteristic
47#137—42no—
48#137—42no—
49#137—42no—
50AGL(1,ℤ/8)—42no—
51C2×Q16—22no—
52C2×Q16—21no—
53C2×Q16—24no—
54C2×Q16—42no—
55C2×Q16—42no—
56C2×SD16—21no—
57C2×SD16—22no—
58C2×SD16—21no—
59C2×SD16—24no—
60C2×SD16—42no—
61C2×SD16—42no—
62C2×D8—21no—
63C2×M16C2311yescharacteristic
64C2×M16C2312yes—
65C2×M16C2311yescharacteristic
66C2×M16C2312yes—
67C22×C8C2312yes—
68C4⋊Q8—22no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71#124—22no—
72#124—21no—
73#124—21no—
74C4×Q8D412yes—
75C4×D4—22no—
76C4×D4D412yes—
77#117—21no—
78#117C2311yescharacteristic
79C2×C4⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C42—21no—
82C2×C42C2311yescharacteristic
83C4⋊C8—24no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86#103—24no—
87#103—22no—
88#102—22no—
89#101—42no—
90#101—42no—
91#101—42no—
92#100—42no—
93C22⋊C8—24no—
94#97—21no—
95#97D412yes—
96#97D412yes—
97C4×C8—21no—
98C24—21no—
99#41—22no—
100#41—22no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8C2×D411yescharacteristic
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4C2×D411yescharacteristic
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4#4112yes—
123C22×C4C2411yescharacteristic
124Q16—48no—
125Q16—42no—
126Q16—44no—
127SD16—48no—
128SD16—42no—
129SD16—44no—
130SD16—42no—
131D8—42no—
132M16—22no—
133M16—22no—
134M16—22no—
135M16—22no—
136M16—22no—
137M16—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8C2×D414yes—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4C2×D412yes—
149C22⋊C4—21no—
150C22⋊C4—22no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—22no—
154C42—21no—
155C42C2×D411yescharacteristic
156C42C2×D411yescharacteristic
157C42C2×D412yes—
158C23—21no—
159C23—41no—
160C23—42no—
161C23—82no—
162C23C2×#4111yescharacteristic
163Q8—21no—
164Q8—21no—
165Q8—42no—
166Q8—41no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—41no—
171Q8—42no—
172D4—21no—
173D4—21no—
174D4—42no—
175D4—41no—
176D4—42no—
177D4—41no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—82no—
188C2×C4#12112yes—
189C2×C4#12112yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C22×D411yescharacteristic
192C2×C4H(5,𝔽2)12yes—
193C2×C4H(5,𝔽2)12yes—
194C8—24no—
195C8—42no—
196C8—41no—
197C8—41no—
198C8—42no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—81no—
203C22—82no—
204C22—162no—
205C22C2×#12111yescharacteristic
206C22#54012yes—
207C4—22no—
208C4—22no—
209C4—41no—
210C4—42no—
211C4—81no—
212C4—82no—
213C4#53512yes—
214C4#53511yescharacteristic
215C4#53511yescharacteristic
216C2—22no—
217C2—81no—
218C2—161no—
219C2#261911yescharacteristic
220C2#306912yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#497C2semidirect
3. #3039C2semidirect
4. #2923C2semidirect
5. #2047C2semidirect
6. #2046C2semidirect
7. #2020C2semidirect
9. #1967C2semidirect
10. #1966C2semidirect
11. #1838C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
#30402
#30392
#29232
#20472
#20462
#20202
#20192
#19672
#19662
#18382