Finite Group Databasebeta

← #33003

Subgroups — #33003

Order 256.

688 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic
3C2×#448C211yesmaximal, characteristic
4#2854C211yesmaximal, characteristic
5#1880C214yesmaximal, complemented
6#1863C214yesmaximal, complemented
7#1823C211yesmaximal, characteristic, complemented
8#1817C211yesmaximal, characteristic, complemented
9#1787C211yesmaximal, characteristic, complemented
10C23×Q8C2211yescharacteristic
11C22×SD16C2212yes—
12C2×#123—21no—
13C2×#121C2211yescharacteristic
14C2×C4×Q8—24no—
15C2×C4×Q8—24no—
16C2×C4×Q8—22no—
17C2×C4×Q8C2214yes—
18C22×C4⋊C4—21no—
19C22×C4⋊C4C2211yescharacteristic
20C22×C42—21no—
21C22×C42C2211yescharacteristic
22#480—24no—
23#448—24no—
24C2×#107C2212yes—
25C2×#103C2212yes—
26C2×#103C2214yes—
27C2×#102C2212yes—
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#390C2212yes—
31#389C2212yes—
32#388—22no—
33#386—21no—
34#380C2211yescharacteristic
35#340C2214yes—
36#336C2214yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—21no—
40C22×Q8D414yes—
41C22×Q8C2312yes—
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×SD16—24no—
50C2×SD16—28no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53#123—44no—
54#121—22no—
55#121—22no—
56C4×Q8—216no—
57C4×Q8—28no—
58C4×Q8—28no—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D414yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—28no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—24no—
76C2×C42D412yes—
77C2×C42C2312yes—
78#107—24no—
79#103—24no—
80#103—28no—
81#102—24no—
82C22⋊C8—22no—
83C22⋊C8—22no—
84#95—21no—
85#95—22no—
86C24—21no—
87C24C2×D411yescharacteristic
88C2×Q8—216no—
89C2×Q8—28no—
90C2×Q8—216no—
91C2×Q8—24no—
92C2×Q8—22no—
93C2×Q8—24no—
94C2×Q8C2×D414yes—
95C2×Q8#4118yes—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—24no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—42no—
102C22×C4—28no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2411yescharacteristic
123SD16—48no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—24no—
127C2×C8—24no—
128C4⋊C4—216no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—24no—
134C4⋊C4—44no—
135C4⋊C4C2×D414yes—
136C22⋊C4—44no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—44no—
140C42—216no—
141C42—24no—
142C42—22no—
143C42—22no—
144C23—22no—
145C23—22no—
146C23—41no—
147C23—42no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23#12012yes—
152C23#13512yes—
153C23AGL(1,ℤ/8)12yes—
154C23C22×D411yescharacteristic
155Q8—216no—
156Q8—24no—
157Q8—28no—
158Q8#12318yescomplemented
159D4—42no—
160D4—42no—
161D4—84no—
162C2×C4—28no—
163C2×C4—216no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—28no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—48no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4#12012yes—
186C2×C4#13714yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C22×D411yescharacteristic
189C2×C4C2×#4114yes—
190C8—42no—
191C8—42no—
192C22—22no—
193C22—24no—
194C22—24no—
195C22—22no—
196C22—81no—
197C22—82no—
198C22—82no—
199C22—81no—
200C22—81no—
201C22#44914yes—
202C22C2×#12011yescharacteristic
203C22#52512yes—
204C22C2×#13511yescharacteristic
205C22C2×AGL(1,ℤ/8)11yescharacteristic
206C22C2×#13712yes—
207C4—28no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—44no—
212C4—82no—
213C4—81no—
214C4C2×#12312yes—
215C4#52512yes—
216C2—24no—
217C2—161no—
218C2#284911yescharacteristic
219C2C2×#44911yescharacteristic
220C2#300411yescharacteristic
221C2#310812yes—
222C2#311212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×Q8C2semidirect
5. #1880C2semidirect
6. #1863C2semidirect
7. #1823C2semidirect
8. #1817C2semidirect
9. #1787C2semidirect
158. Q8#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#4802
C2×#4482
#28542
#18802
#18632
#18232
#18172
#17872