Finite Group Databasebeta

← #33005

Subgroups — #33005

Order 256.

688 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 159 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×#481C211yesmaximal, characteristic
3C2×#448C211yesmaximal, characteristic
4#2854C211yesmaximal, characteristic
5#1880C214yesmaximal, complemented
6#1865C214yesmaximal, complemented
7#1824C211yesmaximal, 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#481—24no—
23#448—24no—
24C2×#106C2212yes—
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#106—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—22no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4SD1614yescomplemented
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2411yescharacteristic
122SD16—48no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—24no—
126C2×C8—24no—
127C4⋊C4—216no—
128C4⋊C4—24no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—24no—
133C4⋊C4—44no—
134C4⋊C4C2×D414yes—
135C22⋊C4—44no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—44no—
139C42—216no—
140C42—24no—
141C42—22no—
142C42—22no—
143C23—22no—
144C23—22no—
145C23—41no—
146C23—42no—
147C23—41no—
148C23—41no—
149C23—42no—
150C23#12012yes—
151C23C2×SD1612yes—
152C23#13712yes—
153C23C22×D411yescharacteristic
154Q8—216no—
155Q8—24no—
156Q8—28no—
157Q8#12318yescomplemented
158D4—42no—
159D4—42no—
160D4—84no—
161C2×C4—216no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—28no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—28no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—48no—
176C2×C4—44no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4#12012yes—
184C2×C4C2×SD1614yes—
185C2×C4C22×D411yescharacteristic
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4114yes—
188C8—42no—
189C8—42no—
190C22—24no—
191C22—22no—
192C22—24no—
193C22—22no—
194C22—82no—
195C22—81no—
196C22—82no—
197C22—81no—
198C22—81no—
199C22#44814yes—
200C22C2×#12011yescharacteristic
201C22#52512yes—
202C22C22×SD1611yescharacteristic
203C22C2×#13711yescharacteristic
204C22#57512yes—
205C4—28no—
206C4—22no—
207C4—24no—
208C4—42no—
209C4—44no—
210C4—82no—
211C4—81no—
212C4C2×#12312yes—
213C4#52512yes—
214C2—24no—
215C2—161no—
216C2#284911yescharacteristic
217C2C2×#44811yescharacteristic
218C2#300511yescharacteristic
219C2#310312yes—
220C2#311712yes—
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. #1865C2semidirect
7. #1824C2semidirect
8. #1817C2semidirect
9. #1787C2semidirect
115. C22×C4SD16semidirect
157. 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×#4812
C2×#4482
#28542
#18802
#18652
#18242
#18172
#17872