Finite Group Databasebeta

← #32995

Subgroups — #32995

Order 256.

623 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 135 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#3428C211yesmaximal, characteristic
2#3095C212yesmaximal, complemented
3#2432C211yesmaximal, characteristic
4#1886C212yesmaximal, complemented
5#1885C212yesmaximal, complemented
6#1857C212yesmaximal, complemented
7#1856C212yesmaximal, complemented
8#1826C212yesmaximal, complemented
9#1785C211yesmaximal, characteristic
10C22×#41C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14#520—22no—
15#520—22no—
16#519C2212yes—
17#518—22no—
18#518C2212yes—
19C4×#41—24no—
20C2×#117—22no—
21C2×#117C2212yes—
22#485—24no—
23#484—22no—
24#483—22no—
25#428C2214yes—
26#419C2214yes—
27C2×#101C2212yes—
28C2×#100C2212yes—
29#407C2212yes—
30#394—22no—
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#385—22no—
34C4×C22⋊C4C2211yescharacteristic
35#339C2214yes—
36#339C2214yes—
37C2×#41—24no—
38C2×#41—24no—
39C2×#41—22no—
40C2×#41—22no—
41C22×Q8C2311yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×D4D412yes—
44C22×D4C2311yescharacteristic
45C23×C4D412yes—
46C23×C4C2311yescharacteristic
47C2×M16C2314yes—
48C4⋊Q8—24no—
49#124—24no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C4×Q8—24no—
53C4×Q8—24no—
54C4×D4—28no—
55C4×D4—28no—
56C4×D4—24no—
57C4×D4—24no—
58#117—24no—
59#117—24no—
60#117—22no—
61#117—24no—
62#117—22no—
63#117D414yes—
64#117C2314yes—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4D412yes—
70C2×C4⋊C4D412yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—24no—
84#103—24no—
85#103—24no—
86#101—24no—
87#100—24no—
88C22⋊C8—22no—
89C22⋊C8—22no—
90#95—21no—
91#95—22no—
92C24C2×D411yescharacteristic
93C24C2×D412yes—
94#41—216no—
95#41—216no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8#4114yes—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—24no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4#4114yes—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2411yescharacteristic
127M16—24no—
128M16—24no—
129C2×C8—24no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—44no—
138C4⋊C4C2×D414yes—
139C4⋊C4C2×D414yes—
140C22⋊C4—28no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—42no—
144C22⋊C4—44no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C42—28no—
148C42—24no—
149C42—24no—
150C42—22no—
151C42—22no—
152C23—22no—
153C23—21no—
154C23—21no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—42no—
159C23—42no—
160C23#12012yes—
161C23#12314yes—
162C23C22×D411yescharacteristic
163Q8—28no—
164Q8—44no—
165D4—216no—
166D4—28no—
167C2×C4—28no—
168C2×C4—28no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—84no—
191C2×C4—84no—
192C2×C4#12011yescharacteristic
193C2×C4#12011yescharacteristic
194C2×C4#12314yes—
195C2×C4C22×D412yes—
196C2×C4C2×#4112yes—
197C2×C4C2×#4112yes—
198C8—44no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—24no—
203C22—22no—
204C22—22no—
205C22—44no—
206C22—42no—
207C22—42no—
208C22—81no—
209C22—82no—
210C22—81no—
211C22C2×#12011yescharacteristic
212C22C2×#12311yescharacteristic
213C22C2×#12311yescharacteristic
214C4—21no—
215C4—21no—
216C4—24no—
217C4—42no—
218C4—44no—
219C4—44no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4#52514yes—
224C2—21no—
225C2—21no—
226C2—42no—
227C2—44no—
228C2#284911yescharacteristic
229C2#301811yescharacteristic
230C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3095C2semidirect
4. #1886C2semidirect
5. #1885C2semidirect
6. #1857C2semidirect
7. #1856C2semidirect
8. #1826C2semidirect

Maximal subgroups

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

subgroupindex
#34282
#30952
#24322
#18862
#18852
#18572
#18562
#18262
#17852