Finite Group Databasebeta

← #32992

Subgroups — #32992

Order 256.

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

normal subgroups only

This group has 652 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, complemented
2#3094C212yesmaximal, complemented
3#2431C211yesmaximal, characteristic
4#1888C212yesmaximal, complemented
5#1887C212yesmaximal, complemented
6#1857C212yesmaximal, complemented
7#1856C212yesmaximal, complemented
8#1826C212yesmaximal, complemented
9#1785C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C2×#127C2211yescharacteristic
12C2×#124C2211yescharacteristic
13C2×#121C2211yescharacteristic
14#520—22no—
15#520—22no—
16#519C2212yes—
17#518—22no—
18#518C2212yes—
19C4×#41—24no—
20C2×#117—22no—
21C2×#117C2212yes—
22#482—24no—
23#481—22no—
24#480—22no—
25#428C2214yes—
26#418C2214yes—
27C2×#101C2212yes—
28C2×#100C2212yes—
29#407C2212yes—
30#392—22no—
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#386—22no—
34C4×C22⋊C4C2211yescharacteristic
35#339C2214yes—
36#339C2214yes—
37C2×#41—24no—
38C2×#41—24no—
39C2×#41—22no—
40C2×#41—22no—
41C22×Q8C2311yescharacteristic
42C22×D4—22no—
43C22×D4—21no—
44C22×D4D412yes—
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C2×M16C2314yes—
50#127—24no—
51#124—24no—
52#121—22no—
53#121—22no—
54C4×Q8—24no—
55C4×Q8—24no—
56C4×D4—28no—
57C4×D4—28no—
58C4×D4—24no—
59C4×D4—24no—
60#117—24no—
61#117—24no—
62#117—22no—
63#117—24no—
64#117—22no—
65#117D414yescomplemented
66#117C2314yes—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4D412yescomplemented
70C2×C4⋊C4D412yescomplemented
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—24no—
84#102—24no—
85#102—24no—
86#101—24no—
87#100—24no—
88C22⋊C8—22no—
89C22⋊C8—22no—
90#95—21no—
91#95—22no—
92C24—21no—
93C24C2×D411yescharacteristic
94C24C2×D412yes—
95#41—216no—
96#41—216no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8#4114yes—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—44no—
109C2×D4—42no—
110C2×D4—44no—
111C2×D4—82no—
112C2×D4—82no—
113C2×D4#4114yes—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2411yescharacteristic
132M16—24no—
133M16—24no—
134C2×C8—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4C2×D414yes—
141C4⋊C4C2×D414yes—
142C22⋊C4—28no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—42no—
146C22⋊C4—41no—
147C22⋊C4—41no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C42—28no—
151C42—24no—
152C42—24no—
153C42—22no—
154C42—22no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23—22no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23#12012yes—
169C23#12314yes—
170C23C22×D411yescharacteristic
171Q8—28no—
172D4—216no—
173D4—28no—
174D4—44no—
175D4—88no—
176D4—84no—
177C2×C4—28no—
178C2×C4—28no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—84no—
198C2×C4—84no—
199C2×C4#12011yescharacteristic
200C2×C4#12011yescharacteristic
201C2×C4#12314yes—
202C2×C4C22×D412yes—
203C2×C4C2×#4112yes—
204C2×C4C2×#4112yes—
205C8—44no—
206C22—21no—
207C22—21no—
208C22—22no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—44no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22C2×#12011yescharacteristic
224C22C2×#12311yescharacteristic
225C22C2×#12311yescharacteristic
226C4—21no—
227C4—21no—
228C4—24no—
229C4—42no—
230C4—44no—
231C4—44no—
232C4—81no—
233C4—81no—
234C4#52514yes—
235C2—21no—
236C2—21no—
237C2—42no—
238C2—44no—
239C2—161no—
240C2#284911yescharacteristic
241C2#301811yescharacteristic
242C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2431C2
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
1. #3428C2semidirect
2. #3094C2semidirect
4. #1888C2semidirect
5. #1887C2semidirect
6. #1857C2semidirect
7. #1856C2semidirect
8. #1826C2semidirect
9. #1785C2semidirect
65. #117D4semidirect
69. C2×C4⋊C4D4semidirect
70. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#34282
#30942
#24312
#18882
#18872
#18572
#18562
#18262
#17852