Finite Group Databasebeta

← #28316

Subgroups — #28316

Order 256.

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

normal subgroups only

This group has 613 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
1C2×#482C212yesmaximal, complemented
2C2×#449C212yesmaximal, complemented
3#2856C211yesmaximal, characteristic, complemented
4#2297C211yesmaximal, characteristic, complemented
5#2096C211yesmaximal, characteristic, complemented
6#2093C211yesmaximal, characteristic, complemented
7#2014C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic, complemented
9#1973C212yesmaximal, complemented
10#1972C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2212yes—
15C2×#127C2211yescharacteristic
16C2×#121—22no—
17C2×#121C2212yes—
18C2×#117C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#482—28no—
21#449—28no—
22C2×#107C2212yes—
23C2×#106C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×C22⋊C8C2212yes—
29C2×#97C2212yes—
30#391C2211yescharacteristic
31#391C2211yescharacteristic
32#390—22no—
33#390C2212yes—
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#386—22no—
37#384C2211yescharacteristic
38C4×C22⋊C4—22no—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—24no—
44C2×#41—42no—
45C22×Q8C2311yescharacteristic
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4C2312yes—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D412yes—
53C23×C4C2311yescharacteristic
54C2×SD16—24no—
55C2×SD16—24no—
56C2×SD16—42no—
57C2×D8—24no—
58C2×D8—24no—
59C2×D8—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62#127—42no—
63#121—24no—
64#121—24no—
65#121—48no—
66#117—24no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42—42no—
78C2×C42D412yes—
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#107—24no—
82#106—24no—
83#103—24no—
84#102—24no—
85#102—24no—
86#102—24no—
87C22⋊C8D418yescomplemented
88#97—42no—
89#95—22no—
90#95—22no—
91C24—21no—
92C24—22no—
93C24C2×D411yescharacteristic
94#41—48no—
95#41—48no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8#4112yes—
100C2×D4—24no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—48no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4#4112yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D414yes—
134C22×C4C2×D412yes—
135C22×C4C2411yescharacteristic
136SD16—48no—
137D8—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8C2×D414yes—
143C2×C8C2×D414yes—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—44no—
149C22⋊C4—44no—
150C22⋊C4—48no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C42—22no—
155C42—22no—
156C42—42no—
157C42—42no—
158C23—21no—
159C23—21no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—41no—
164C23—41no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—44no—
171C23—44no—
172C23#12712yes—
173C23C22×D411yescharacteristic
174Q8—24no—
175Q8—42no—
176D4—24no—
177D4—48no—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—88no—
184D4—88no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—48no—
196C2×C4—48no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4#12712yes—
206C2×C4AGL(1,ℤ/8)14yes—
207C2×C4C22×D412yes—
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C8—24no—
212C8—24no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—42no—
217C22—42no—
218C22—44no—
219C22—42no—
220C22—81no—
221C22—81no—
222C22—82no—
223C22—84no—
224C22—82no—
225C22—82no—
226C22—84no—
227C22—82no—
228C22#52511yescharacteristic
229C22C2×#12711yescharacteristic
230C22#53411yescharacteristic
231C22C2×AGL(1,ℤ/8)12yes—
232C22#57512yes—
233C4—24no—
234C4—22no—
235C4—42no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4—82no—
240C4#54612yes—
241C4#54612yes—
242C2—42no—
243C2—82no—
244C2—162no—
245C2#253911yescharacteristic
246C2#311512yes—
247C2#315011yescharacteristic
248C2#315111yescharacteristic
249C2#319412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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. C2×#482C2semidirect
2. C2×#449C2semidirect
3. #2856C2semidirect
4. #2297C2semidirect
5. #2096C2semidirect
6. #2093C2semidirect
7. #2014C2semidirect
8. #2011C2semidirect
9. #1973C2semidirect
10. #1972C2semidirect
11. #1749C2semidirect
87. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4492
#28562
#22972
#20962
#20932
#20142
#20112
#19732
#19722
#17492