Finite Group Databasebeta

← #30245

Subgroups — #30245

Order 256.

566 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 131 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×#469C212yesmaximal, complemented
2C2×#468C212yesmaximal, complemented
3#2856C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic, complemented
5#2096C211yesmaximal, characteristic, complemented
6#2093C211yesmaximal, characteristic, complemented
7#2092C212yesmaximal, complemented
8#2014C211yesmaximal, characteristic, complemented
9#2012C212yesmaximal, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1728C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C22×M16C2212yes—
15C2×#127C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121—22no—
19C2×#121C2211yescharacteristic
20C2×#121C2212yes—
21C22×C42C2211yescharacteristic
22#469—28no—
23#468—28no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#103C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30#391C2211yescharacteristic
31#390—22no—
32#390C2211yescharacteristic
33#390C2212yes—
34#388—22no—
35#386—22no—
36#384C2211yescharacteristic
37#382C2211yescharacteristic
38#336C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8C2311yescharacteristic
42C22×D4—22no—
43C22×D4—21no—
44C22×D4—22no—
45C22×D4C2311yescharacteristic
46C22×D4C2312yes—
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C2×SD16—24no—
50C2×SD16—42no—
51C2×SD16—42no—
52C2×D8—24no—
53C2×D8—42no—
54C2×D8—42no—
55C2×M16—42no—
56C2×M16—42no—
57C2×M16D418yescomplemented
58C22×C8C2312yes—
59C22×C8C2312yes—
60#127—42no—
61#124—42no—
62C22⋊Q8—24no—
63#121—24no—
64#121—24no—
65#121—24no—
66#121—48no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C42—42no—
79C2×C42D412yes—
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—24no—
84#103—24no—
85#102—24no—
86#102—24no—
87#102—24no—
88#95—22no—
89#95—22no—
90C24—21no—
91C24—22no—
92C24C2×D411yescharacteristic
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—48no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—42no—
113C2×D4—44no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D414yes—
127C22×C4C2×D412yes—
128C22×C4C2411yescharacteristic
129SD16—48no—
130D8—48no—
131M16—48no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8C2×D414yes—
137C2×C8C2×D414yes—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C22⋊C4—48no—
145C22⋊C4—44no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C42—22no—
151C42—22no—
152C42—42no—
153C23—21no—
154C23—21no—
155C23—22no—
156C23—22no—
157C23—42no—
158C23—41no—
159C23—41no—
160C23—42no—
161C23—41no—
162C23—42no—
163C23—44no—
164C23—42no—
165C23—42no—
166C23—44no—
167C23#12712yes—
168C23C22×D411yescharacteristic
169Q8—42no—
170Q8—42no—
171D4—42no—
172D4—42no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—88no—
178D4—88no—
179D4—84no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—48no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4#12712yes—
200C2×C4AGL(1,ℤ/8)14yes—
201C2×C4C22×D412yes—
202C2×C4H(5,𝔽2)11yescharacteristic
203C2×C4H(5,𝔽2)12yes—
204C2×C42−1+411yescharacteristic
205C8—24no—
206C8—24no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—82no—
213C22—81no—
214C22—81no—
215C22—82no—
216C22—81no—
217C22—82no—
218C22—84no—
219C22—82no—
220C22—84no—
221C22—82no—
222C22C2×#12711yescharacteristic
223C22#53411yescharacteristic
224C22#53711yescharacteristic
225C22C2×AGL(1,ℤ/8)12yes—
226C22#57512yes—
227C4—22no—
228C4—24no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4—81no—
233C4—81no—
234C4#53412yes—
235C4#53712yes—
236C2—42no—
237C2—162no—
238C2—161no—
239C2#265611yescharacteristic
240C2#315111yescharacteristic
241C2#315311yescharacteristic
242C2#319412yes—
243C2#325112yes—
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×#469C2semidirect
2. C2×#468C2semidirect
3. #2856C2semidirect
4. #2850C2semidirect
5. #2096C2semidirect
6. #2093C2semidirect
7. #2092C2semidirect
8. #2014C2semidirect
9. #2012C2semidirect
10. #2011C2semidirect
11. #1728C2semidirect
57. C2×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4682
#28562
#28502
#20962
#20932
#20922
#20142
#20122
#20112
#17282