Finite Group Databasebeta

← #29726

Subgroups — #29726

Order 256.

485 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 119 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×#452C212yesmaximal
2#2621C211yesmaximal, characteristic, complemented
3#2311C211yesmaximal, characteristic
4#2074C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic, complemented
6#2056C212yesmaximal, complemented
7#2055C212yesmaximal, complemented
8#2016C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#1872C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C22⋊Q8C2212yes—
17C2×#117C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#452—28no—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25C2×C22⋊C8C2212yes—
26#398—22no—
27#397—22no—
28#391C2212yes—
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#386—22no—
34#382C2211yescharacteristic
35#380—22no—
36#340C2212yes—
37#340C2212yes—
38#336C2211yescharacteristic
39#336C2211yescharacteristic
40C2×#41—22no—
41C2×#41—42no—
42C2×#41—42no—
43C2×#41D414yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4—21no—
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4—21no—
50C23×C4C2311yescharacteristic
51C2×Q16—24no—
52C2×Q16—42no—
53C2×Q16—42no—
54C2×SD16—24no—
55C2×SD16—42no—
56C2×SD16—42no—
57C22×C8C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—42no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62#117—42no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4D412yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C4⋊C8—42no—
78#103—24no—
79#103—42no—
80#103—42no—
81#102—24no—
82C22⋊C8—24no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—24no—
87C24—21no—
88C24C2×D411yescharacteristic
89#41—48no—
90#41—48no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8C2×D412yes—
101C2×D4—24no—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4C2×D412yes—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—82no—
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D412yes—
124C22×C4#4112yes—
125C22×C4C2411yescharacteristic
126Q16—48no—
127SD16—48no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C22⋊C4—42no—
143C22⋊C4—44no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C42—22no—
147C42—22no—
148C23—21no—
149C23—21no—
150C23—22no—
151C23—22no—
152C23—22no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23—42no—
157C23#12112yes—
158C23C22×D411yescharacteristic
159Q8—24no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165D4—24no—
166D4—48no—
167D4—42no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—48no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—84no—
193C2×C4#12112yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4H(5,𝔽2)12yes—
197C2×C4H(5,𝔽2)12yes—
198C8—42no—
199C8—42no—
200C22—22no—
201C22—24no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22—44no—
207C22—82no—
208C22—81no—
209C22—81no—
210C22C2×#12111yescharacteristic
211C22#53411yescharacteristic
212C22#53511yescharacteristic
213C22#57712yes—
214C22#57812yes—
215C4—22no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—82no—
220C4—81no—
221C4—81no—
222C4—82no—
223C4#53412yes—
224C4#53812yes—
225C2—42no—
226C2—82no—
227C2#261711yescharacteristic
228C2#304612yes—
229C2#320112yes—
230C2#322211yescharacteristic
231C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2621C2semidirect
4. #2074C2semidirect
5. #2073C2semidirect
6. #2056C2semidirect
7. #2055C2semidirect
8. #2016C2semidirect
9. #2013C2semidirect
10. #1872C2semidirect
11. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
#26212
#23112
#20742
#20732
#20562
#20552
#20162
#20132
#18722
#17982