Finite Group Databasebeta

← #24646

Subgroups — #24646

Order 256.

432 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 127 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×#461C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic, complemented
4#2357C211yesmaximal, characteristic
5#2055C212yesmaximal
6#2013C212yesmaximal
7#1981C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1928C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#461—24no—
19#440—24no—
20C2×#107C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#398—21no—
31#398—21no—
32#391C2212yes—
33#390C2212yes—
34#389—22no—
35#388—22no—
36#382C2211yescharacteristic
37#381—21no—
38C4×C4⋊C4C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40C4×C22⋊C4—21no—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—24no—
50C2×SD16—22no—
51C2×SD16—42no—
52C2×SD16—42no—
53C22×C8C2312yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C4⋊Q8—24no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—42no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×D4—22no—
63C4×D4D414yes—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D412yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42—22no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80#107—22no—
81C4⋊C8—22no—
82C4⋊C8—42no—
83#103—24no—
84#103—22no—
85#103—24no—
86#103—42no—
87#103—42no—
88#102—22no—
89#97—22no—
90#95—21no—
91#95—22no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95D411yescharacteristic
96#95D411yescharacteristic
97C24—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8#4112yes—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C2×D412yes—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—82no—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2411yescharacteristic
128SD16—28no—
129SD16—48no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8#4112yes—
139C4⋊C4—24no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4C2×D412yes—
151C22⋊C4—24no—
152C22⋊C4—44no—
153C22⋊C4—82no—
154C42—22no—
155C42—22no—
156C42—42no—
157C42—82no—
158C42C2×D412yes—
159C42#4112yes—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—42no—
164C23—42no—
165C23C22×D411yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174D4—22no—
175D4—22no—
176D4—41no—
177D4—41no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—84no—
199C2×C4#12112yes—
200C2×C4#12112yes—
201C2×C4#13714yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)11yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C8—22no—
209C8—41no—
210C8—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—82no—
215C22—82no—
216C22C2×#12111yescharacteristic
217C22#54611yescharacteristic
218C22#55011yescharacteristic
219C22C2×#13712yes—
220C22#57712yes—
221C4—22no—
222C4—22no—
223C4—24no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—82no—
228C4—82no—
229C4—81no—
230C4#52511yescharacteristic
231C4#53511yescharacteristic
232C4#55311yescharacteristic
233C4#55511yescharacteristic
234C2—82no—
235C2#241411yescharacteristic
236C2#303911yescharacteristic
237C2#304211yescharacteristic
238C2#332012yes—
239C2#340012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2546C2semidirect
7. #1981C2semidirect
8. #1960C2semidirect
9. #1940C2semidirect
10. #1928C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×#4402
#25462
#23572
#20552
#20132
#19812
#19602
#19402
#19282
#18052