Finite Group Databasebeta

← #33282

Subgroups — #33282

Order 256.

360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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#2861C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic, complemented
3#1992C211yesmaximal, characteristic, complemented
4#1992C211yesmaximal, characteristic, complemented
5#1959C211yesmaximal, characteristic
6#1959C211yesmaximal, characteristic
7#1936C212yesmaximal
8#1935C212yesmaximal, complemented
9#1934C212yesmaximal
10#1933C212yesmaximal, complemented
11C4×#97C211yesmaximal, characteristic, complemented
12C2×#125—21no—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×#103C2212yes—
16C2×#103C2212yes—
17C2×#102C2212yes—
18C2×#102C2212yes—
19C2×#97C2212yes—
20C2×#97C2211yescharacteristic
21C2×#97C2211yescharacteristic
22C2×C4×C8C2212yes—
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#390C2211yescharacteristic
26#390C2211yescharacteristic
27#389—21no—
28#389—21no—
29#389C2212yes—
30#389C2212yes—
31#388—21no—
32#388—21no—
33#388—21no—
34#388—21no—
35#384C2211yescharacteristic
36#384C2211yescharacteristic
37#382—21no—
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—22no—
48#125—22no—
49#125—44no—
50#124—22no—
51#124—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#103—24no—
78#103—24no—
79#102—24no—
80#102—24no—
81#97—24no—
82#97—22no—
83#97—22no—
84C4×C8—24no—
85#95—22no—
86#95—22no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8#4114yes—
117C2×C8#4114yes—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—44no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—41no—
145C42—41no—
146C42#4112yes—
147C42#4112yes—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23C22×D411yescharacteristic
156Q8—42no—
157Q8—42no—
158D4—42no—
159D4—42no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4#12412yes—
190C2×C4#12412yes—
191C2×C4AGL(1,ℤ/8)12yes—
192C2×C4#13712yes—
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4C2×#4112yes—
196C2×C4C2×#4112yes—
197C8—24no—
198C8—24no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22C2×#12411yescharacteristic
207C22#53211yescharacteristic
208C22#53211yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22C2×#13711yescharacteristic
211C22#57511yescharacteristic
212C22#57511yescharacteristic
213C4—22no—
214C4—22no—
215C4—22no—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—81no—
220C4—81no—
221C4—81no—
222C4—82no—
223C4—82no—
224C4#52912yes—
225C4#52912yes—
226C2—161no—
227C2#286111yescharacteristic
228C2#314111yescharacteristic
229C2#314211yescharacteristic
230C2#317111yescharacteristic
231C2#317211yescharacteristic
232C2#318011yescharacteristic
233C2#318111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2859C2semidirect
3. #1992C2semidirect
4. #1992C2semidirect
8. #1935C2semidirect
10. #1933C2semidirect
11. C4×#97C2semidirect

Maximal subgroups

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

subgroupindex
#28612
#28592
#19922
#19592
#19362
#19352
#19342
#19332
C4×#972