Finite Group Databasebeta

← #29912

Subgroups — #29912

Order 256.

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

normal subgroups only

This group has 398 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
1#2854C211yesmaximal, characteristic
2C2×#382C211yesmaximal, characteristic, complemented
3#2101C212yesmaximal, complemented
4#2092C212yesmaximal, complemented
5#2085C212yesmaximal, complemented
6#2073C212yesmaximal, complemented
7#1874C211yesmaximal, characteristic
8#1874C211yesmaximal, characteristic
9#1853C211yesmaximal, characteristic, complemented
10#1852C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic, complemented
12C2×#123—21no—
13C2×#121C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23#390C2211yescharacteristic
24#390C2211yescharacteristic
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#388—21no—
28#388—21no—
29#386—21no—
30#382—21no—
31#382—21no—
32#382C2212yes—
33#382C2212yes—
34#380C2211yescharacteristic
35C2×#95—21no—
36C2×#95—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52#123—44no—
53#121—22no—
54#121—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80#102—42no—
81#102—42no—
82C22⋊C8—22no—
83C22⋊C8—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—42no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4#4112yes—
125C22×C4#4112yes—
126C22×C4C2411yescharacteristic
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23#12312yes—
169C23#13512yes—
170C23AGL(1,ℤ/8)12yes—
171C23C22×D411yescharacteristic
172D4—42no—
173D4—42no—
174D4—84no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4#12312yes—
211C2×C4C2×#4111yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)12yes—
214C2×C42−1+412yes—
215C8—42no—
216C8—42no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22#48214yes—
231C22C2×#12311yescharacteristic
232C22#53511yescharacteristic
233C22#53611yescharacteristic
234C22C2×#13511yescharacteristic
235C22C2×AGL(1,ℤ/8)11yescharacteristic
236C22#57511yescharacteristic
237C22#57511yescharacteristic
238C4—22no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—82no—
245C4—81no—
246C4—82no—
247C4—81no—
248C4#53912yes—
249C4#53912yes—
250C2—24no—
251C2—161no—
252C2#262311yescharacteristic
253C2C2×#48211yescharacteristic
254C2#309311yescharacteristic
255C2#321111yescharacteristic
256C2#321311yescharacteristic
257C2#323211yescharacteristic
258C2#323411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#382C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2085C2semidirect
6. #2073C2semidirect
9. #1853C2semidirect
10. #1852C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28542
C2×#3822
#21012
#20922
#20852
#20732
#18742
#18532
#18522
#18172