Finite Group Databasebeta

← #29256

Subgroups — #29256

Order 256.

442 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C4×#127C211yesmaximal, characteristic
2C4×#125C211yesmaximal, characteristic, complemented
3#2092C212yesmaximal, complemented
4#2025C212yesmaximal, complemented
5#1982C211yesmaximal, characteristic
6#1982C211yesmaximal, characteristic
7#1948C211yesmaximal, characteristic, complemented
8#1947C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1831C211yesmaximal, characteristic, complemented
12C2×#127C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×D4—21no—
15C2×C4×D4—21no—
16C2×C4×D4C2212yes—
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25#390—21no—
26#390—21no—
27#390C2212yes—
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4—21no—
45C22×D4C2312yes—
46C23×C4—22no—
47C22×C8C2312yes—
48C22×C8C2312yes—
49#127—22no—
50#127Q814yes—
51#125—22no—
52#125—22no—
53C4×D4—42no—
54C4×D4—42no—
55C4×D4—44no—
56C4×D4—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81#106—42no—
82C4⋊C8—22no—
83C4⋊C8—22no—
84#102—24no—
85#102—24no—
86#102—42no—
87#102—42no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24—22no—
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4C2×Q814yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4#4112yes—
143C4⋊C4#4112yes—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—41no—
155C42—42no—
156C42—42no—
157C42C2×Q812yes—
158C42#4112yes—
159C23—24no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23C22×D411yescharacteristic
167D4—24no—
168D4—24no—
169D4—48no—
170D4—42no—
171D4—42no—
172D4—48no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—88no—
202C2×C4C22⋊Q812yes—
203C2×C4C22⋊Q812yes—
204C2×C4#13512yes—
205C2×C4AGL(1,ℤ/8)12yes—
206C2×C4AGL(1,ℤ/8)12yes—
207C2×C4C22×Q811yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4H(5,𝔽2)11yescharacteristic
212C2×C4H(5,𝔽2)11yescharacteristic
213C8—42no—
214C8—42no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—82no—
221C22—82no—
222C22C2×C22⋊Q811yescharacteristic
223C22#52511yescharacteristic
224C22#53511yescharacteristic
225C22C2×#13511yescharacteristic
226C22C2×AGL(1,ℤ/8)11yescharacteristic
227C22C2×AGL(1,ℤ/8)11yescharacteristic
228C22#57511yescharacteristic
229C4—22no—
230C4—22no—
231C4—41no—
232C4—42no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4#47814yes—
240C4#55411yescharacteristic
241C4#55411yescharacteristic
242C4#55911yescharacteristic
243C4#55911yescharacteristic
244C2—84no—
245C2#258011yescharacteristic
246C2C2×#47811yescharacteristic
247C2#307711yescharacteristic
248C2#310611yescharacteristic
249C2#311511yescharacteristic
250C2#321211yescharacteristic
251C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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. C4×#125C2semidirect
3. #2092C2semidirect
4. #2025C2semidirect
7. #1948C2semidirect
8. #1947C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1831C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1272
C4×#1252
#20922
#20252
#19822
#19482
#19472
#18642
#18622
#18312