Finite Group Databasebeta

← #27778

Subgroups — #27778

Order 256.

439 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 131 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×#489C212yesmaximal, complemented
2#2551C211yesmaximal, characteristic, complemented
3#2548C211yesmaximal, characteristic
4#2042C212yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1982C211yesmaximal, characteristic
7#1957C211yesmaximal, characteristic
8#1936C212yesmaximal, complemented
9#1803C211yesmaximal, characteristic, complemented
10#1770C211yesmaximal, characteristic, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×Q8C2211yescharacteristic
17#489—28no—
18C2×C4⋊C8C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2211yescharacteristic
21C2×#103C2211yescharacteristic
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#97C2212yes—
27C2×C4×C8C2211yescharacteristic
28#395—22no—
29#394—22no—
30#390—21no—
31#390—21no—
32#390C2212yes—
33#389—21no—
34#389C2212yes—
35#384C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8—22no—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4—21no—
46C22×D4C2312yes—
47C22×C8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C4⋊Q8—24no—
51C4⋊Q8—42no—
52C4⋊Q8—44no—
53#127—22no—
54#127—22no—
55#124—24no—
56#124—42no—
57C4×Q8—22no—
58C4×Q8—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C2312yes—
77C2×C42C2311yescharacteristic
78C4⋊C8—22no—
79#103—24no—
80#103—22no—
81#103D414yescomplemented
82#102—24no—
83#102—24no—
84#102—24no—
85#102—42no—
86#97—24no—
87C4×C8—22no—
88#95—21no—
89#95—22no—
90#95D412yes—
91C24—22no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—82no—
100C2×Q8C2×D412yes—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—81no—
112C2×D4—81no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8C2×D412yes—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—81no—
145C4⋊C4C2×D412yes—
146C4⋊C4#4112yes—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C22⋊C4—82no—
151C42—22no—
152C42—22no—
153C42—42no—
154C42—42no—
155C42#4112yes—
156C42#4112yes—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23C22×D411yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169Q8—88no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—42no—
174D4—84no—
175D4—84no—
176C2×C4—21no—
177C2×C4—21no—
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—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4#12412yes—
200C2×C4#12412yes—
201C2×C4AGL(1,ℤ/8)12yes—
202C2×C4#13712yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C42−1+411yescharacteristic
209C8—22no—
210C8—42no—
211C8—41no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22C2×#12411yescharacteristic
220C22D4211yescharacteristic
221C22#54811yescharacteristic
222C22C2×AGL(1,ℤ/8)11yescharacteristic
223C22C2×#13711yescharacteristic
224C22#57611yescharacteristic
225C22#57711yescharacteristic
226C4—22no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—81no—
233C4—82no—
234C4#48914yescomplemented
235C4#53211yescharacteristic
236C4#54211yescharacteristic
237C4#54711yescharacteristic
238C4#55511yescharacteristic
239C2—162no—
240C2#249111yescharacteristic
241C2C2×#48911yescharacteristic
242C2#313611yescharacteristic
243C2#327411yescharacteristic
244C2#327811yescharacteristic
245C2#337311yescharacteristic
246C2#337511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
1. C2×#489C2semidirect
2. #2551C2semidirect
4. #2042C2semidirect
5. #2023C2semidirect
8. #1936C2semidirect
9. #1803C2semidirect
10. #1770C2semidirect
11. C4×#103C2semidirect
81. #103D4semidirect
234. C4#489semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
#25512
#25482
#20422
#20232
#19822
#19572
#19362
#18032
#17702
C4×#1032