Finite Group Databasebeta

← #29122

Subgroups — #29122

Order 256.

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

normal subgroups only

This group has 377 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
1#2859C211yesmaximal, characteristic, complemented
2C4×#124C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2042C212yesmaximal
5#1934C212yesmaximal
6#1933C212yesmaximal, complemented
7#1803C211yesmaximal, characteristic, complemented
8#1803C211yesmaximal, characteristic, complemented
9#1802C211yesmaximal, characteristic
10#1802C211yesmaximal, characteristic
11C4×C4⋊C8C211yesmaximal, characteristic, complemented
12C2×#125—21no—
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16C2×C4⋊C8C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C4×C8C2212yes—
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#389—21no—
26#389—21no—
27#389C2212yes—
28#389C2212yes—
29#388—21no—
30#388—21no—
31#384C2211yescharacteristic
32#384C2211yescharacteristic
33#382—21no—
34#382C2211yescharacteristic
35C4×C22⋊C4—21no—
36C4×C22⋊C4—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48#125—44no—
49#124—22no—
50#124—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×D4—22no—
54C4×D4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C42—21no—
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C4⋊C8—22no—
80C4⋊C8—22no—
81#103—24no—
82#103—42no—
83#102—24no—
84#102—42no—
85C4×C8—42no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92C24—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8#4112yes—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4#4112yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—42no—
155C42#4112yes—
156C42#4112yes—
157C23—22no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23C22×D411yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169D4—22no—
170D4—22no—
171D4—41no—
172D4—41no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—84no—
201C2×C4#12412yes—
202C2×C4#12412yes—
203C2×C4#13512yes—
204C2×C4#13512yes—
205C2×C4AGL(1,ℤ/8)12yes—
206C2×C4#13712yes—
207C2×C4C2×#4111yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4H(5,𝔽2)11yescharacteristic
212C2×C42−1+411yescharacteristic
213C8—42no—
214C8—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22#52511yescharacteristic
223C22C2×#12411yescharacteristic
224C22#53711yescharacteristic
225C22C2×#13511yescharacteristic
226C22C2×#13511yescharacteristic
227C22C2×AGL(1,ℤ/8)11yescharacteristic
228C22C2×#13711yescharacteristic
229C4—22no—
230C4—22no—
231C4—22no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—82no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4#48814yes—
240C4#55211yescharacteristic
241C4#55211yescharacteristic
242C4#55311yescharacteristic
243C4#55311yescharacteristic
244C2—82no—
245C2#256211yescharacteristic
246C2#310611yescharacteristic
247C2#310711yescharacteristic
248C2C2×#48811yescharacteristic
249C2#314111yescharacteristic
250C2#324711yescharacteristic
251C2#324811yescharacteristic
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
1. #2859C2semidirect
3. #2101C2semidirect
6. #1933C2semidirect
7. #1803C2semidirect
8. #1803C2semidirect
11. C4×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#28592
C4×#1242
#21012
#20422
#19342
#19332
#18032
#18022
C4×C4⋊C82