Finite Group Databasebeta

← #27646

Subgroups — #27646

Order 256.

447 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 135 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×#482C212yesmaximal
2C4×#121C211yesmaximal, characteristic
3#2285C211yesmaximal, characteristic, complemented
4#2085C212yesmaximal, complemented
5#2073C212yesmaximal, complemented
6#1948C211yesmaximal, characteristic, complemented
7#1947C211yesmaximal, characteristic, complemented
8#1825C212yesmaximal, complemented
9#1802C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×#121C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C2×#117C2212yes—
16C22×C42C2211yescharacteristic
17#482—28no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#390C2211yescharacteristic
26#389C2211yescharacteristic
27#388—21no—
28#386—21no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382C2212yes—
32#380—21no—
33#380—21no—
34#380C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52#121—22no—
53#121D414yes—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—44no—
57#117—24no—
58#117—24no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#107—24no—
84#106—24no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87#102—24no—
88#102—24no—
89C22⋊C8—24no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94#95—21no—
95C24—21no—
96C24C2×D411yescharacteristic
97C2×D4—24no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4C2×D412yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4#4112yes—
131C22×C4C2411yescharacteristic
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4C2×D412yes—
153C22⋊C4—24no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—44no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—21no—
162C42—21no—
163C42—44no—
164C42—42no—
165C42#4112yes—
166C42#4112yes—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23#12112yes—
181C23C22×D411yescharacteristic
182D4—22no—
183D4—22no—
184D4—48no—
185D4—41no—
186D4—41no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—84no—
219C2×C4#12112yes—
220C2×C4#12312yes—
221C2×C4#12312yes—
222C2×C4#13512yes—
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4111yescharacteristic
226C2×C4C2×#4111yescharacteristic
227C2×C4C2×#4111yescharacteristic
228C2×C42−1+411yescharacteristic
229C2×C42−1+411yescharacteristic
230C8—42no—
231C8—42no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22C2×#12111yescharacteristic
247C22C2×#12311yescharacteristic
248C22#53611yescharacteristic
249C22C2×#13511yescharacteristic
250C22C2×AGL(1,ℤ/8)11yescharacteristic
251C22#57511yescharacteristic
252C22#57511yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—42no—
258C4—81no—
259C4—81no—
260C4—82no—
261C4—82no—
262C4#48214yescomplemented
263C4#54711yescharacteristic
264C4#54711yescharacteristic
265C4#55211yescharacteristic
266C4#55211yescharacteristic
267C2—42no—
268C2—82no—
269C2#248611yescharacteristic
270C2#303511yescharacteristic
271C2#303611yescharacteristic
272C2C2×#48211yescharacteristic
273C2#309311yescharacteristic
274C2#323311yescharacteristic
275C2#323511yescharacteristic
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
3. #2285C2semidirect
4. #2085C2semidirect
5. #2073C2semidirect
6. #1948C2semidirect
7. #1947C2semidirect
8. #1825C2semidirect
9. #1802C2semidirect
10. #1802C2semidirect
11. #1797C2semidirect
262. C4#482semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C4×#1212
#22852
#20852
#20732
#19482
#19472
#18252
#18022
#17972