Finite Group Databasebeta

← #26231

Subgroups — #26231

Order 256.

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

normal subgroups only

This group has 431 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×#485C212yesmaximal, complemented
2#2542C211yesmaximal, characteristic
3#2354C211yesmaximal, characteristic
4#2036C212yesmaximal, complemented
5#2035C212yesmaximal, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic
9#1948C211yesmaximal, characteristic, complemented
10#1873C212yesmaximal, complemented
11C4×#107C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×C4×D4C2211yescharacteristic
16C2×#117C2212yes—
17#485—28no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2212yes—
27C2×C22⋊C8C2212yes—
28C2×C4×C8C2211yescharacteristic
29#399—22no—
30#390—21no—
31#390—21no—
32#388—22no—
33#384—21no—
34#384C2211yescharacteristic
35#380C2212yes—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C4⋊C4C2211yescharacteristic
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42C22×Q8C2312yes—
43C22×D4D411yescharacteristic
44C23×C4C2312yes—
45C22×C8C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52C22⋊Q8—42no—
53C22⋊Q8—42no—
54C4×D4—22no—
55C4×D4—22no—
56#117—24no—
57#117—42no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C42—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C2311yescharacteristic
79#107—22no—
80#107D414yescomplemented
81#106—24no—
82C4⋊C8—22no—
83#103—24no—
84#103—24no—
85#103—42no—
86#103—42no—
87C22⋊C8—24no—
88C22⋊C8—42no—
89C4×C8—22no—
90#95—22no—
91#95—21no—
92#95—21no—
93C24C2×D412yes—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8C2×D412yes—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—81no—
142C4⋊C4C2×D412yes—
143C4⋊C4C2×D412yes—
144C4⋊C4#4112yes—
145C22⋊C4—24no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C42—44no—
151C42—42no—
152C42—42no—
153C42#4112yes—
154C42#4112yes—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23C22×D411yescharacteristic
163Q8—42no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167D4—44no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4#12312yes—
196C2×C4#12312yes—
197C2×C4#13512yes—
198C2×C4#13712yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4H(5,𝔽2)11yescharacteristic
204C2×C4H(5,𝔽2)11yescharacteristic
205C8—22no—
206C8—41no—
207C8—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22C2×#12311yescharacteristic
216C22D4211yescharacteristic
217C22#54611yescharacteristic
218C22C2×#13511yescharacteristic
219C22C2×#13711yescharacteristic
220C22#57711yescharacteristic
221C22#57811yescharacteristic
222C4—22no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—82no—
228C4—82no—
229C4—81no—
230C4#48514yescomplemented
231C4#53211yescharacteristic
232C4#54011yescharacteristic
233C4#55011yescharacteristic
234C4#55911yescharacteristic
235C2—82no—
236C2#244511yescharacteristic
237C2C2×#48511yescharacteristic
238C2#309711yescharacteristic
239C2#327811yescharacteristic
240C2#328311yescharacteristic
241C2#330311yescharacteristic
242C2#332811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#485C2semidirect
4. #2036C2semidirect
5. #2035C2semidirect
6. #1981C2semidirect
7. #1956C2semidirect
9. #1948C2semidirect
10. #1873C2semidirect
11. C4×#107C2semidirect
80. #107D4semidirect
230. C4#485semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
#25422
#23542
#20362
#20352
#19812
#19562
#19542
#19482
#18732
C4×#1072