Finite Group Databasebeta

← #28540

Subgroups — #28540

Order 256.

457 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 151 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×#500C211yesmaximal, characteristic, complemented
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic
5#2568C211yesmaximal, characteristic, complemented
6#2542C211yesmaximal, characteristic
7#2095C212yesmaximal, complemented
8#2092C212yesmaximal
9#1973C212yesmaximal
10#1847C212yesmaximal, complemented
11#1770C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C22⋊Q8—22no—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#500—24no—
19#475—24no—
20#474—24no—
21C4×SD16—24no—
22C2×#107C2212yes—
23C2×#107C2212yes—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×#97C2212yes—
33C2×C4×C8C2211yescharacteristic
34#400—22no—
35#399—22no—
36#391—22no—
37#391C2212yes—
38#390C2212yes—
39#384C2212yes—
40#382C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×SD16—22no—
49C2×SD16—22no—
50C2×SD16Q814yescomplemented
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2311yescharacteristic
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C22⋊Q8—48no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×D4—22no—
62C4×D4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D412yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—24no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—24no—
78#107—24no—
79#107—24no—
80#106—22no—
81#106—24no—
82C4⋊C8D414yes—
83#103—22no—
84#103—24no—
85#102—22no—
86#102—24no—
87#97—24no—
88C4×C8D414yes—
89#95—24no—
90#95—24no—
91#95D412yes—
92C24—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—44no—
101C2×Q8—44no—
102C2×Q8C2×Q812yes—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4C2×Q812yes—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4C2×D414yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120SD16—28no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8C2×D412yes—
126C2×C8C2×D414yes—
127C2×C8C2×D412yes—
128C2×C8C2×Q812yes—
129C2×C8#4112yes—
130C4⋊C4—24no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—48no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C22⋊C4—48no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C42—42no—
148C42C2×D412yes—
149C42C2×D412yes—
150C23—22no—
151C23—41no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23C22×D411yescharacteristic
156Q8—22no—
157Q8—22no—
158Q8—41no—
159Q8—41no—
160Q8—84no—
161Q8—84no—
162D4—22no—
163D4—22no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—84no—
182C2×C4#12712yes—
183C2×C4#12712yes—
184C2×C4AGL(1,ℤ/8)14yes—
185C2×C4C22×D411yescharacteristic
186C2×C4C22×D411yescharacteristic
187C2×C4C22×Q811yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C42−1+411yescharacteristic
190C2×C42−1+411yescharacteristic
191C8—22no—
192C8—24no—
193C8C22⋊Q814yescomplemented
194C22—44no—
195C22—42no—
196C22—82no—
197C22—81no—
198C22—81no—
199C22C2×#12711yescharacteristic
200C22#54711yescharacteristic
201C22D4×Q811yescharacteristic
202C22C2×AGL(1,ℤ/8)12yes—
203C22#57812yes—
204C4—22no—
205C4—44no—
206C4—42no—
207C4—81no—
208C4—82no—
209C4—82no—
210C4—81no—
211C4#49614yes—
212C4C2×C22⋊Q811yescharacteristic
213C4#53611yescharacteristic
214C4#54711yescharacteristic
215C4D4×Q811yescharacteristic
216C2—82no—
217C2#254211yescharacteristic
218C2C2×#49611yescharacteristic
219C2#315611yescharacteristic
220C2#335312yes—
221C2#338612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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×#500C2semidirect
2. C2×#475C2semidirect
5. #2568C2semidirect
7. #2095C2semidirect
10. #1847C2semidirect
11. #1770C2semidirect
50. C2×SD16Q8semidirect
193. C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5002
C2×#4752
C2×#4742
C2×C4×SD162
#25682
#25422
#20952
#20922
#19732
#18472
#17702