Finite Group Databasebeta

← #18778

Subgroups — #18778

Order 256.

466 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 129 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×#557C211yesmaximal, characteristic, complemented
2#3227C211yesmaximal, characteristic
3#3103C211yesmaximal, characteristic
4#3037C211yesmaximal, characteristic
5#1653C214yesmaximal, complemented
6#1626C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1487C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C2×#137—21no—
11#557—22no—
12#557—22no—
13#557—22no—
14#557C2214yes—
15#532C2211yescharacteristic
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—22no—
19C2×#125—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24#498C2212yes—
25#486C2212yes—
26#481—21no—
27#481—22no—
28#476—22no—
29#475C2212yes—
30#464—22no—
31#462C2212yes—
32#460C2212yes—
33#460C2212yes—
34#448—21no—
35C4×SD16—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#332C2214yes—
41Q8⋊C8C2214yes—
42Q8⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—21no—
45C22×Q8D411yescharacteristic
46C22×Q8D411yescharacteristic
47#137—44no—
48C2×Q16—22no—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×M16—21no—
52C4⋊Q8—24no—
53C4⋊Q8—24no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8D412yes—
57C4⋊Q8C2311yescharacteristic
58C4⋊Q8C2312yes—
59#127C2311yescharacteristic
60#125—28no—
61#125—24no—
62#125—28no—
63#125—24no—
64#124—21no—
65#121—21no—
66#121—21no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×Q8—24no—
70C4×Q8—24no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8D412yes—
76C4×Q8D412yes—
77C4×Q8C2312yes—
78C4×Q8C2312yes—
79C4×D4—21no—
80C4×D4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D412yes—
90C2×C42—22no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94#106—22no—
95#106—24no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—22no—
101#103—22no—
102#102—22no—
103#102—22no—
104#102—24no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C4×C8—21no—
108C4×C8C2312yes—
109#41—42no—
110#41—42no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8—44no—
119C2×Q8SD1614yescomplemented
120C2×Q8C2×D412yes—
121C2×Q8C2×D412yes—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—41no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—41no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135Q16—44no—
136SD16—44no—
137SD16—44no—
138M16—42no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—28no—
143C4⋊C4—28no—
144C4⋊C4—28no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D414yes—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C42—28no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—22no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2411yescharacteristic
170C23—41no—
171C23#12011yescharacteristic
172Q8—22no—
173Q8—24no—
174Q8—22no—
175Q8—22no—
176Q8—42no—
177Q8—44no—
178Q8—44no—
179Q8—41no—
180Q8—41no—
181Q8C2×SD1614yes—
182D4—42no—
183D4—41no—
184D4—41no—
185D4—42no—
186D4—82no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4C2×SD1612yes—
205C2×C4#13712yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C8—42no—
210C8—42no—
211C8—42no—
212C22—21no—
213C22—21no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22#44812yes—
218C22C2×#12011yescharacteristic
219C4—24no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—44no—
225C4—42no—
226C4—42no—
227C4—81no—
228C4#44812yes—
229C4C22×SD1611yescharacteristic
230C4C2×#13711yescharacteristic
231C4#57512yes—
232C2—22no—
233C2—161no—
234C2C2×#44811yescharacteristic
235C2#300511yescharacteristic
236C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#557C2semidirect
5. #1653C2semidirect
6. #1626C2semidirect
7. #1549C2semidirect
8. #1487C2semidirect
9. #1479C2semidirect
119. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
#32272
#31032
#30372
#16532
#16262
#15492
#14872
#14792