Finite Group Databasebeta

← #21702

Subgroups — #21702

Order 256.

547 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 199 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×#497C211yesmaximal, characteristic
2C2×#441C212yesmaximal, complemented
3C2×#440C212yesmaximal
4C4×C4⋊Q8C211yesmaximal, characteristic, complemented
5C4×#124C211yesmaximal, characteristic
6#1969C212yesmaximal, complemented
7#1968C212yesmaximal
8#1960C211yesmaximal, characteristic, complemented
9#1959C211yesmaximal, characteristic
10#1847C211yesmaximal, characteristic, complemented
11C4×#97C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#497C418yescomplemented
20#441—28no—
21#440—28no—
22C2×#107C2212yes—
23C2×#106C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#102C2212yes—
28C2×#97C2211yescharacteristic
29C2×#97C2211yescharacteristic
30C2×#97C2212yes—
31C2×C4×C8C2212yes—
32#391C2212yes—
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#389—22no—
36#388—22no—
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39C4×C4⋊C4—22no—
40C4×C22⋊C4—22no—
41C43C2211yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16C2×C418yescomplemented
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16C2×C418yes—
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8C2×C414yescomplemented
56#124—22no—
57#124C2×C414yes—
58C4×Q8—24no—
59C4×Q8—24no—
60C4×Q8—24no—
61C4×D4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—22no—
74C2×C42D412yes—
75C2×C42D412yes—
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—24no—
80#106—24no—
81#103—24no—
82#103—24no—
83#103—24no—
84#102—24no—
85#97—22no—
86#97C2×C414yescomplemented
87#97D418yes—
88C4×C8—24no—
89#95—22no—
90#95—22no—
91#95—24no—
92C24—21no—
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C22×C412yes—
102C2×Q8C22×C414yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4C22×C412yes—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2411yescharacteristic
119Q16—216no—
120SD16—216no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8C22×C414yes—
124C2×C8C2×D414yes—
125C2×C8C2×D414yes—
126C2×C8#4114yes—
127C4⋊C4—28no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—44no—
133C4⋊C4—42no—
134C4⋊C4—44no—
135C22⋊C4—28no—
136C22⋊C4—42no—
137C22⋊C4—44no—
138C42—24no—
139C42—22no—
140C42—42no—
141C42—44no—
142C42—42no—
143C42C22×C412yes—
144C42C2×D414yes—
145C42#4112yes—
146C23—22no—
147C23—41no—
148C23—41no—
149C23—42no—
150C23—42no—
151C23C22×D411yescharacteristic
152Q8—24no—
153Q8—24no—
154Q8—24no—
155Q8—42no—
156Q8—42no—
157Q8—42no—
158D4—24no—
159D4—42no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—84no—
179C2×C4C4×D414yes—
180C2×C4#12712yes—
181C2×C4#12712yes—
182C2×C4#13714yes—
183C2×C4C23×C411yescharacteristic
184C2×C4C22×D412yes—
185C2×C4C2×#4112yes—
186C2×C4C2×#4111yescharacteristic
187C8—24no—
188C8C4×D418yes—
189C22—42no—
190C22—44no—
191C22—81no—
192C22—81no—
193C22—82no—
194C22C2×C4×D411yescharacteristic
195C22C2×#12711yescharacteristic
196C22#53211yescharacteristic
197C22C2×#13712yes—
198C22#57512yes—
199C4—22no—
200C4—22no—
201C4—24no—
202C4—24no—
203C4—42no—
204C4—44no—
205C4—81no—
206C4—82no—
207C4—81no—
208C4C2×C4×D412yes—
209C4#53212yes—
210C2—82no—
211C2C4×#12711yescharacteristic
212C2C4×#13712yes—
213C2#315211yescharacteristic
214C2#315311yescharacteristic
215C2#318112yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×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
2. C2×#441C2semidirect
4. C4×C4⋊Q8C2semidirect
6. #1969C2semidirect
8. #1960C2semidirect
10. #1847C2semidirect
11. C4×#97C2semidirect
19. #497C4semidirect
48. C2×Q16C2×C4semidirect
55. C4⋊Q8C2×C4semidirect
86. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4412
C2×#4402
C4×C4⋊Q82
C4×#1242
#19692
#19682
#19602
#19592
#18472
C4×#972