Finite Group Databasebeta

← #18753

Subgroups — #18753

Order 256.

483 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 125 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×#554C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic, complemented
3#3101C211yesmaximal, characteristic
4#3078C211yesmaximal, characteristic, complemented
5#1643C214yesmaximal, complemented
6#1632C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic
9#1479C211yesmaximal, characteristic, complemented
10#554—22no—
11#554—22no—
12#554—22no—
13#554C2214yes—
14#533C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4—21no—
23#494C2212yescomplemented
24#487C2212yescomplemented
25#483—21no—
26#479—22no—
27#477C2212yescomplemented
28#476C2212yes—
29#475C2212yescomplemented
30#462—22no—
31#461C2212yes—
32#451—22no—
33#450—21no—
34C4×SD16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#419—21no—
40#333C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C22×Q8D411yescharacteristic
44C22×Q8D412yes—
45C22×D4D411yescharacteristic
46C23×C4—22no—
47C23×C4—21no—
48C2×Q16—24no—
49C2×SD16—22no—
50C2×M16—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D412yes—
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2312yes—
57#125—24no—
58#125—24no—
59#125—21no—
60C22⋊Q8—24no—
61C22⋊Q8—28no—
62C22⋊Q8—24no—
63C22⋊Q8—28no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8D412yes—
71C4×Q8C2312yes—
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4D412yes—
77C4×D4C2312yes—
78#117—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#106—22no—
94#106—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#103—22no—
101#103—24no—
102#103—22no—
103#102—22no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C4×C8—21no—
107C4×C8C2312yes—
108C24—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—82no—
116C2×Q8C2×D412yes—
117C2×Q8C2×D414yes—
118C2×D4—24no—
119C2×D4—21no—
120C2×D4C2×D412yes—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—28no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—84no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134Q16—84no—
135SD16—44no—
136M16—42no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—28no—
141C4⋊C4—28no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—28no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—41no—
162C42—22no—
163C42—22no—
164C42—41no—
165C42C2×D411yescharacteristic
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2411yescharacteristic
169C23—24no—
170C23—22no—
171C23—42no—
172C23—42no—
173C23#12011yescharacteristic
174Q8—22no—
175Q8—22no—
176Q8—44no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180Q8—82no—
181D4—24no—
182D4—22no—
183D4#13714yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—88no—
200C2×C4#12011yescharacteristic
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#13512yes—
204C2×C4#13712yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C8—42no—
209C8—42no—
210C8—42no—
211C22—21no—
212C22—21no—
213C22—24no—
214C22—48no—
215C22—42no—
216C22—42no—
217C22#45212yes—
218C22C2×#12011yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—44no—
226C4—81no—
227C4—81no—
228C4#45212yes—
229C4C2×#13511yescharacteristic
230C4C2×#13711yescharacteristic
231C4C2×#13712yes—
232C2—22no—
233C2—44no—
234C2C2×#45211yescharacteristic
235C2#300411yescharacteristic
236C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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
2. #3209C2semidirect
4. #3078C2semidirect
5. #1643C2semidirect
6. #1632C2semidirect
7. #1548C2semidirect
9. #1479C2semidirect
23. #494C22semidirect
24. #487C22semidirect
27. #477C22semidirect
29. #475C22semidirect
40. #333C22semidirect
41. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#32092
#31012
#30782
#16432
#16322
#15482
#14862
#14792