Finite Group Databasebeta

← #19177

Subgroups — #19177

Order 256.

325 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#3141C211yesmaximal, characteristic
2#3103C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4Q8×M16C211yesmaximal, characteristic, complemented
5#1702C212yesmaximal, complemented
6#1700C212yesmaximal, complemented
7#1682C212yesmaximal, complemented
8#1680C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic, complemented
10#1549C211yesmaximal, characteristic, complemented
11#1547C211yesmaximal, characteristic
12C22×Q16—21no—
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#489—22no—
17#487C2212yes—
18#486C2212yes—
19#481—21no—
20#475C2211yescharacteristic
21#475C2211yescharacteristic
22#462—21no—
23#462—21no—
24#462C2212yes—
25#462C2212yes—
26#460C2211yescharacteristic
27#460C2211yescharacteristic
28#448—21no—
29#446—22no—
30#446C2212yes—
31C8×Q8C2212yes—
32C4×SD16—21no—
33C4×SD16—21no—
34#431C2211yescharacteristic
35#423—21no—
36#423C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38#417—21no—
39#417—21no—
40C2×#103—21no—
41C4×M16—21no—
42C4×M16C2211yescharacteristic
43#335C2212yes—
44#334C2212yes—
45#332C2212yes—
46#332C2212yes—
47#332C2212yes—
48#331C2212yes—
49C2×#41—21no—
50C22×Q8—21no—
51C22×Q8D411yescharacteristic
52C2×Q16—22no—
53C2×Q16—22no—
54C2×Q16—44no—
55C2×SD16—21no—
56C2×SD16—21no—
57C2×M16—21no—
58C2×M16D411yescharacteristic
59C2×M16D411yescharacteristic
60C22×C8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8C2312yes—
65C4⋊Q8C2311yescharacteristic
66#127C2311yescharacteristic
67#124—21no—
68#121—21no—
69#121—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8D411yescharacteristic
73C4×Q8D411yescharacteristic
74C4×Q8C2311yescharacteristic
75C4×Q8C2311yescharacteristic
76C4×D4—21no—
77C4×D4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C42—21no—
83C2×C42C2311yescharacteristic
84#106—21no—
85#106—21no—
86C4⋊C8—22no—
87C4⋊C8—22no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8C2312yes—
91C4⋊C8C2312yes—
92C4⋊C8C2311yescharacteristic
93C4⋊C8C2311yescharacteristic
94#103—21no—
95#103—21no—
96#103—22no—
97#103—22no—
98#103—22no—
99#103—22no—
100#102—22no—
101#102—22no—
102#102—21no—
103#102—21no—
104C22⋊C8—21no—
105#97—22no—
106#97D412yes—
107C4×C8—22no—
108C4×C8—21no—
109C4×C8C2312yes—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—41no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—41no—
120C2×Q8C2×D411yescharacteristic
121C2×Q8C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—41no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4#4111yescharacteristic
133Q16—44no—
134Q16—44no—
135SD16—42no—
136SD16—42no—
137M16—22no—
138M16—22no—
139M16—42no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—42no—
145C2×C8C2×D412yes—
146C2×C8C2×D412yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C2×D411yescharacteristic
159C4⋊C4C2×D411yescharacteristic
160C22⋊C4—41no—
161C22⋊C4—41no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42#4111yescharacteristic
169C42C2411yescharacteristic
170C23—41no—
171C23#12111yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—21no—
175Q8—42no—
176Q8—41no—
177Q8—42no—
178Q8—41no—
179Q8—42no—
180Q8—82no—
181Q8AGL(1,ℤ/8)12yes—
182D4—41no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—81no—
187D4—82no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#12111yescharacteristic
202C2×C4#12111yescharacteristic
203C2×C4#12111yescharacteristic
204C2×C4#13512yes—
205C2×C4#13712yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C8—22no—
210C8—22no—
211C8—42no—
212C8—42no—
213C8—41no—
214C8—41no—
215C22—21no—
216C22—21no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22#46412yes—
221C22C2×#12111yescharacteristic
222C4—21no—
223C4—22no—
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—81no—
230C4—82no—
231C4#46412yes—
232C4C2×#13511yescharacteristic
233C4C2×AGL(1,ℤ/8)11yescharacteristic
234C4C2×#13711yescharacteristic
235C4#57511yescharacteristic
236C2—22no—
237C2—161no—
238C2C2×#46411yescharacteristic
239C2#305511yescharacteristic
240C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
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
3. C2×#462C2semidirect
4. Q8×M16C2semidirect
5. #1702C2semidirect
6. #1700C2semidirect
7. #1682C2semidirect
8. #1680C2semidirect
9. #1565C2semidirect
10. #1549C2semidirect

Maximal subgroups

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

subgroupindex
#31412
#31032
C2×#4622
Q8×M162
#17022
#17002
#16822
#16802
#15652
#15492
#15472