Finite Group Databasebeta

← #18758

Subgroups — #18758

Order 256.

596 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 133 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×D4×Q8C211yesmaximal, characteristic, complemented
2#3227C211yesmaximal, characteristic, complemented
3#3103C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic
5#1651C214yesmaximal, complemented
6#1627C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1487C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C23×Q8—21no—
11C2×AGL(1,ℤ/8)—21no—
12D4×Q8—22no—
13D4×Q8—22no—
14D4×Q8—22no—
15D4×Q8C2214yes—
16#532C2211yescharacteristic
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8—22no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#498C2212yescomplemented
25#486C2212yescomplemented
26#481—21no—
27#481—22no—
28#476—22no—
29#475C2212yescomplemented
30#463—22no—
31#461C2212yes—
32#460C2212yescomplemented
33#459C2212yes—
34#448—21no—
35C4×SD16—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—24no—
47C22×Q8—21no—
48C22×Q8—48no—
49C22×Q8D411yescharacteristic
50C22×D4D411yescharacteristic
51C23×C4—22no—
52C23×C4—21no—
53AGL(1,ℤ/8)—44no—
54C2×SD16—22no—
55C2×SD16—22no—
56C2×D8—22no—
57C2×M16—21no—
58C4⋊Q8—24no—
59C4⋊Q8—24no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8D412yes—
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2312yes—
65#127C2311yescharacteristic
66#124—21no—
67C22⋊Q8—24no—
68C22⋊Q8—28no—
69C22⋊Q8—24no—
70C22⋊Q8—28no—
71#121—21no—
72#121—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8D412yes—
76C4×Q8C2312yes—
77C4×D4—24no—
78C4×D4—24no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4D412yes—
84C4×D4C2312yes—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D412yes—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95#106—22no—
96#106—24no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8C2312yes—
100C4⋊C8C2312yes—
101#103—22no—
102#103—22no—
103#102—22no—
104#102—22no—
105#102—24no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—21no—
111#41—42no—
112#41—42no—
113C2×Q8—24no—
114C2×Q8—28no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—48no—
120C2×Q8—44no—
121C2×Q8—416no—
122C2×Q8—48no—
123C2×Q8—48no—
124C2×Q8C2×D412yes—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—41no—
131C2×D4SD1614yescomplemented
132C2×D4C2×D412yes—
133C22×C4—21no—
134C22×C4—28no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—48no—
141C22×C4—44no—
142C22×C4—41no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146SD16—44no—
147SD16—44no—
148D8—44no—
149M16—42no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C4⋊C4—28no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D414yes—
166C22⋊C4—28no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C42—22no—
172C42—22no—
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C42C2×D411yescharacteristic
176C42C2411yescharacteristic
177C23—22no—
178C23—24no—
179C23—42no—
180C23—42no—
181C23—41no—
182C23#12011yescharacteristic
183Q8—24no—
184Q8—22no—
185Q8—416no—
186Q8—44no—
187Q8—44no—
188Q8—41no—
189Q8—41no—
190Q8AGL(1,ℤ/8)14yescomplemented
191D4—22no—
192D4—24no—
193D4—42no—
194D4—41no—
195D4—42no—
196D4—41no—
197D4—82no—
198D4C2×SD1614yes—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—416no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4#12011yescharacteristic
217C2×C4C2×SD1612yes—
218C2×C4#13712yes—
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C8—42no—
223C8—42no—
224C8—42no—
225C22—21no—
226C22—21no—
227C22—24no—
228C22—48no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22#44812yes—
235C22C2×#12011yescharacteristic
236C4—24no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—44no—
241C4—42no—
242C4—42no—
243C4—81no—
244C4#44812yes—
245C4C22×SD1611yescharacteristic
246C4C2×AGL(1,ℤ/8)12yes—
247C4C2×#13711yescharacteristic
248C2—22no—
249C2—44no—
250C2—161no—
251C2C2×#44811yescharacteristic
252C2#300311yescharacteristic
253C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3036C2
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×D4×Q8C2semidirect
2. #3227C2semidirect
3. #3103C2semidirect
5. #1651C2semidirect
6. #1627C2semidirect
7. #1549C2semidirect
8. #1487C2semidirect
9. #1478C2semidirect
24. #498C22semidirect
25. #486C22semidirect
29. #475C22semidirect
32. #460C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect
131. C2×D4SD16semidirect
190. Q8AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
#32272
#31032
#30362
#16512
#16272
#15492
#14872
#14782