Finite Group Databasebeta

← #18840

Subgroups — #18840

Order 256.

446 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 113 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#3507C211yesmaximal, characteristic, complemented
2#3254C211yesmaximal, characteristic, complemented
3#3207C211yesmaximal, characteristic
4#3112C211yesmaximal, characteristic, complemented
5#1636C212yesmaximal, complemented
6#1630C212yesmaximal, complemented
7#1626C212yesmaximal, complemented
8#1620C212yesmaximal, complemented
9#1557C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1487C211yesmaximal, characteristic, complemented
12#559C2212yes—
13#557C2212yes—
14#556—21no—
15#556—22no—
16#554—21no—
17#552—22no—
18#539—22no—
19#536—21no—
20#533C2211yescharacteristic
21#532C2211yescharacteristic
22#532C2211yescharacteristic
23#529—22no—
24#520—21no—
25C2×C4×Q8C2211yescharacteristic
26#493C2211yescharacteristic
27#492C2211yescharacteristic
28#487C2211yescharacteristic
29#486C2211yescharacteristic
30#480—21no—
31#477C2212yes—
32#477C2212yes—
33#474—21no—
34#468—22no—
35#463—22no—
36#461—21no—
37#460C2212yes—
38#460C2212yes—
39#449—22no—
40#448—21no—
41#440—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44#423C2211yescharacteristic
45#332C2212yes—
46#331C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C2×#41—21no—
52C2×#41—21no—
53C22×Q8D411yescharacteristic
54C2×SD16—22no—
55C2×SD16—22no—
56C2×SD16—22no—
57C2×D8—22no—
58C2×M16—21no—
59C4⋊Q8—22no—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8D411yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64#127C2311yescharacteristic
65#127C2311yescharacteristic
66#126—24no—
67#126—24no—
68#125—22no—
69#125—24no—
70#125—21no—
71#124—22no—
72#124—22no—
73#124—21no—
74#124—21no—
75#123—24no—
76#123—24no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80C22⋊Q8—21no—
81C22⋊Q8D412yes—
82#121—21no—
83#121—21no—
84#121—21no—
85#121D412yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8D412yes—
92C4×Q8C2312yes—
93C4×Q8C2312yes—
94C4×D4—24no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4D412yes—
101#117—22no—
102#117—24no—
103#117—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126#41—44no—
127#41—82no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—42no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—41no—
142C2×D4C2×D412yes—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151SD16—44no—
152SD16—82no—
153SD16—82no—
154D8—82no—
155M16—81no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—22no—
159C2×C8—22no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D412yes—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C42—24no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2411yescharacteristic
193C23—22no—
194C23—41no—
195C23#12011yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201Q8—42no—
202Q8—42no—
203Q8—81no—
204D4—42no—
205D4—42no—
206D4—44no—
207D4—42no—
208D4—42no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—41no—
213D4—82no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—84no—
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C8—41no—
236C8—41no—
237C8—42no—
238C8—42no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22C2×#12011yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4#57512yes—
260C4#57711yescharacteristic
261C4#57811yescharacteristic
262C2—41no—
263C2—82no—
264C2—161no—
265C2#300511yescharacteristic
266C2#301311yescharacteristic
267C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3207C2
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. #3507C2semidirect
2. #3254C2semidirect
4. #3112C2semidirect
5. #1636C2semidirect
6. #1630C2semidirect
7. #1626C2semidirect
8. #1620C2semidirect
9. #1557C2semidirect
10. #1497C2semidirect
11. #1487C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32542
#32072
#31122
#16362
#16302
#16262
#16202
#15572
#14972
#14872