Finite Group Databasebeta

← #18612

Subgroups — #18612

Order 256.

438 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 117 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#3502C211yesmaximal, characteristic, complemented
2#3216C211yesmaximal, characteristic
3#3182C211yesmaximal, characteristic
4#3103C211yesmaximal, characteristic
5#1648C212yesmaximal, complemented
6#1645C212yesmaximal, complemented
7#1626C212yesmaximal, complemented
8#1623C212yesmaximal, complemented
9#1554C211yesmaximal, characteristic, complemented
10#1491C211yesmaximal, characteristic, complemented
11#1485C211yesmaximal, characteristic, complemented
12Q82—21no—
13Q82C2212yes—
14#557—21no—
15#557—22no—
16#557—22no—
17#557C2212yes—
18#544—22no—
19#544—21no—
20#533—22no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23#532C2211yescharacteristic
24#519—21no—
25C2×C4×Q8C2211yescharacteristic
26#500C2211yescharacteristic
27#497—21no—
28#492C2211yescharacteristic
29#489—21no—
30#487C2211yescharacteristic
31#486C2211yescharacteristic
32#481—21no—
33#475C2212yes—
34#468—22no—
35#462C2212yes—
36#460C2212yes—
37#460C2212yes—
38#452—22no—
39#448—21no—
40#440—22no—
41C4×SD16—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#333C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8D411yescharacteristic
53C2×Q16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C2×SD16—22no—
57C2×M16—21no—
58C4⋊Q8—24no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68C4⋊Q8C2311yescharacteristic
69#127C2311yescharacteristic
70#125—24no—
71#125—24no—
72#125—22no—
73#125—24no—
74#125—22no—
75#125—21no—
76#125—21no—
77#124—21no—
78C22⋊Q8—24no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8D412yes—
82C22⋊Q8D412yes—
83#121—21no—
84#121—21no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—24no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8D412yes—
97C4×Q8D412yes—
98C4×Q8C2312yes—
99C4×Q8C2312yes—
100C4×D4—21no—
101C4×D4—21no—
102#117—24no—
103#117—21no—
104#117—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#106—22no—
111C4⋊C8C2312yes—
112C4⋊C8C2312yes—
113#103—22no—
114#103—22no—
115#103—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119C22⋊C8—21no—
120C22⋊C8—21no—
121#97—21no—
122#97—21no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126#41—82no—
127C2×Q8—24no—
128C2×Q8—24no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—41no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146Q16—82no—
147SD16—44no—
148SD16—44no—
149SD16—82no—
150M16—42no—
151C2×C8—21no—
152C2×C8—22no—
153C2×C8—21no—
154C2×C8—22no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D412yes—
177C4⋊C4C2×D412yes—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—24no—
183C42—24no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2411yescharacteristic
192C23—41no—
193C23#12011yescharacteristic
194Q8—24no—
195Q8—22no—
196Q8—22no—
197Q8—22no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—81no—
202Q8—84no—
203Q8#13714yes—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—22no—
228C8—41no—
229C8—42no—
230C8—42no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22C2×#12011yescharacteristic
238C4—24no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4C2×#13712yes—
250C4#57712yes—
251C2—41no—
252C2—161no—
253C2#300411yescharacteristic
254C2#301011yescharacteristic
255C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
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
1. #3502C2semidirect
5. #1648C2semidirect
6. #1645C2semidirect
7. #1626C2semidirect
8. #1623C2semidirect
9. #1554C2semidirect
10. #1491C2semidirect
11. #1485C2semidirect

Maximal subgroups

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

subgroupindex
#35022
#32162
#31822
#31032
#16482
#16452
#16262
#16232
#15542
#14912
#14852