Finite Group Databasebeta

← #18828

Subgroups — #18828

Order 256.

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

normal subgroups only

This group has 453 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#3503C211yesmaximal, characteristic, complemented
2#3226C211yesmaximal, characteristic, complemented
3#3216C211yesmaximal, characteristic, complemented
4#3115C211yesmaximal, characteristic
5#1649C212yesmaximal, complemented
6#1648C212yesmaximal, complemented
7#1627C212yesmaximal, complemented
8#1626C212yesmaximal, complemented
9#1556C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, characteristic, complemented
12Q82—21no—
13#557C2212yes—
14#554—21no—
15#554—22no—
16D4×Q8—22no—
17D4×Q8C2212yes—
18#544—22no—
19#544—21no—
20#533—21no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#500C2212yes—
28#497—21no—
29#489—21no—
30#486C2212yes—
31#482—21no—
32#480—22no—
33#475C2211yescharacteristic
34#474—22no—
35#474C2211yescharacteristic
36#462C2212yes—
37#460C2211yescharacteristic
38#460C2212yes—
39#459C2211yescharacteristic
40#452—22no—
41#449—21no—
42#442—21no—
43#440—21no—
44#432C2211yescharacteristic
45#432C2211yescharacteristic
46#431C2211yescharacteristic
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41—21no—
54C2×#41D411yescharacteristic
55C22×Q8—22no—
56C22×Q8—21no—
57C2×Q16—22no—
58C2×SD16—21no—
59C2×SD16—22no—
60C2×D8—21no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8—24no—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8D412yes—
70C4⋊Q8D411yescharacteristic
71C4⋊Q8C2311yescharacteristic
72C4⋊Q8C2312yes—
73#127C2311yescharacteristic
74#125—22no—
75#125—24no—
76#125—22no—
77#125—21no—
78#124—21no—
79C22⋊Q8—22no—
80C22⋊Q8—24no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—22no—
84C22⋊Q8—24no—
85C22⋊Q8D412yes—
86#121—21no—
87#121—21no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8D411yescharacteristic
93C4×Q8C2311yescharacteristic
94C4×Q8C2312yes—
95C4×D4—24no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4D411yescharacteristic
102C4×D4C2311yescharacteristic
103#117—22no—
104#117—21no—
105#117—21no—
106#117D411yescharacteristic
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4D412yes—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113#107—21no—
114#107—24no—
115#106—21no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—22no—
122#102—21no—
123#102—21no—
124#102—21no—
125#102—22no—
126#102—24no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130#97—21no—
131#97—21no—
132C4×C8C2312yes—
133#41—22no—
134#41—22no—
135#41—82no—
136C2×Q8—24no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—21no—
141C2×Q8—82no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8C2×D412yes—
145C2×Q8C2×D411yescharacteristic
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—41no—
151C2×D4C2×D411yescharacteristic
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161Q16—82no—
162SD16—42no—
163SD16—82no—
164D8—42no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D411yescharacteristic
187C4⋊C4C2×D414yes—
188C4⋊C4C2×D411yescharacteristic
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C42—24no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—22no—
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2411yescharacteristic
205C23—21no—
206C23—41no—
207C23#12011yescharacteristic
208Q8—21no—
209Q8—22no—
210Q8—22no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214Q8—42no—
215Q8—82no—
216Q8—84no—
217Q8#13712yes—
218D4—24no—
219D4—21no—
220D4—42no—
221D4—41no—
222D4—41no—
223D4—42no—
224D4—82no—
225D4#13712yes—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—84no—
243C2×C4#12011yescharacteristic
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C8—42no—
250C8—41no—
251C8—41no—
252C8—42no—
253C22—21no—
254C22—22no—
255C22—21no—
256C22—21no—
257C22—41no—
258C22—41no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22C2×#12011yescharacteristic
263C4—21no—
264C4—22no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—42no—
270C4—44no—
271C4—41no—
272C4—42no—
273C4—81no—
274C4C2×#13711yescharacteristic
275C4C2×#13711yescharacteristic
276C4#57612yes—
277C2—41no—
278C2—42no—
279C2—161no—
280C2#300411yescharacteristic
281C2#300811yescharacteristic
282C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3115C2
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. #3503C2semidirect
2. #3226C2semidirect
3. #3216C2semidirect
5. #1649C2semidirect
6. #1648C2semidirect
7. #1627C2semidirect
8. #1626C2semidirect
9. #1556C2semidirect
10. #1497C2semidirect
11. #1488C2semidirect

Maximal subgroups

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

subgroupindex
#35032
#32262
#32162
#31152
#16492
#16482
#16272
#16262
#15562
#14972
#14882