Finite Group Databasebeta

← #18646

Subgroups — #18646

Order 256.

512 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 123 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#3511C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic, complemented
5#1628C214yesmaximal, complemented
6#1625C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)—21no—
11#556C2212yescomplemented
12#555—21no—
13#554—21no—
14#553—22no—
15#548—22no—
16#547C2212yescomplemented
17#542—22no—
18#537—21no—
19#533—21no—
20#532—21no—
21#532C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#127C2211yescharacteristic
24#520—21no—
25C4×#41C2211yescharacteristic
26#493—21no—
27#486—22no—
28#486—21no—
29#486C2214yes—
30#460—21no—
31#460C2212yes—
32#459—21no—
33#459C2212yes—
34#459C2214yes—
35#447—22no—
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38#418—21no—
39C2×#102—22no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43D4⋊C8C2212yescomplemented
44D4⋊C8C2214yescomplemented
45C2×#41—22no—
46C2×#41—21no—
47C2×#41D411yescharacteristic
48C22×Q8—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52AGL(1,ℤ/8)—44no—
53C2×SD16—22no—
54C2×D8—22no—
55C2×D8—24no—
56C2×M16—21no—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D412yes—
61C4⋊Q8D411yescharacteristic
62C4⋊Q8C2312yes—
63#127—21no—
64#127—21no—
65#127—21no—
66#127D411yescharacteristic
67#127C2312yes—
68#126—24no—
69#125—22no—
70#124—24no—
71#124—22no—
72#124—21no—
73#123—24no—
74C22⋊Q8—22no—
75C22⋊Q8—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78#121—22no—
79#121—22no—
80#121—24no—
81#121—22no—
82#121—21no—
83#121D412yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8D411yescharacteristic
87C4×Q8C2311yescharacteristic
88C4×D4—22no—
89C4×D4—24no—
90C4×D4—22no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4D411yescharacteristic
95C4×D4C2311yescharacteristic
96C4×D4C2312yes—
97#117—22no—
98#117—22no—
99#117—21no—
100#117D411yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4D412yes—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C4⋊C8—21no—
107C4⋊C8C2312yes—
108C4⋊C8C2312yes—
109#102—28no—
110#102—22no—
111#102—22no—
112#102—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117C24—41no—
118#41—22no—
119#41—22no—
120#41—42no—
121#41—42no—
122#41—84no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—82no—
127C2×Q8C2×D411yescharacteristic
128C2×D4—21no—
129C2×D4—22no—
130C2×D4—24no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—44no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—82no—
143C2×D4C2×D411yescharacteristic
144C2×D4C2×D412yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154SD16—44no—
155D8—44no—
156D8—84no—
157M16—42no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—42no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4C2×D411yescharacteristic
171C4⋊C4C2×D414yes—
172C4⋊C4C2×D411yescharacteristic
173C4⋊C4C2×D412yes—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—22no—
180C22⋊C4—21no—
181C22⋊C4—22no—
182C42—24no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2411yescharacteristic
190C23—21no—
191C23—22no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—82no—
196C23#12011yescharacteristic
197Q8—21no—
198Q8—21no—
199Q8—44no—
200Q8—42no—
201Q8—82no—
202D4—21no—
203D4—21no—
204D4—24no—
205D4—44no—
206D4—42no—
207D4—44no—
208D4—42no—
209D4—42no—
210D4—44no—
211D4—42no—
212D4—42no—
213D4—82no—
214D4—84no—
215D4—82no—
216D4—82no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—84no—
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4#12011yescharacteristic
234C2×C4#13512yes—
235C2×C4AGL(1,ℤ/8)12yes—
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C8—42no—
240C8—42no—
241C8—42no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—42no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—82no—
252C22—82no—
253C22—164no—
254C22#44912yes—
255C22C2×#12011yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—44no—
262C4—41no—
263C4#44912yes—
264C4#45412yes—
265C4C2×#13511yescharacteristic
266C4C2×AGL(1,ℤ/8)11yescharacteristic
267C4#57612yes—
268C2—22no—
269C2—81no—
270C2—82no—
271C2—162no—
272C2C2×#44911yescharacteristic
273C2#300811yescharacteristic
274C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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. #3511C2semidirect
2. #3205C2semidirect
3. C2×#486C2semidirect
4. #3038C2semidirect
5. #1628C2semidirect
6. #1625C2semidirect
7. #1549C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
11. #556C22semidirect
16. #547C22semidirect
41. #332C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35112
#32052
C2×#4862
#30382
#16282
#16252
#15492
#14922
#14762