Finite Group Databasebeta

← #23587

Subgroups — #23587

Order 256.

637 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 129 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#3460C211yesmaximal, characteristic, complemented
2C4×AGL(1,ℤ/8)C212yesmaximal, complemented
3C4×#127C211yesmaximal, characteristic, complemented
4#2060C211yesmaximal, characteristic, complemented
5#2039C211yesmaximal, characteristic, complemented
6#2037C212yesmaximal, complemented
7#2027C211yesmaximal, characteristic, complemented
8#2006C211yesmaximal, characteristic, complemented
9#2002C212yesmaximal, complemented
10C4×#104C212yesmaximal, complemented
11C4×#100C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)C2212yes—
13#544C2211yescharacteristic
14#542—22no—
15#540—22no—
16#538—22no—
17#537C2211yescharacteristic
18#535C2212yes—
19#532—22no—
20C2×#127C2211yescharacteristic
21#529—22no—
22#525—22no—
23C4×#41C2212yes—
24C2×C4×D4—22no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#442—22no—
28#440—22no—
29C4×SD16—22no—
30C4×D8—22no—
31#428C2212yes—
32C2×#104C2212yes—
33C2×#104C2212yes—
34#418C2212yes—
35#417C2212yes—
36C2×#100C2212yes—
37C4×M16C2212yes—
38#390—22no—
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#384C2211yescharacteristic
42C43C2211yescharacteristic
43#343C2211yescharacteristic
44#343C2211yescharacteristic
45#339C2212yes—
46#339C2212yes—
47C2×#41—22no—
48C2×#41C2312yes—
49C22×D4—22no—
50C22×D4C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4D412yes—
54AGL(1,ℤ/8)—24no—
55AGL(1,ℤ/8)—24no—
56C2×SD16—22no—
57C2×D8—22no—
58C2×M16C2312yes—
59C2×M16C2312yes—
60C4⋊Q8—22no—
61#127—22no—
62#127—22no—
63#127—22no—
64#126—24no—
65#125—22no—
66#124—24no—
67#124—22no—
68#123—24no—
69#123—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73#121—22no—
74#121—24no—
75#121—22no—
76#121—22no—
77#121—22no—
78#121—22no—
79#120—24no—
80C4×Q8D412yescomplemented
81C4×Q8D412yescomplemented
82C4×D4—24no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—48no—
90C4×D4D412yescomplemented
91C4×D4D412yescomplemented
92#117—22no—
93#117—22no—
94#117C2312yes—
95C2×C4⋊C4—22no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4D412yes—
102C2×C42—21no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#106—22no—
110#104—24no—
111#104—24no—
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#100—24no—
117#97—22no—
118C4×C8—22no—
119C24—21no—
120C24C2×D412yes—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125#41—84no—
126C2×Q8—22no—
127C2×Q8C2×D412yes—
128C2×D4—28no—
129C2×D4—22no—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4C2×D412yes—
143C2×D4#4112yescomplemented
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160SD16—28no—
161D8—28no—
162M16—22no—
163M16—22no—
164M16—22no—
165M16#4114yescomplemented
166C2×C8—22no—
167C2×C8—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—44no—
176C4⋊C4C2×D412yes—
177C4⋊C4C2×D412yes—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—24no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—42no—
188C42—24no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—21no—
193C42—22no—
194C42—42no—
195C42C2×D412yes—
196C42C2×D412yes—
197C42#4112yescomplemented
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23C22×D411yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—82no—
213D4—22no—
214D4—24no—
215D4—22no—
216D4—416no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—84no—
223D4—82no—
224D4—84no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4#12012yes—
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4C22×D412yes—
252C2×C4C2×#4111yescharacteristic
253C2×C4C2×#4112yes—
254C2×C4C2×#4111yescharacteristic
255C8—24no—
256C8—42no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—44no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22—82no—
270C22—81no—
271C22—81no—
272C22C2×#12011yescharacteristic
273C22#52511yescharacteristic
274C22#53211yescharacteristic
275C4—24no—
276C4—21no—
277C4—22no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—81no—
284C4#45312yes—
285C4#52512yes—
286C4#53212yes—
287C2—21no—
288C2—21no—
289C2—82no—
290C2—82no—
291C2—82no—
292C2#238811yescharacteristic
293C2C2×#45311yescharacteristic
294C2#301811yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC22×C4C24
socleC22C2×#120
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3460C2semidirect
2. C4×AGL(1,ℤ/8)C2semidirect
3. C4×#127C2semidirect
4. #2060C2semidirect
5. #2039C2semidirect
6. #2037C2semidirect
7. #2027C2semidirect
8. #2006C2semidirect
9. #2002C2semidirect
10. C4×#104C2semidirect
11. C4×#100C2semidirect
80. C4×Q8D4semidirect
81. C4×Q8D4semidirect
90. C4×D4D4semidirect
91. C4×D4D4semidirect
143. C2×D4#41semidirect
165. M16#41semidirect
197. C42#41semidirect

Maximal subgroups

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

subgroupindex
#34602
C4×AGL(1,ℤ/8)2
C4×#1272
#20602
#20392
#20372
#20272
#20062
#20022
C4×#1042
C4×#1002