Finite Group Databasebeta

← #18194

Subgroups — #18194

Order 256.

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

normal subgroups only

This group has 825 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#3496C211yesmaximal, characteristic, complemented
2C2×#447C212yesmaximal, complemented
3#2408C211yesmaximal, characteristic, complemented
4#1598C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1519C212yesmaximal, complemented
7#1459C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)—21no—
9C23×D4C2211yescharacteristic, complemented
10C22×D8—22no—
11#547—22no—
12#546—24no—
13D42—22no—
14#537C2212yescomplemented
15#534—21no—
16#534C2212yescomplemented
17#532—21no—
18C2×#121—21no—
19C2×#121—21no—
20C2×#121C2212yes—
21C2×#120—21no—
22C2×C4×D4C2211yescharacteristic
23#447—24no—
24#447C2218yescomplemented
25C2×#102—22no—
26C2×C22⋊C8C2212yescomplemented
27#394C2212yescomplemented
28#392C2212yescomplemented
29#390—21no—
30#380C2211yescharacteristic, complemented
31#379—21no—
32#327C2218yescomplemented
33#324C2212yescomplemented
34#323C2212yescomplemented
35C25—21no—
36H(5,𝔽2)—44no—
37H(5,𝔽2)—44no—
38C2×#41—24no—
39C2×#41—22no—
40C22×D4—24no—
41C22×D4—22no—
42C22×D4—22no—
43C22×D4—21no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—44no—
49C22×D4D411yescharacteristic
50C22×D4D412yescomplemented
51C22×D4C2312yes—
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C2×D8—28no—
55C2×D8—48no—
56C22×C8—22no—
57C4⋊Q8—21no—
58#127—21no—
59#124—22no—
60#123—24no—
61#123—24no—
62C22⋊Q8—24no—
63#121—22no—
64#121—22no—
65#121—24no—
66#121—22no—
67#121—22no—
68#121—22no—
69#121—22no—
70#121—44no—
71#121D414yescomplemented
72#121C2314yes—
73#120—24no—
74#120—24no—
75#120—44no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D412yescomplemented
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4D411yescharacteristic
91C2×C22⋊C4C2311yescharacteristic
92C2×C42D411yescharacteristic
93#102—28no—
94C22⋊C8—22no—
95C22⋊C8C2314yes—
96#95C2312yes—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—44no—
101C24—42no—
102C24—42no—
103C24—42no—
104C24—82no—
105C24C2×D411yescharacteristic
106C24C2×D411yescharacteristic
107#41—44no—
108#41—416no—
109#41—44no—
110C2×Q8—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—28no—
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—24no—
123C2×D4—42no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—48no—
128C2×D4—416no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—44no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—82no—
138C2×D4—82no—
139C2×D4D814yescomplemented
140C2×D4C2×D414yes—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D412yes—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D412yes—
155C22×C4C2411yescharacteristic
156D8—416no—
157C2×C8—24no—
158C2×C8—44no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4C2×D414yes—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C22⋊C4—44no—
174C22⋊C4—82no—
175C42—21no—
176C42—21no—
177C23—21no—
178C23—21no—
179C23—24no—
180C23—24no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—42no—
186C23—48no—
187C23—42no—
188C23—42no—
189C23—44no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23—82no—
195C23—82no—
196C23—84no—
197C23—82no—
198C23—82no—
199C23—88no—
200C23—88no—
201C23—84no—
202C23#12012yes—
203C23C2×D812yes—
204C23C2×D812yescomplemented
205C23AGL(1,ℤ/8)12yes—
206C23C22×D411yescharacteristic
207Q8—44no—
208D4—28no—
209D4—28no—
210D4—44no—
211D4—44no—
212D4—44no—
213D4—44no—
214D4—48no—
215D4—44no—
216D4—44no—
217D4—48no—
218D4—44no—
219D4—44no—
220D4—82no—
221D4—82no—
222D4—88no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—48no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4#12011yescharacteristic
243C2×C4#12011yescharacteristic
244C2×C4C2×D812yescomplemented
245C2×C4C22×D412yes—
246C8—44no—
247C22—21no—
248C22—22no—
249C22—21no—
250C22—22no—
251C22—22no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—44no—
256C22—44no—
257C22—41no—
258C22—41no—
259C22—44no—
260C22—44no—
261C22—44no—
262C22—44no—
263C22—81no—
264C22—82no—
265C22—82no—
266C22—816no—
267C22—82no—
268C22—82no—
269C22—164no—
270C22—164no—
271C22#44714yes—
272C22C2×#12011yescharacteristic
273C22C22×D811yescharacteristic
274C22C2×AGL(1,ℤ/8)11yescharacteristic
275C4—22no—
276C4—22no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—82no—
282C4B(4,𝔽2)12yes—
283C2—21no—
284C2—21no—
285C2—24no—
286C2—42no—
287C2—84no—
288C2—84no—
289C2C2×#44711yescharacteristic
290C2#302111yescharacteristic
291C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3496C2semidirect
2. C2×#447C2semidirect
3. #2408C2semidirect
4. #1598C2semidirect
5. #1595C2semidirect
6. #1519C2semidirect
7. #1459C2semidirect
9. C23×D4C22semidirect
14. #537C22semidirect
16. #534C22semidirect
24. #447C22semidirect
26. C2×C22⋊C8C22semidirect
27. #394C22semidirect
28. #392C22semidirect
30. #380C22semidirect
32. #327C22semidirect
33. #324C22semidirect
34. #323C22semidirect
50. C22×D4D4semidirect
71. #121D4semidirect
82. C2×C4⋊C4D4semidirect
139. C2×D4D8semidirect
204. C23C2×D8semidirect
244. C2×C4C2×D8semidirect

Maximal subgroups

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

subgroupindex
#34962
C2×#4472
#24082
#15982
#15952
#15192
#14592