Finite Group Databasebeta

← #18912

Subgroups — #18912

Order 256.

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

normal subgroups only

This group has 609 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#3497C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic, complemented
4C2×#460C211yesmaximal, characteristic, complemented
5#1627C214yesmaximal, complemented
6#1621C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C2×2−1+4—21no—
11C22×SD16—21no—
12#550—21no—
13#550C2212yescomplemented
14D4×Q8—21no—
15D4×Q8C2212yescomplemented
16#548—22no—
17#548—22no—
18#537—22no—
19#537—21no—
20#532—22no—
21#532C2211yescharacteristic
22#532C2211yescharacteristic
23C2×#127C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#493—21no—
26#493C2212yes—
27#486—21no—
28#486C2212yes—
29#460—22no—
30#460C2214yes—
31#459—22no—
32#459C2214yes—
33#447—22no—
34#447—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#332C2212yescomplemented
40#331C2212yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
432−1+4—44no—
442−1+4—44no—
45C2×#41—24no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41—21no—
49C2×#41—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8D411yescharacteristic
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—21no—
56C2×SD16—24no—
57C2×SD16—44no—
58C2×D8—24no—
59C2×D8—24no—
60C22×C8—21no—
61C4⋊Q8—22no—
62C4⋊Q8D411yescharacteristic
63C4⋊Q8C2311yescharacteristic
64#127—22no—
65#127—21no—
66#127—21no—
67#127—21no—
68#127D411yescharacteristic
69#127C2312yes—
70#127C2311yescharacteristic
71#124—24no—
72#124—21no—
73#124—21no—
74C22⋊Q8—22no—
75C22⋊Q8—24no—
76C22⋊Q8—22no—
77C22⋊Q8—24no—
78#121—24no—
79#121—22no—
80#121—24no—
81#121—24no—
82#121—21no—
83#121—21no—
84#121—22no—
85#121D412yes—
86#121D412yes—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8D412yes—
90C4×Q8C2312yes—
91C4×D4—24no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4D412yes—
97C4×D4C2312yes—
98C2×C4⋊C4—21no—
99C2×C4⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8C2312yes—
105C4⋊C8C2312yes—
106#102—24no—
107#102—24no—
108#102—24no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C4×C8—21no—
112C4×C8C2312yes—
113C24—41no—
114#41—416no—
115#41—44no—
116#41—48no—
117#41—44no—
118#41—42no—
119#41—42no—
120#41—42no—
121#41—42no—
122C2×Q8—24no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—21no—
127C2×Q8—44no—
128C2×Q8—48no—
129C2×Q8—44no—
130C2×Q8D814yescomplemented
131C2×Q8C2×D412yes—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—82no—
149C2×D4—82no—
150C2×D4C2×D412yes—
151C2×D4C2×D412yes—
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161SD16—48no—
162D8—84no—
163D8—84no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—42no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D412yes—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C42—22no—
184C42—22no—
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2411yescharacteristic
189C23—22no—
190C23—22no—
191C23—42no—
192C23—82no—
193C23—82no—
194C23—82no—
195C23#12011yescharacteristic
196Q8—22no—
197Q8—24no—
198Q8—48no—
199Q8—41no—
200Q8—41no—
201Q8—42no—
202Q8C2×D814yes—
203D4—24no—
204D4—42no—
205D4—48no—
206D4—44no—
207D4—44no—
208D4—44no—
209D4—41no—
210D4—42no—
211D4—41no—
212D4—42no—
213D4—42no—
214D4—44no—
215D4—42no—
216D4—81no—
217D4—82no—
218D4—82no—
219D4—84no—
220D4—82no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—48no—
229C2×C4—48no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—81no—
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4#12011yescharacteristic
237C2×C4C2×D812yes—
238C2×C4AGL(1,ℤ/8)12yes—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C8—42no—
243C8—42no—
244C8—42no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—164no—
257C22#44712yes—
258C22C2×#12011yescharacteristic
259C4—24no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4#44712yes—
267C4C22×D811yescharacteristic
268C4C2×AGL(1,ℤ/8)11yescharacteristic
269C4#57712yes—
270C2—22no—
271C2—82no—
272C2—82no—
273C2—162no—
274C2C2×#44711yescharacteristic
275C2#301011yescharacteristic
276C2#302111yescharacteristic
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. #3497C2semidirect
2. #3205C2semidirect
3. #3205C2semidirect
4. C2×#460C2semidirect
5. #1627C2semidirect
6. #1621C2semidirect
7. #1540C2semidirect
8. #1480C2semidirect
9. #1479C2semidirect
13. #550C22semidirect
15. D4×Q8C22semidirect
39. #332C22semidirect
40. #331C22semidirect
41. Q8⋊C8C22semidirect
42. D4⋊C8C22semidirect
130. C2×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#34972
#32052
C2×#4602
#16272
#16212
#15402
#14802
#14792