Finite Group Databasebeta

← #23590

Subgroups — #23590

Order 256.

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

normal subgroups only

This group has 537 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#3461C211yesmaximal, characteristic
2C4×#137C212yesmaximal, complemented
3C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#2060C211yesmaximal, characteristic, complemented
5#2039C211yesmaximal, characteristic
6#2038C212yesmaximal, complemented
7#2028C211yesmaximal, characteristic, complemented
8#2007C211yesmaximal, characteristic
9#2005C212yesmaximal, complemented
10C4×#104C212yesmaximal, complemented
11C4×#101C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13#544C2211yescharacteristic
14#542—22no—
15#541—22no—
16#539—22no—
17#537C2211yescharacteristic
18#536C2212yes—
19#533—22no—
20C2×C4⋊Q8C2211yescharacteristic
21#529—22no—
22#529—22no—
23C4×#41C2212yes—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26#441—22no—
27#440—22no—
28C4×Q16—22no—
29C4×SD16—22no—
30#428C2212yes—
31C2×#104C2212yes—
32C2×#104C2212yes—
33#419C2212yes—
34#417C2212yes—
35C2×#101C2212yes—
36C4×M16C2212yes—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#389—22no—
40#384C2211yescharacteristic
41C4×C4⋊C4—22no—
42C43C2211yescharacteristic
43#343C2211yescharacteristic
44#343C2211yescharacteristic
45#339C2212yes—
46#339C2212yes—
47C2×#41C2312yes—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50#137—24no—
51#137—24no—
52C2×Q16—22no—
53C2×SD16—22no—
54C2×M16C2312yes—
55C2×M16C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60#126—24no—
61#126—24no—
62#125—22no—
63#125—24no—
64#125—22no—
65#124—22no—
66#124—22no—
67#123—24no—
68#123—24no—
69C22⋊Q8—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74#121—22no—
75#121—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8D412yes—
82C4×Q8D412yes—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4D412yes—
87C4×D4D412yes—
88#117—24no—
89#117—22no—
90#117C2312yes—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4D412yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42D412yes—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—22no—
107#104—24no—
108#104—24no—
109#103—22no—
110#103—22no—
111#103—22no—
112#102—22no—
113#101—24no—
114#97—22no—
115C4×C8—22no—
116#95—22no—
117#95—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121#41—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8C2×D412yes—
131C2×Q8#4112yes—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4C2×D412yes—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4C2×D412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147Q16—28no—
148SD16—28no—
149M16—22no—
150M16—22no—
151M16—22no—
152M16#4114yes—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—28no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C42—24no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—21no—
183C42—22no—
184C42—42no—
185C42—41no—
186C42—41no—
187C42C2×D412yes—
188C42C2×D412yes—
189C42#4112yes—
190C23—22no—
191C23C22×D411yescharacteristic
192Q8—24no—
193Q8—22no—
194Q8—22no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198Q8—82no—
199D4—22no—
200D4—22no—
201D4—42no—
202D4—42no—
203D4—82no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—82no—
229C2×C4#12011yescharacteristic
230C2×C4#12012yes—
231C2×C4#12011yescharacteristic
232C2×C4C22×D412yes—
233C2×C4C2×#4111yescharacteristic
234C2×C4C2×#4112yes—
235C2×C4C2×#4111yescharacteristic
236C8—24no—
237C8—42no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22C2×#12011yescharacteristic
244C22#52511yescharacteristic
245C22#53211yescharacteristic
246C4—24no—
247C4—21no—
248C4—22no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—81no—
257C4#45612yes—
258C4#52512yes—
259C4#53212yes—
260C2—21no—
261C2—21no—
262C2—82no—
263C2#238811yescharacteristic
264C2C2×#45611yescharacteristic
265C2#301811yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC22×C4C24
socleC22C2×#120
prime radical#537C22
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
2. C4×#137C2semidirect
3. C4×C4⋊Q8C2semidirect
4. #2060C2semidirect
6. #2038C2semidirect
7. #2028C2semidirect
9. #2005C2semidirect
10. C4×#104C2semidirect
11. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
#34612
C4×#1372
C4×C4⋊Q82
#20602
#20392
#20382
#20282
#20072
#20052
C4×#1042
C4×#1012