Finite Group Databasebeta

← #23593

Subgroups — #23593

Order 256.

559 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 127 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
2C4×#137C212yesmaximal, complemented
3#2326C211yesmaximal, characteristic, complemented
4#2060C211yesmaximal, characteristic, complemented
5#2039C211yesmaximal, characteristic, complemented
6#2037C212yesmaximal, complemented
7#2028C211yesmaximal, characteristic, complemented
8#2007C211yesmaximal, characteristic, complemented
9#2004C212yesmaximal, complemented
10#1759C212yesmaximal, complemented
11C4×#100C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13#544C2211yescharacteristic
14#542—22no—
15#540—22no—
16#538—22no—
17#537C2211yescharacteristic
18#535C2212yes—
19#532—22no—
20#529—22no—
21C2×#124C2211yescharacteristic
22#525—22no—
23C4×#41C2212yes—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#441—22no—
27#440—22no—
28C4×Q16—22no—
29C4×SD16—22no—
30#428C2212yes—
31C2×#104C2212yes—
32C2×#104C2212yes—
33#419C2212yes—
34#417C2212yes—
35C2×#100C2212yes—
36C4×M16C2212yes—
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#388—22no—
40#385—22no—
41#384C2211yescharacteristic
42#376C2211yescharacteristic
43#343C2211yescharacteristic
44#343C2211yescharacteristic
45#339C2212yes—
46#339C2212yes—
47C2×#41—22no—
48C2×#41C2312yes—
49C22×Q8C2311yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4D412yes—
52#137—24no—
53#137—24no—
54C2×Q16—22no—
55C2×SD16—22no—
56C2×M16C2312yes—
57C2×M16C2312yes—
58C4⋊Q8—22no—
59#127—22no—
60#126—24no—
61#125—22no—
62#124—24no—
63#124—22no—
64#124—22no—
65#124—22no—
66#123—24no—
67#123—24no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C22⋊Q8—22no—
71#121—22no—
72#121—24no—
73#121—22no—
74#121—22no—
75#121—22no—
76#121—22no—
77#120—24no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8D412yes—
81C4×Q8D412yes—
82C4×D4—24no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4D412yes—
88C4×D4D412yes—
89#117—22no—
90#117—22no—
91#117C2312yes—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4D412yes—
99C2×C42—22no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#106—22no—
106#104—24no—
107#104—24no—
108#103—22no—
109#103—22no—
110#103—22no—
111#102—22no—
112#100—24no—
113#97—22no—
114C4×C8—22no—
115#95—21no—
116#95—21no—
117#95—22no—
118C24C2×D412yes—
119#41—22no—
120#41—22no—
121#41—22no—
122#41—22no—
123#41—84no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8C2×D412yes—
129C2×Q8#4112yescomplemented
130C2×D4—22no—
131C2×D4—24no—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C2×D412yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—82no—
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2411yescharacteristic
153Q16—28no—
154SD16—28no—
155M16—22no—
156M16—22no—
157M16—22no—
158M16#4114yescomplemented
159C2×C8—22no—
160C2×C8—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—28no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—82no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—21no—
186C42—22no—
187C42—42no—
188C42C2×D412yes—
189C42C2×D412yes—
190C42#4112yescomplemented
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—42no—
196C23—42no—
197C23C22×D411yescharacteristic
198Q8—24no—
199Q8—22no—
200Q8—22no—
201Q8—42no—
202Q8—82no—
203D4—22no—
204D4—22no—
205D4—42no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—84no—
210D4—82no—
211D4—84no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#12011yescharacteristic
237C2×C4#12012yes—
238C2×C4#12011yescharacteristic
239C2×C4C22×D412yes—
240C2×C4C2×#4111yescharacteristic
241C2×C4C2×#4112yes—
242C2×C4C2×#4111yescharacteristic
243C8—24no—
244C8—42no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—82no—
255C22C2×#12011yescharacteristic
256C22#52511yescharacteristic
257C22#53211yescharacteristic
258C4—21no—
259C4—21no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—81no—
268C4—81no—
269C4#52512yes—
270C4#53212yes—
271C2—21no—
272C2—21no—
273C2—82no—
274C2—82no—
275C2#238811yescharacteristic
276C2#301911yescharacteristic
277C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #2326C2semidirect
4. #2060C2semidirect
5. #2039C2semidirect
6. #2037C2semidirect
7. #2028C2semidirect
8. #2007C2semidirect
9. #2004C2semidirect
10. #1759C2semidirect
11. C4×#100C2semidirect
129. C2×Q8#41semidirect
158. M16#41semidirect
190. C42#41semidirect

Maximal subgroups

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

subgroupindex
#34602
C4×#1372
#23262
#20602
#20392
#20372
#20282
#20072
#20042
#17592
C4×#1002