Finite Group Databasebeta

← #21475

Subgroups — #21475

Order 256.

685 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C4×2−1+4C211yesmaximal, characteristic, complemented
2C2×#454C211yesmaximal, characteristic
3C4×AGL(1,ℤ/8)C212yesmaximal, complemented
4#2326C211yesmaximal, characteristic, complemented
5#1888C211yesmaximal, characteristic, complemented
6#1883C212yesmaximal, complemented
7#1859C212yesmaximal, complemented
8#1857C211yesmaximal, characteristic, complemented
9#1793C211yesmaximal, characteristic, complemented
10#1759C212yesmaximal, complemented
11C4×#101C211yesmaximal, characteristic, complemented
12C2×2−1+4C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2212yes—
14C2×#124C2211yescharacteristic
15#520—22no—
16#520—22no—
17#520C2212yes—
18#519—22no—
19#519C2211yescharacteristic
20C4×#41—22no—
21C4×#41—22no—
22C4×#41C2212yes—
23C2×C4×Q8—22no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#454C418yescomplemented
27#442—22no—
28#440—22no—
29C4×SD16—22no—
30C4×D8—22no—
31#428C2212yes—
32C2×#104C2212yes—
33C2×#104C2212yes—
34#418C2212yes—
35#417C2212yes—
36C2×#101C2211yescharacteristic
37C2×#101C2211yescharacteristic
38C4×M16C2212yes—
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#388—22no—
42#385—22no—
43#384C2211yescharacteristic
44#376C2211yescharacteristic
45#343C2212yes—
46#339C2212yes—
47#339C2212yes—
482−1+4—24no—
492−1+4—22no—
502−1+4C2×C414yescomplemented
51C2×#41—22no—
52C2×#41—22no—
53C2×#41C2312yes—
54C22×Q8—22no—
55C22×Q8C2311yescharacteristic
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58AGL(1,ℤ/8)—24no—
59AGL(1,ℤ/8)C2×C418yescomplemented
60C2×SD16—22no—
61C2×D8—22no—
62C2×M16C2312yes—
63C2×M16C2312yes—
64#124—22no—
65#124C2×C414yescomplemented
66C4×Q8—22no—
67C4×Q8—24no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×Q8D412yes—
75C4×Q8D412yes—
76C4×D4—22no—
77C4×D4—24no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—24no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4D412yes—
86C4×D4D412yes—
87#117—24no—
88#117—24no—
89#117—22no—
90#117—22no—
91#117—22no—
92#117C2312yes—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4D412yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—22no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C42—24no—
103C2×C42—22no—
104C2×C42D412yes—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#106—22no—
110#104—24no—
111#104C2×C418yescomplemented
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#101—22no—
117#101C2×C414yescomplemented
118#97—22no—
119C4×C8—22no—
120#95—21no—
121#95—21no—
122#95—22no—
123C24—21no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C22×C414yes—
131#41#4114yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C22×C412yes—
138C2×Q8C2×D412yes—
139C2×Q8#4112yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4C22×C412yes—
146C2×D4C2×D412yes—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—82no—
157C22×C4C2×D412yes—
158C22×C4C2×D412yes—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162SD16—28no—
163D8—28no—
164M16—22no—
165M16—22no—
166M16—22no—
167M16C22×C414yes—
168C2×C8—22no—
169C2×C8—22no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—82no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C22⋊C4—24no—
182C22⋊C4—22no—
183C22⋊C4—28no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—82no—
189C42—24no—
190C42—24no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—21no—
197C42—22no—
198C42C22×C412yes—
199C42C2×D412yes—
200C42C2×D412yes—
201C23—22no—
202C23—22no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—41no—
207C23C22×D411yescharacteristic
208Q8—28no—
209Q8—24no—
210Q8—22no—
211Q8—24no—
212Q8C4×D414yes—
213D4—28no—
214D4—24no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—42no—
219D4C4×D414yes—
220C2×C4—216no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—82no—
242C2×C4—84no—
243C2×C4—84no—
244C2×C4C4×D414yes—
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4#12012yes—
248C2×C4C23×C411yescharacteristic
249C2×C4C22×D412yes—
250C2×C4C2×#4112yes—
251C2×C4C2×#4111yescharacteristic
252C8—24no—
253C8—42no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—82no—
262C22—81no—
263C22—81no—
264C22C2×C4×D411yescharacteristic
265C22C2×#12011yescharacteristic
266C22#52511yescharacteristic
267C4—21no—
268C4—21no—
269C4—22no—
270C4—24no—
271C4—24no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4C2×C4×D412yes—
279C4#52512yes—
280C2—21no—
281C2—21no—
282C2—44no—
283C2—82no—
284C2C4×#12011yescharacteristic
285C2#302011yescharacteristic
286C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#454C2
LCSG ▷ C2×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
1. C4×2−1+4C2semidirect
3. C4×AGL(1,ℤ/8)C2semidirect
4. #2326C2semidirect
5. #1888C2semidirect
6. #1883C2semidirect
7. #1859C2semidirect
8. #1857C2semidirect
9. #1793C2semidirect
10. #1759C2semidirect
11. C4×#101C2semidirect
26. #454C4semidirect
50. 2−1+4C2×C4semidirect
59. AGL(1,ℤ/8)C2×C4semidirect
65. #124C2×C4semidirect
111. #104C2×C4semidirect
117. #101C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+42
C2×#4542
C4×AGL(1,ℤ/8)2
#23262
#18882
#18832
#18592
#18572
#17932
#17592
C4×#1012