Finite Group Databasebeta

← #21476

Subgroups — #21476

Order 256.

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

normal subgroups only

This group has 661 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
2C2×#456C211yesmaximal, characteristic
3C4×#137C212yesmaximal, complemented
4#2327C211yesmaximal, characteristic, complemented
5#1886C211yesmaximal, characteristic, complemented
6#1882C212yesmaximal, complemented
7#1860C212yesmaximal, complemented
8#1857C211yesmaximal, characteristic, complemented
9#1793C211yesmaximal, characteristic
10#1759C212yesmaximal, complemented
11C4×#101C211yesmaximal, characteristic, complemented
12C2×2−1+4C2211yescharacteristic
13C2×#137C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
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×Q8C2211yescharacteristic
26#456C418yescomplemented
27#441—22no—
28#440—22no—
29C4×Q16—22no—
30C4×SD16—22no—
31#428C2212yes—
32C2×#104C2212yes—
33C2×#104C2212yes—
34#419C2212yes—
35#417C2212yes—
36C2×#101C2211yescharacteristic
37C2×#101C2211yescharacteristic
38C4×M16C2212yes—
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#387—22no—
42#387—22no—
43#382C2211yescharacteristic
44#376C2211yescharacteristic
45#343C2212yes—
46#339C2212yes—
47#339C2212yes—
482−1+4—24no—
492−1+4—22no—
502−1+4C2×C414yes—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41C2312yes—
54C22×Q8—22no—
55C22×Q8C2311yescharacteristic
56C22×Q8C2311yescharacteristic
57#137—24no—
58#137C2×C418yescomplemented
59C2×Q16—22no—
60C2×SD16—22no—
61C2×M16C2312yes—
62C2×M16C2312yes—
63C4⋊Q8—22no—
64C4⋊Q8C2×C414yescomplemented
65C4×Q8—22no—
66C4×Q8—24no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×Q8D412yes—
76C4×Q8D412yes—
77C4×D4—22no—
78C4×D4—24no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—24no—
83C4×D4—22no—
84C4×D4D412yes—
85C4×D4D412yes—
86#117—24no—
87#117—24no—
88#117—22no—
89#117—22no—
90#117—22no—
91#117C2312yes—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D412yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42D412yes—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—22no—
107#106—22no—
108#104—24no—
109#104C2×C418yescomplemented
110#103—22no—
111#103—22no—
112#103—22no—
113#102—22no—
114#101—22no—
115#101C2×C414yescomplemented
116#97—22no—
117C4×C8—22no—
118#95—21no—
119#95—21no—
120#95—22no—
121#95—22no—
122#95—22no—
123#41—28no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—22no—
128#41—22no—
129#41C22×C414yes—
130#41#4114yes—
131C2×Q8—22no—
132C2×Q8—28no—
133C2×Q8—28no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—21no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8C22×C412yes—
140C2×Q8C22×C412yes—
141C2×Q8C2×D412yes—
142C2×Q8#4112yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C2×D412yes—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160Q16—28no—
161SD16—28no—
162M16—22no—
163M16—22no—
164M16—22no—
165M16C22×C414yes—
166C2×C8—22no—
167C2×C8—22no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—28no—
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C22⋊C4—24no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C42—24no—
186C42—24no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—21no—
193C42—22no—
194C42—42no—
195C42C22×C412yes—
196C42C2×D412yes—
197C42C2×D412yes—
198C23—22no—
199C23C22×D411yescharacteristic
200Q8—28no—
201Q8—24no—
202Q8—22no—
203Q8—24no—
204Q8—24no—
205Q8—42no—
206Q8C4×D414yes—
207D4—28no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4C4×D414yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4C4×D414yes—
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4#12012yes—
242C2×C4C23×C411yescharacteristic
243C2×C4C22×D412yes—
244C2×C4C2×#4112yes—
245C2×C4C2×#4111yescharacteristic
246C8—24no—
247C8—42no—
248C22—21no—
249C22—21no—
250C22—24no—
251C22—42no—
252C22—42no—
253C22C2×C4×D411yescharacteristic
254C22C2×#12011yescharacteristic
255C22#52511yescharacteristic
256C4—21no—
257C4—21no—
258C4—22no—
259C4—24no—
260C4—24no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—81no—
267C4—81no—
268C4C2×C4×D412yes—
269C4#52512yes—
270C2—21no—
271C2—21no—
272C2—44no—
273C2C4×#12011yescharacteristic
274C2#302011yescharacteristic
275C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×2−1+4C22
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
3. C4×#137C2semidirect
4. #2327C2semidirect
5. #1886C2semidirect
6. #1882C2semidirect
7. #1860C2semidirect
8. #1857C2semidirect
10. #1759C2semidirect
11. C4×#101C2semidirect
26. #456C4semidirect
58. #137C2×C4semidirect
64. C4⋊Q8C2×C4semidirect
109. #104C2×C4semidirect
115. #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×#4562
C4×#1372
#23272
#18862
#18822
#18602
#18572
#17932
#17592
C4×#1012