Finite Group Databasebeta

← #19007

Subgroups — #19007

Order 256.

431 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#563C211yesmaximal, characteristic
2#3236C211yesmaximal, characteristic, complemented
3#3215C211yesmaximal, characteristic
4#3166C211yesmaximal, characteristic, complemented
5#1664C214yesmaximal, complemented
6#1660C214yesmaximal, complemented
7#1534C211yesmaximal, characteristic, complemented
8#1528C211yesmaximal, characteristic, complemented
9#1527C211yesmaximal, characteristic
10#563—22no—
11#563—22no—
12#563—22no—
13#563C2214yes—
14#533C2211yescharacteristic
15C2×C4⋊Q8—21no—
16C2×#126—21no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24#501C2212yescomplemented
25#501C2212yescomplemented
26#501C2212yescomplemented
27#499—22no—
28#497C2212yes—
29#491C2212yescomplemented
30#489C2212yes—
31#484—21no—
32#483—21no—
33#479—22no—
34#464—22no—
35#451—21no—
36#450—21no—
37#432C2211yescharacteristic
38#432C2211yescharacteristic
39#428—21no—
40#428—21no—
41C4×M16C2211yescharacteristic
42#330C2214yescomplemented
43#330C2214yescomplemented
44#329C2214yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×D4D411yescharacteristic
48C23×C4—21no—
49C2×Q16—22no—
50C2×SD16—22no—
51C2×M16—21no—
52C4⋊Q8—24no—
53C4⋊Q8—24no—
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56#126—24no—
57#126—24no—
58#126—24no—
59#126—24no—
60#125—21no—
61#125—21no—
62#125—21no—
63#125—21no—
64#125—21no—
65#125D412yes—
66#125D412yes—
67#125C2312yes—
68#125C2312yes—
69#124—21no—
70#124—21no—
71#124D412yes—
72#124C2312yes—
73#123—24no—
74#123—24no—
75C22⋊Q8—24no—
76C22⋊Q8—24no—
77C22⋊Q8—24no—
78C22⋊Q8—24no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C4×Q8—21no—
82C4×Q8—21no—
83#117—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#107—22no—
104#106—22no—
105#106—22no—
106#106—22no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109#103—22no—
110#103—22no—
111#103—22no—
112#102—22no—
113C22⋊C8—21no—
114C22⋊C8—21no—
115#97C2312yes—
116#97C2312yes—
117#97C2312yes—
118C4×C8—21no—
119C24—21no—
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—81no—
125C2×Q8—82no—
126C2×Q8C2×D412yes—
127C2×D4—21no—
128C2×D4—81no—
129C2×D4C2×D412yes—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—82no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140Q16—82no—
141SD16—82no—
142M16—24no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—41no—
178C42—24no—
179C42—41no—
180C42C2×D411yescharacteristic
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2411yescharacteristic
184C23—22no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23#12011yescharacteristic
189Q8—42no—
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—84no—
194D4—42no—
195D4—81no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—84no—
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C8—24no—
219C8—42no—
220C8—42no—
221C22—21no—
222C22—21no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—84no—
227C22#45612yes—
228C22C2×#12011yescharacteristic
229C4—21no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C2—22no—
243C2—82no—
244C2C2×#45611yescharacteristic
245C2#301911yescharacteristic
246C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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
2. #3236C2semidirect
4. #3166C2semidirect
5. #1664C2semidirect
6. #1660C2semidirect
7. #1534C2semidirect
8. #1528C2semidirect
24. #501C22semidirect
25. #501C22semidirect
26. #501C22semidirect
29. #491C22semidirect
42. #330C22semidirect
43. #330C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5632
#32362
#32152
#31662
#16642
#16602
#15342
#15282
#15272