Finite Group Databasebeta

← #19002

Subgroups — #19002

Order 256.

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

normal subgroups only

This group has 442 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, complemented
2#3234C211yesmaximal, characteristic, complemented
3#3213C211yesmaximal, characteristic
4#3142C211yesmaximal, characteristic, complemented
5#1664C214yesmaximal, complemented
6#1657C214yesmaximal, complemented
7#1534C211yesmaximal, characteristic, complemented
8#1528C211yesmaximal, characteristic, complemented
9#1527C211yesmaximal, characteristic, complemented
10#563—22no—
11#563—22no—
12#563—22no—
13#563C2214yes—
14#532C2211yescharacteristic
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#496C2212yes—
26#491C2212yescomplemented
27#490C2212yescomplemented
28#490C2212yescomplemented
29#489C2212yes—
30#488—22no—
31#481—21no—
32#480—21no—
33#478—22no—
34#463—22no—
35#448—21no—
36#447—21no—
37#432C2211yescharacteristic
38#432C2211yescharacteristic
39#417—21no—
40#417—21no—
41C4×M16C2211yescharacteristic
42#330C2214yescomplemented
43#330C2214yescomplemented
44#329C2214yes—
45C2×#41—21no—
46C22×Q8—21no—
47C22×Q8D411yescharacteristic
48C22×D4D411yescharacteristic
49C23×C4—21no—
50C2×SD16—22no—
51C2×D8—22no—
52C2×M16—21no—
53C4⋊Q8—24no—
54C4⋊Q8—24no—
55C4⋊Q8C2311yescharacteristic
56#127C2311yescharacteristic
57#126—24no—
58#126—24no—
59#126—24no—
60#126—24no—
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#124—21no—
72#124D412yes—
73#124C2312yes—
74#123—24no—
75#123—24no—
76C22⋊Q8—24no—
77C22⋊Q8—24no—
78C22⋊Q8—24no—
79C22⋊Q8—24no—
80#121—21no—
81#121—21no—
82C4×D4—21no—
83C4×D4—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#106—22no—
103C4⋊C8—21no—
104C4⋊C8—21no—
105#103—22no—
106#103—22no—
107#103—22no—
108#102—22no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113C22⋊C8—21no—
114C22⋊C8—21no—
115#97C2312yes—
116#97C2312yes—
117#97C2312yes—
118C4×C8—21no—
119C24—21no—
120#41—44no—
121C2×Q8—24no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—81no—
125C2×Q8—82no—
126C2×Q8C2×D412yes—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—41no—
132C2×D4—81no—
133C2×D4C2×D412yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—41no—
141C22×C4—82no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145SD16—82no—
146D8—82no—
147M16—42no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C42—24no—
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2411yescharacteristic
185C23—22no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23#12011yescharacteristic
191Q8—42no—
192Q8—42no—
193Q8—81no—
194Q8—84no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—42no—
199D4—81no—
200D4—82no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C8—42no—
224C8—42no—
225C8—42no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#45412yes—
236C22C2×#12011yescharacteristic
237C4—21no—
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—81no—
249C2—22no—
250C2—82no—
251C2—161no—
252C2C2×#45411yescharacteristic
253C2#301911yescharacteristic
254C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3213C2
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
1. C2×#563C2semidirect
2. #3234C2semidirect
4. #3142C2semidirect
5. #1664C2semidirect
6. #1657C2semidirect
7. #1534C2semidirect
8. #1528C2semidirect
9. #1527C2semidirect
24. #501C22semidirect
26. #491C22semidirect
27. #490C22semidirect
28. #490C22semidirect
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
#32342
#32132
#31422
#16642
#16572
#15342
#15282
#15272