Finite Group Databasebeta

← #18998

Subgroups — #18998

Order 256.

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

normal subgroups only

This group has 463 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×#562C211yesmaximal, characteristic
2#3236C211yesmaximal, characteristic, complemented
3#3215C211yesmaximal, characteristic, complemented
4#3140C211yesmaximal, characteristic, complemented
5#1662C214yesmaximal, complemented
6#1661C214yesmaximal, complemented
7#1533C211yesmaximal, characteristic, complemented
8#1528C211yesmaximal, characteristic, complemented
9#1527C211yesmaximal, characteristic, complemented
10#562—22no—
11#562—22no—
12#562—22no—
13#562C2214yes—
14#533C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8—21no—
22C2×#121—21no—
23C2×#121—21no—
24#501C2212yescomplemented
25#497C2212yescomplemented
26#491C2212yescomplemented
27#489C2212yescomplemented
28#489C2212yescomplemented
29#489C2212yescomplemented
30#488—22no—
31#484—21no—
32#483—21no—
33#479—22no—
34#464—22no—
35#451—21no—
36#450—21no—
37#432C2211yescharacteristic
38#432C2211yescharacteristic
39#419—21no—
40#418—21no—
41C4×M16C2211yescharacteristic
42#330C2214yescomplemented
43#329C2214yescomplemented
44#329C2214yescomplemented
45C22×Q8D411yescharacteristic
46C22×Q8D411yescharacteristic
47C22×D4—21no—
48C22×D4—21no—
49C22×D4D411yescharacteristic
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C2×Q16—22no—
54C2×SD16—22no—
55C2×M16—21no—
56C4⋊Q8C2311yescharacteristic
57C4⋊Q8C2311yescharacteristic
58#125—21no—
59#125—21no—
60#125—21no—
61#125D412yes—
62#125C2312yes—
63#124—21no—
64#124—21no—
65#124—21no—
66#124—21no—
67#124D412yes—
68#124D412yes—
69#124C2312yes—
70#124C2312yes—
71#123—24no—
72#123—24no—
73#123—24no—
74#123—24no—
75C22⋊Q8—24no—
76C22⋊Q8—24no—
77C22⋊Q8—24no—
78C22⋊Q8—24no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81#121—24no—
82#121—24no—
83#121—24no—
84#121—24no—
85C4×Q8—21no—
86C4×Q8—21no—
87#117—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
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×C22⋊C4—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#103—22no—
109#103—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—
120C24—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—42no—
126C2×Q8—81no—
127C2×Q8C2×D412yes—
128C2×Q8C2×D412yes—
129C2×D4—24no—
130C2×D4—24no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—81no—
135C2×D4—82no—
136C2×D4—82no—
137C2×D4C2×D412yes—
138C2×D4C2×D412yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150Q16—82no—
151SD16—82no—
152M16—42no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—41no—
183C42—41no—
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2411yescharacteristic
188C23—22no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23#12011yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—81no—
202Q8—81no—
203D4—42no—
204D4—42no—
205D4—81no—
206D4—81no—
207D4—84no—
208D4—84no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4#12011yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C8—42no—
231C8—42no—
232C8—42no—
233C22—21no—
234C22—21no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—84no—
243C22#45512yes—
244C22C2×#12011yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—22no—
258C2—82no—
259C2—82no—
260C2C2×#45511yescharacteristic
261C2#301911yescharacteristic
262C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#562C2
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
3. #3215C2semidirect
4. #3140C2semidirect
5. #1662C2semidirect
6. #1661C2semidirect
7. #1533C2semidirect
8. #1528C2semidirect
9. #1527C2semidirect
24. #501C22semidirect
25. #497C22semidirect
26. #491C22semidirect
27. #489C22semidirect
28. #489C22semidirect
29. #489C22semidirect
42. #330C22semidirect
43. #329C22semidirect
44. #329C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5622
#32362
#32152
#31402
#16622
#16612
#15332
#15282
#15272