Finite Group Databasebeta

← #18783

Subgroups — #18783

Order 256.

585 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 125 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×#547C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3101C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1647C214yesmaximal, complemented
6#1635C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#547—22no—
12#547—22no—
13#547—22no—
14#547C2214yes—
15#533C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#123—22no—
18C2×C22⋊Q8—21no—
19C2×#121—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4—21no—
23#500C2212yescomplemented
24#487C2212yescomplemented
25#484—22no—
26#483—21no—
27#478—22no—
28#477—22no—
29#476C2212yescomplemented
30#476C2212yescomplemented
31#474C2212yescomplemented
32#461C2212yescomplemented
33#450—21no—
34C4×SD16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#418—21no—
40#332C2214yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—48no—
46C22×Q8—21no—
47C22×D4—22no—
48C22×D4D411yescharacteristic
49C22×D4D411yescharacteristic
50C23×C4—22no—
51C23×C4—21no—
52C23×C4—21no—
53C2×SD16—22no—
54C2×M16—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8D412yes—
58C4⋊Q8C2311yescharacteristic
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2312yes—
61#125—21no—
62#123—28no—
63#123—28no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66C22⋊Q8—21no—
67C22⋊Q8—21no—
68#121—24no—
69#121—24no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4D412yes—
77C4×D4D412yes—
78C4×D4C2312yes—
79C4×D4C2312yes—
80#117—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D412yes—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#107—24no—
94#106—22no—
95#106—22no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—22no—
101#103—24no—
102#102—22no—
103#102—22no—
104#102—22no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C4×C8—21no—
108C4×C8C2312yes—
109C24—21no—
110C24—21no—
111#41—416no—
112#41—48no—
113#41—48no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—44no—
119C2×D4—24no—
120C2×D4—28no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—48no—
126C2×D4C2×D412yes—
127C2×D4C2×D412yes—
128C22×C4—28no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—44no—
137C22×C4—48no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143SD16—44no—
144M16—42no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C4⋊C4—28no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D414yes—
162C4⋊C4C2×D412yes—
163C22⋊C4—28no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—41no—
169C42—41no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2411yescharacteristic
174C23—24no—
175C23—22no—
176C23—22no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23#12011yescharacteristic
183Q8—44no—
184Q8—42no—
185Q8—44no—
186Q8—42no—
187D4—24no—
188D4—22no—
189D4—22no—
190D4—22no—
191D4—416no—
192D4—42no—
193D4AGL(1,ℤ/8)14yes—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—41no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—416no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4#12011yescharacteristic
211C2×C4#12011yescharacteristic
212C2×C4#12011yescharacteristic
213C2×C4#13512yes—
214C2×C4#13712yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C8—42no—
219C8—42no—
220C8—42no—
221C22—21no—
222C22—21no—
223C22—24no—
224C22—48no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—84no—
231C22#45212yes—
232C22C2×#12011yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—44no—
239C4—81no—
240C4—81no—
241C4#45212yes—
242C4C2×#13511yescharacteristic
243C4C2×AGL(1,ℤ/8)12yes—
244C4C2×#13711yescharacteristic
245C2—22no—
246C2—44no—
247C2—82no—
248C2C2×#45211yescharacteristic
249C2#300311yescharacteristic
250C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#547C2
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. #3230C2semidirect
3. #3101C2semidirect
4. #3077C2semidirect
5. #1647C2semidirect
6. #1635C2semidirect
7. #1549C2semidirect
8. #1486C2semidirect
9. #1478C2semidirect
23. #500C22semidirect
24. #487C22semidirect
29. #476C22semidirect
30. #476C22semidirect
31. #474C22semidirect
32. #461C22semidirect
40. #332C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
#32302
#31012
#30772
#16472
#16352
#15492
#14862
#14782