Finite Group Databasebeta

← #18288

Subgroups — #18288

Order 256.

517 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 119 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×#539C211yesmaximal, characteristic, complemented
2#3187C211yesmaximal, characteristic, complemented
3#3186C211yesmaximal, characteristic
4#2618C211yesmaximal, characteristic, complemented
5#1609C214yesmaximal, complemented
6#1597C214yesmaximal, complemented
7#1520C211yesmaximal, characteristic, complemented
8#1519C211yesmaximal, characteristic, complemented
9C2×#324C211yesmaximal, characteristic, complemented
10#539—22no—
11#539—22no—
12#539—22no—
13#539C2214yes—
14C2×#126—21no—
15C2×#125—21no—
16C2×#123—21no—
17C2×#123—21no—
18C2×#123—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4—21no—
24C22×C4⋊C4C2211yescharacteristic
25#483C2212yescomplemented
26#481—21no—
27#480—21no—
28#480C2212yes—
29#466—22no—
30#465—22no—
31#450C2212yescomplemented
32#448—21no—
33#447—21no—
34#447C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#398C2212yescomplemented
38#396—22no—
39#394C2212yescomplemented
40#386—21no—
41#380C2211yescharacteristic
42#328C2214yescomplemented
43#327C2214yes—
44#324C2214yescomplemented
45C22×Q8D411yescharacteristic
46C22×D4—21no—
47C22×D4—21no—
48C22×D4D411yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C2×SD16—22no—
55C2×D8—22no—
56C22×C8—21no—
57C22×C8—21no—
58#126—24no—
59#126—24no—
60#125—24no—
61#125—24no—
62#123—24no—
63#123—24no—
64#123—24no—
65#123—24no—
66#123—44no—
67C22⋊Q8—24no—
68C22⋊Q8—24no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8D412yes—
72C22⋊Q8C2312yes—
73#121—21no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121—21no—
78#121—21no—
79#121D412yes—
80#121C2312yes—
81C4×D4—24no—
82C4×D4—24no—
83C2×C4⋊C4—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⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D412yes—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2312yes—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C42—21no—
108#107—22no—
109#106—22no—
110#103—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114C22⋊C8—21no—
115C22⋊C8—21no—
116C22⋊C8C2312yes—
117C22⋊C8C2312yes—
118#95—22no—
119#95C2312yes—
120C24—21no—
121C24—21no—
122C24C2×D411yescharacteristic
123C2×Q8—21no—
124C2×Q8—81no—
125C2×Q8C2×D412yes—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—81no—
138C2×D4—81no—
139C2×D4C2×D412yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—82no—
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2411yescharacteristic
163SD16—82no—
164D8—82no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—81no—
168C2×C8—81no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—44no—
183C4⋊C4—81no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—44no—
194C22⋊C4—42no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—82no—
198C42—24no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23#12011yescharacteristic
211C23#12011yescharacteristic
212C23#13512yes—
213C23AGL(1,ℤ/8)12yes—
214C23C22×D411yescharacteristic
215Q8—42no—
216Q8—81no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—44no—
223D4—81no—
224D4—82no—
225D4—82no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—84no—
254C2×C4#12011yescharacteristic
255C2×C4#12011yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C8—42no—
259C8—42no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—84no—
272C22—82no—
273C22—82no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22#44912yes—
278C22#44912yes—
279C22C2×#12011yescharacteristic
280C22C2×#13511yescharacteristic
281C22C2×AGL(1,ℤ/8)11yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—82no—
292C4—81no—
293C2—21no—
294C2—21no—
295C2—22no—
296C2—22no—
297C2—82no—
298C2—161no—
299C2C2×#44911yescharacteristic
300C2#302011yescharacteristic
301C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3186C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#539C2semidirect
2. #3187C2semidirect
4. #2618C2semidirect
5. #1609C2semidirect
6. #1597C2semidirect
7. #1520C2semidirect
8. #1519C2semidirect
9. C2×#324C2semidirect
25. #483C22semidirect
31. #450C22semidirect
37. #398C22semidirect
39. #394C22semidirect
42. #328C22semidirect
44. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#31872
#31862
#26182
#16092
#15972
#15202
#15192
C2×#3242