Finite Group Databasebeta

← #24500

Subgroups — #24500

Order 256.

635 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 141 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×#447C211yesmaximal, characteristic
2C2×#431C211yesmaximal, characteristic, complemented
3#2854C211yesmaximal, characteristic, complemented
4C2×#390C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#1970C212yesmaximal, complemented
8#1934C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1852C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C2×#123—21no—
15C2×#121C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4—21no—
18C22×C22⋊C4—21no—
19C22×C42C2211yescharacteristic
20#447—24no—
21#431—24no—
22C2×#107C2212yescomplemented
23C2×C4⋊C8C2212yescomplemented
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yescomplemented
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C4×C8C2212yescomplemented
31#390—21no—
32#390—21no—
33#390C2212yes—
34#390C2212yes—
35#390C2211yescharacteristic, complemented
36#390C2211yescharacteristic, complemented
37#389C2211yescharacteristic, complemented
38#389C2211yescharacteristic, complemented
39#388—21no—
40#388—21no—
41#386—21no—
42#380C2211yescharacteristic
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25—21no—
46C22×D4—24no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4C2311yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×D8—24no—
59C2×D8—24no—
60C2×D8—42no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63#123—44no—
64#121—22no—
65#121—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4—44no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42D411yescharacteristic
89C2×C42D411yescharacteristic
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92#107—42no—
93C4⋊C8—24no—
94#102—24no—
95#102—24no—
96#102—42no—
97#102—42no—
98C22⋊C8—22no—
99C22⋊C8D414yes—
100C4×C8—24no—
101#95—21no—
102#95—21no—
103#95—21no—
104C24—24no—
105C24—22no—
106C24—21no—
107C24—42no—
108C24—42no—
109C24—42no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24C2×D411yescharacteristic
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4#4114yes—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4D814yescomplemented
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4#4112yes—
154C22×C4C2411yescharacteristic
155D8—48no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8C2×D414yes—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—44no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—44no—
181C42—22no—
182C42—22no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—24no—
192C23—48no—
193C23—48no—
194C23—48no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—84no—
209C23—84no—
210C23—84no—
211C23—84no—
212C23#12112yes—
213C23C2×D812yes—
214C23#13512yes—
215C23C22×D411yescharacteristic
216D4—24no—
217D4—24no—
218D4—48no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—42no—
223D4—84no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—28no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4#12112yes—
253C2×C4C2×D812yes—
254C2×C4C2×D812yes—
255C2×C4AGL(1,ℤ/8)12yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C2×#4112yes—
258C2×C4C2×#4111yescharacteristic
259C2×C4H(5,𝔽2)12yes—
260C8—24no—
261C8—42no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—816no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22#46614yes—
284C22C2×#12111yescharacteristic
285C22#52511yescharacteristic
286C22#53511yescharacteristic
287C22C22×D811yescharacteristic
288C22C2×#13511yescharacteristic
289C22C2×AGL(1,ℤ/8)11yescharacteristic
290C22#57511yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—82no—
299C4—81no—
300C4#54612yes—
301C4#55112yes—
302C2—24no—
303C2—84no—
304C2—161no—
305C2#241111yescharacteristic
306C2C2×#46611yescharacteristic
307C2#305511yescharacteristic
308C2#310011yescharacteristic
309C2#310511yescharacteristic
310C2#320511yescharacteristic
311C2#321111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#431C2semidirect
3. #2854C2semidirect
4. C2×#390C2semidirect
5. #2067C2semidirect
6. #2054C2semidirect
7. #1970C2semidirect
8. #1934C2semidirect
9. #1874C2semidirect
10. #1852C2semidirect
11. #1786C2semidirect
22. C2×#107C22semidirect
23. C2×C4⋊C8C22semidirect
27. C2×#102C22semidirect
30. C2×C4×C8C22semidirect
35. #390C22semidirect
36. #390C22semidirect
37. #389C22semidirect
38. #389C22semidirect
43. #340C22semidirect
44. #340C22semidirect
146. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C2×#4312
#28542
C2×#3902
#20672
#20542
#19702
#19342
#18742
#18522
#17862