Finite Group Databasebeta

← #28135

Subgroups — #28135

Order 256.

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

normal subgroups only

This group has 448 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
1#3092C211yesmaximal, characteristic
2#3004C211yesmaximal, characteristic
3#2929C211yesmaximal, characteristic
4C22×#103C211yesmaximal, characteristic, complemented
5#2064C212yesmaximal, complemented
6#2052C212yesmaximal, complemented
7#1930C212yesmaximal, complemented
8#1908C212yesmaximal, complemented
9#1821C211yesmaximal, characteristic, complemented
10#1810C211yesmaximal, characteristic, complemented
11#1784C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C2×#137C2212yes—
14C2×#131C2211yescharacteristic
15C23×C8C2211yescharacteristic
16#525C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#484—21no—
19#483—21no—
20#481—21no—
21#480—21no—
22#451—21no—
23#451—21no—
24#448—21no—
25#448—21no—
26#435—21no—
27#435—21no—
28C8×D4—21no—
29C8×D4—21no—
30C2×#108C2212yes—
31#428C2212yes—
32#424C2211yescharacteristic
33#419C2212yes—
34#417C2212yes—
35C2×#103—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103C2212yes—
40C2×#103C2212yes—
41C2×#101C2212yes—
42C2×#101C2212yes—
43#408C2211yescharacteristic
44#407C2211yescharacteristic
45#407C2211yescharacteristic
46#407C2211yescharacteristic
47#405C2211yescharacteristic
48#343C2212yes—
49#343C2212yes—
50#341C2212yes—
51C2×#41C2311yescharacteristic
52C22×Q8—21no—
53C22×Q8—24no—
54C22×Q8—21no—
55C22×Q8—21no—
56C22×Q8—21no—
57C22×Q8—21no—
58C22×Q8—42no—
59C22×Q8C2312yes—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63#137—24no—
64#137—42no—
65C2×Q16—22no—
66C2×SD16—22no—
67#131—42no—
68C2×M16—21no—
69C2×M16C2311yescharacteristic
70C2×M16C2312yes—
71C2×M16C2312yes—
72C2×M16C2311yescharacteristic
73C22×C8—21no—
74C22×C8—42no—
75C22×C8D411yescharacteristic
76C22×C8D411yescharacteristic
77C22×C8D411yescharacteristic
78C22×C8D411yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#123—21no—
82C22⋊Q8D411yescharacteristic
83C22⋊Q8D411yescharacteristic
84#121D411yescharacteristic
85#121D411yescharacteristic
86#120—21no—
87C4×D4—21no—
88C4×D4—21no—
89#117C2311yescharacteristic
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2312yes—
96#108—42no—
97#107—22no—
98#106—22no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101#103—22no—
102#103—22no—
103#103—22no—
104#103—22no—
105#103—22no—
106#103—22no—
107#103—22no—
108#102—22no—
109#101—24no—
110#101—42no—
111C22⋊C8—21no—
112C22⋊C8—21no—
113C22⋊C8—21no—
114C22⋊C8—21no—
115C22⋊C8—21no—
116C22⋊C8—21no—
117C22⋊C8—21no—
118C22⋊C8—21no—
119#97—21no—
120C4×C8—21no—
121C24#4111yescharacteristic
122#41—22no—
123#41—22no—
124C2×Q8—22no—
125C2×Q8—24no—
126C2×Q8—24no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8C2×D411yescharacteristic
139C2×Q8#4114yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C2×D411yescharacteristic
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—44no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—82no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4#4111yescharacteristic
160C22×C4C2411yescharacteristic
161Q16—44no—
162SD16—44no—
163M16—22no—
164M16—22no—
165M16—22no—
166M16—22no—
167M16—22no—
168M16—22no—
169M16—42no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—21no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—21no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C42—21no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—21no—
203C23#12111yescharacteristic
204C23#12111yescharacteristic
205C23C2×#4111yescharacteristic
206Q8—24no—
207Q8—24no—
208Q8—22no—
209Q8—44no—
210Q8—44no—
211Q8—42no—
212Q8—42no—
213D4—22no—
214D4—42no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#12111yescharacteristic
239C2×C4#12111yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C2×C4C2×#4112yes—
243C2×C4H(5,𝔽2)12yes—
244C8—21no—
245C8—21no—
246C8—21no—
247C8—21no—
248C8—41no—
249C8—41no—
250C8—42no—
251C8—42no—
252C22—28no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22#47112yes—
263C22#47312yes—
264C22C2×#12111yescharacteristic
265C22#55112yes—
266C4—21no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#52511yescharacteristic
274C4#53411yescharacteristic
275C4#54612yes—
276C2—22no—
277C2—22no—
278C2—22no—
279C2—81no—
280C2#253411yescharacteristic
281C2C2×#47111yescharacteristic
282C2C2×#47311yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#103C2semidirect
5. #2064C2semidirect
6. #2052C2semidirect
7. #1930C2semidirect
8. #1908C2semidirect
9. #1821C2semidirect
10. #1810C2semidirect
11. #1784C2semidirect

Maximal subgroups

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

subgroupindex
#30922
#30042
#29292
C22×#1032
#20642
#20522
#19302
#19082
#18212
#18102
#17842