Finite Group Databasebeta

← #23946

Subgroups — #23946

Order 256.

611 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 131 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×#542C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic, complemented
3C2×#448C211yesmaximal, characteristic
4#2310C211yesmaximal, characteristic, complemented
5#2024C212yesmaximal, complemented
6#2015C212yesmaximal, complemented
7#1878C212yesmaximal, complemented
8#1877C212yesmaximal, complemented
9#1875C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic, complemented
11#1818C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15#542—24no—
16#542—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#126—21no—
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#451—24no—
30#448—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#391—21no—
38#391—21no—
39#391C2212yes—
40#391C2212yes—
41#381—21no—
42#381—21no—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—24no—
48C22×Q8—21no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2311yescharacteristic
54C22×Q8C2312yes—
55C22×D4—21no—
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C2311yescharacteristic
60C2×Q16—24no—
61C2×Q16—24no—
62C2×Q16—42no—
63C2×SD16—24no—
64C2×SD16—24no—
65C2×SD16—42no—
66C22×C8C2312yes—
67C22×C8C2312yes—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70#126—28no—
71#124—28no—
72#124—22no—
73#124—22no—
74#123—28no—
75C22⋊Q8—28no—
76C22⋊Q8—22no—
77C22⋊Q8D414yes—
78#121—22no—
79#121D414yes—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×D4—22no—
83C4×D4—22no—
84#117—22no—
85#117—22no—
86#117—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#103—24no—
108#103—24no—
109#103—42no—
110#102—42no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—48no—
130C2×Q8—48no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—82no—
134C2×Q8C2×D412yes—
135C2×Q8#4114yes—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—82no—
141C2×D4C2×D412yes—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—82no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2411yescharacteristic
162Q16—48no—
163SD16—48no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C42—24no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—42no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23#12012yes—
205C23C22×D411yescharacteristic
206Q8—24no—
207Q8—24no—
208Q8—24no—
209Q8—48no—
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—84no—
214D4—24no—
215D4—42no—
216D4—84no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—48no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4#12012yes—
245C2×C4C22×D411yescharacteristic
246C2×C4C22×D411yescharacteristic
247C2×C4C2×#4112yes—
248C2×C4H(5,𝔽2)12yes—
249C8—42no—
250C8—42no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—82no—
260C22—82no—
261C22C2×#12011yescharacteristic
262C22#52511yescharacteristic
263C22#53511yescharacteristic
264C22#57712yes—
265C22#57812yes—
266C4—22no—
267C4—44no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—42no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—82no—
276C4#54612yes—
277C4#54612yes—
278C2—42no—
279C2—82no—
280C2#240411yescharacteristic
281C2#300911yescharacteristic
282C2#301111yescharacteristic
283C2#312112yes—
284C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#451C2semidirect
4. #2310C2semidirect
5. #2024C2semidirect
6. #2015C2semidirect
7. #1878C2semidirect
8. #1877C2semidirect
10. #1873C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4512
C2×#4482
#23102
#20242
#20152
#18782
#18772
#18752
#18732
#18182