Finite Group Databasebeta

← #25643

Subgroups — #25643

Order 256.

554 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 133 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×#556C211yesmaximal, characteristic
2C2×#497C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4#2568C211yesmaximal, characteristic, complemented
5#2016C212yesmaximal, complemented
6#2015C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal
11#1763C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15#556—24no—
16#556—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#126—21no—
20C2×#126—21no—
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117—21no—
29#497—24no—
30#462—24no—
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2212yes—
36C2×#97C2211yescharacteristic
37#400—21no—
38#400—21no—
39#391—21no—
40#391C2212yes—
41#391C2212yes—
42#382—21no—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8D411yescharacteristic
48C22×Q8C2312yes—
49C22×Q8C2311yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C2×Q16—24no—
54C2×Q16—24no—
55C2×Q16—24no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×Q16—42no—
59C2×SD16—24no—
60C2×SD16—24no—
61C2×SD16—42no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68#126—28no—
69#126—28no—
70#124—22no—
71#124—22no—
72#124—22no—
73#124D414yes—
74C22⋊Q8—28no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8D414yes—
81C4×D4—22no—
82C4×D4—22no—
83#117—28no—
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⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C4⋊C8—22no—
107#103—24no—
108#103—24no—
109#103—42no—
110#102—42no—
111#97—22no—
112#95—21no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117C24—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—21no—
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—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—41no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4C2×D412yes—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154Q16—48no—
155Q16—48no—
156SD16—48no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8#4114yes—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4C2×D412yes—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C42—24no—
182C42—24no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—42no—
188C42C2×D412yes—
189C42C2×D412yes—
190C23—22no—
191C23—42no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23C22×D411yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—24no—
199Q8—44no—
200Q8—41no—
201Q8—41no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205Q8—42no—
206Q8—42no—
207D4—24no—
208D4—41no—
209D4—41no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4#12012yes—
237C2×C4#12012yes—
238C2×C4#13712yes—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C2×#4112yes—
242C2×C4H(5,𝔽2)12yes—
243C8—24no—
244C8—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—82no—
249C22—82no—
250C22C2×#12011yescharacteristic
251C22#55012yes—
252C22C2×#13711yescharacteristic
253C22#57511yescharacteristic
254C22#57812yes—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4—82no—
266C4#53211yescharacteristic
267C4#53511yescharacteristic
268C4#54612yes—
269C2—82no—
270C2#243111yescharacteristic
271C2#300711yescharacteristic
272C2#301111yescharacteristic
273C2#339812yes—
274C2#340212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#462C2semidirect
4. #2568C2semidirect
5. #2016C2semidirect
7. #1985C2semidirect
9. #1878C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4972
C2×#4622
#25682
#20162
#20152
#19852
#19832
#18782
#18772
#17632