Finite Group Databasebeta

← #25644

Subgroups — #25644

Order 256.

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

normal subgroups only

This group has 606 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, complemented
2C2×#496C211yesmaximal, characteristic
3C2×#460C211yesmaximal, characteristic, complemented
4#2548C211yesmaximal, characteristic, complemented
5#2013C212yesmaximal, complemented
6#2012C212yesmaximal, complemented
7#1985C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×SD16C2212yes—
14C22×D8C2212yes—
15#556—24no—
16#556—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#126—21no—
20C2×#126—21no—
21C2×#124—21no—
22C2×#124C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30#496—24no—
31#460—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×#97C2211yescharacteristic
38#394—21no—
39#394—21no—
40#390—21no—
41#390C2212yes—
42#390C2212yes—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8D411yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C2×SD16—24no—
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—42no—
59C2×SD16—42no—
60C2×SD16—42no—
61C2×D8—24no—
62C2×D8—24no—
63C2×D8—42no—
64C22×C8C2312yes—
65C22×C8C2312yes—
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68#127—22no—
69#127—22no—
70#126—28no—
71#126—28no—
72#124—22no—
73#124—22no—
74#124—22no—
75#124—44no—
76#124D414yes—
77C22⋊Q8—28no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8D414yes—
84C4×D4—22no—
85C4×D4—22no—
86#117—28no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C4⋊C8—22no—
112#103—24no—
113#103—42no—
114#102—24no—
115#102—42no—
116#97—22no—
117#95—21no—
118C24—21no—
119C24—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—41no—
129C2×Q8C2×D412yes—
130C2×Q8C2×D412yes—
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—82no—
145C2×D4C2×D412yes—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160SD16—48no—
161SD16—48no—
162D8—48no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8#4114yes—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4C2×D412yes—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—82no—
188C22⋊C4—82no—
189C42—24no—
190C42—24no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—42no—
196C42C2×D412yes—
197C42C2×D412yes—
198C23—22no—
199C23—42no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23C22×D411yescharacteristic
211Q8—22no—
212Q8—22no—
213Q8—24no—
214Q8—44no—
215Q8—41no—
216Q8—41no—
217Q8—42no—
218D4—24no—
219D4—41no—
220D4—42no—
221D4—41no—
222D4—42no—
223D4—42no—
224D4—42no—
225D4—84no—
226D4—84no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#12012yes—
251C2×C4#12012yes—
252C2×C4AGL(1,ℤ/8)12yes—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4112yes—
256C2×C4H(5,𝔽2)12yes—
257C8—24no—
258C8—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22C2×#12011yescharacteristic
272C22#55012yes—
273C22C2×AGL(1,ℤ/8)11yescharacteristic
274C22#57511yescharacteristic
275C22#57712yes—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—41no—
282C4—42no—
283C4—41no—
284C4—41no—
285C4—82no—
286C4#53211yescharacteristic
287C4#53511yescharacteristic
288C4#54612yes—
289C2—82no—
290C2—162no—
291C2#243111yescharacteristic
292C2#300611yescharacteristic
293C2#301011yescharacteristic
294C2#339712yes—
295C2#340112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#496C2
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
1. C2×#556C2semidirect
3. C2×#460C2semidirect
4. #2548C2semidirect
5. #2013C2semidirect
6. #2012C2semidirect
7. #1985C2semidirect
8. #1983C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
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×#4962
C2×#4602
#25482
#20132
#20122
#19852
#19832
#18802
#18792
#17632