Finite Group Databasebeta

← #25501

Subgroups — #25501

Order 256.

694 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 137 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×#532C211yesmaximal, characteristic
2C2×#468C212yesmaximal, complemented
3#2855C211yesmaximal, characteristic, complemented
4#2057C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#2014C211yesmaximal, characteristic, complemented
8#1895C212yesmaximal, complemented
9#1877C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1728C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×SD16C2212yes—
14C22×M16C2212yes—
15#532—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×D4—21no—
23C2×C4×D4C2212yes—
24C22×C42C2211yescharacteristic
25#468—28no—
26C2×#107C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31#391C2211yescharacteristic
32#391C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#387—21no—
36#387—21no—
37#385—21no—
38#381—21no—
39#380C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2212yes—
44#336C2211yescharacteristic
45C2×#41—28no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2312yes—
54C23×C4—22no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×SD16—24no—
59C2×SD16—24no—
60C2×SD16—42no—
61C2×M16—24no—
62C2×M16—24no—
63C2×M16—42no—
64C22×C8C2312yes—
65C22×C8C2312yes—
66C4⋊Q8—24no—
67#127—24no—
68#124—28no—
69C22⋊Q8—24no—
70#121—28no—
71#121—24no—
72#121D418yes—
73C4×D4—28no—
74C4×D4—24no—
75C4×D4—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—24no—
89C2×C42D411yescharacteristic
90C2×C42D411yescharacteristic
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93#107—24no—
94#103—24no—
95#102—24no—
96#102—24no—
97#102—42no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105C24—22no—
106C24C2×D411yescharacteristic
107#41—432no—
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—28no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4C2×D414yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—28no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D414yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154SD16—48no—
155M16—48no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8#4114yescomplemented
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4C2×D414yes—
172C22⋊C4—28no—
173C22⋊C4—28no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—24no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23#12012yes—
194C23C22×D411yescharacteristic
195Q8—48no—
196Q8—42no—
197Q8—42no—
198D4—24no—
199D4—24no—
200D4—48no—
201D4—48no—
202D4—416no—
203D4—42no—
204D4—42no—
205C2×C4—28no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—416no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4#12012yes—
237C2×C4AGL(1,ℤ/8)12yes—
238C2×C4AGL(1,ℤ/8)12yes—
239C2×C4#13712yes—
240C2×C4C22×D412yes—
241C2×C4C2×#4112yes—
242C2×C4H(5,𝔽2)11yescharacteristic
243C2×C42−1+411yescharacteristic
244C8—24no—
245C8—42no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—44no—
254C22—44no—
255C22—44no—
256C22—82no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22C2×#12011yescharacteristic
261C22#53211yescharacteristic
262C22#53711yescharacteristic
263C22C2×AGL(1,ℤ/8)11yescharacteristic
264C22C2×AGL(1,ℤ/8)11yescharacteristic
265C22C2×#13711yescharacteristic
266C22#57511yescharacteristic
267C4—22no—
268C4—22no—
269C4—22no—
270C4—22no—
271C4—44no—
272C4—41no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4#54612yes—
278C4#54712yes—
279C2—42no—
280C2—84no—
281C2#243011yescharacteristic
282C2#300311yescharacteristic
283C2#300711yescharacteristic
284C2#317111yescharacteristic
285C2#318211yescharacteristic
286C2#325111yescharacteristic
287C2#325411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
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×#468C2semidirect
3. #2855C2semidirect
4. #2057C2semidirect
5. #2041C2semidirect
6. #2024C2semidirect
7. #2014C2semidirect
8. #1895C2semidirect
9. #1877C2semidirect
10. #1862C2semidirect
11. #1728C2semidirect
161. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4682
#28552
#20572
#20412
#20242
#20142
#18952
#18772
#18622
#17282