Finite Group Databasebeta

← #25502

Subgroups — #25502

Order 256.

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

normal subgroups only

This group has 725 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, complemented
2C2×#469C212yesmaximal, complemented
3#2854C211yesmaximal, characteristic, complemented
4#2058C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2011C211yesmaximal, characteristic, complemented
8#1895C212yesmaximal, complemented
9#1879C212yesmaximal, complemented
10#1864C212yesmaximal, complemented
11#1728C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×D8C2212yes—
14C22×M16C2212yes—
15#532—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124—21no—
19C2×#123—21no—
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2212yes—
23C2×C4×D4—21no—
24C2×C4×D4C2212yes—
25C22×C42C2211yescharacteristic
26#469—28no—
27C2×#106C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#388—21no—
38#386—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×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2311yescharacteristic
54C22×D4C2312yes—
55C23×C4—22no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4C2311yescharacteristic
59C2×D8—24no—
60C2×D8—24no—
61C2×D8—42no—
62C2×M16—24no—
63C2×M16—24no—
64C2×M16—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67C4⋊Q8—24no—
68#127—24no—
69#124—28no—
70#123—44no—
71#121—28no—
72#121—24no—
73#121—24no—
74#121D418yes—
75C4×D4—28no—
76C4×D4—24no—
77C4×D4—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—24no—
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#106—24no—
99#102—24no—
100#102—24no—
101#102—24no—
102#102—42no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—22no—
107C24—21no—
108C24C2×D411yescharacteristic
109#41—432no—
110C2×Q8—24no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—28no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4C2×D414yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—28no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D414yes—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2411yescharacteristic
158D8—48no—
159M16—48no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8#4114yescomplemented
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4C2×D414yes—
175C22⋊C4—28no—
176C22⋊C4—28no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—22no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—24no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23#12012yes—
208C23C22×D411yescharacteristic
209Q8—48no—
210D4—24no—
211D4—24no—
212D4—48no—
213D4—48no—
214D4—416no—
215D4—42no—
216D4—42no—
217D4—42no—
218D4—42no—
219D4—84no—
220C2×C4—28no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—44no—
236C2×C4—416no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4#12012yes—
249C2×C4AGL(1,ℤ/8)12yes—
250C2×C4AGL(1,ℤ/8)12yes—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4C22×D412yes—
253C2×C4C2×#4112yes—
254C2×C4H(5,𝔽2)11yescharacteristic
255C2×C42−1+411yescharacteristic
256C8—24no—
257C8—42no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—44no—
266C22—44no—
267C22—44no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22C2×#12011yescharacteristic
280C22#53211yescharacteristic
281C22#53711yescharacteristic
282C22C2×AGL(1,ℤ/8)11yescharacteristic
283C22C2×AGL(1,ℤ/8)11yescharacteristic
284C22C2×AGL(1,ℤ/8)11yescharacteristic
285C22#57511yescharacteristic
286C4—22no—
287C4—22no—
288C4—22no—
289C4—22no—
290C4—44no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4#54612yes—
296C4#54712yes—
297C2—42no—
298C2—84no—
299C2—161no—
300C2#243011yescharacteristic
301C2#300311yescharacteristic
302C2#300611yescharacteristic
303C2#317111yescharacteristic
304C2#318411yescharacteristic
305C2#325111yescharacteristic
306C2#325311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#532C2semidirect
2. C2×#469C2semidirect
3. #2854C2semidirect
4. #2058C2semidirect
5. #2041C2semidirect
6. #2023C2semidirect
7. #2011C2semidirect
8. #1895C2semidirect
9. #1879C2semidirect
10. #1864C2semidirect
11. #1728C2semidirect
165. 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×#4692
#28542
#20582
#20412
#20232
#20112
#18952
#18792
#18642
#17282