Finite Group Databasebeta

← #24088

Subgroups — #24088

Order 256.

677 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 677 conjugacy classes of subgroups — too many to draw. Only its 147 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×#547C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic, complemented
3C22×#107C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2057C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1869C211yesmaximal, characteristic, complemented
9#1862C212yesmaximal, complemented
10#1852C212yesmaximal, complemented
11#1823C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C23×C8C2211yescharacteristic
14#547—24no—
15#547—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#123—21no—
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#480—24no—
29C2×#107—24no—
30C2×#107C2212yes—
31C2×#107C2212yes—
32C2×#107C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37#398—21no—
38#397—21no—
39#389C2212yes—
40#381—21no—
41#380C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—28no—
48C2×#41—44no—
49C2×#41—44no—
50C22×Q8—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4D411yescharacteristic
54C22×D4C2311yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C22×C8—22no—
62C22×C8—24no—
63C22×C8C2312yes—
64C22×C8C2312yes—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67#123—28no—
68#123—28no—
69#123—44no—
70C22⋊Q8—28no—
71#121—22no—
72#121—22no—
73#121—22no—
74#121D414yes—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D414yes—
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42C2311yescharacteristic
104#107—24no—
105#107—24no—
106#107—24no—
107#102—24no—
108#102—24no—
109C22⋊C8—22no—
110C22⋊C8—24no—
111#95—22no—
112#95—21no—
113#95—22no—
114C24—21no—
115C24C2×D411yescharacteristic
116C24C2×D411yescharacteristic
117#41—432no—
118C2×Q8—24no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4D814yes—
133C2×D4C2×D412yes—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4C2411yescharacteristic
164C2×C8—28no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8#4114yescomplemented
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—44no—
194C42—21no—
195C42—21no—
196C23—22no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23—42no—
207C23#12012yes—
208C23C2×D812yes—
209C23C2×D812yes—
210C23AGL(1,ℤ/8)12yes—
211C23C22×D411yescharacteristic
212Q8—48no—
213D4—28no—
214D4—24no—
215D4—48no—
216D4—48no—
217D4—41no—
218D4—41no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—48no—
233C2×C4—48no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—48no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—84no—
251C2×C4#12012yes—
252C2×C4C2×D812yes—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4112yes—
256C2×C42−1+412yes—
257C8—24no—
258C8—42no—
259C22—24no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—22no—
265C22—42no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22#44714yes—
274C22C2×#12011yescharacteristic
275C22#54712yes—
276C22C22×D811yescharacteristic
277C22C2×AGL(1,ℤ/8)11yescharacteristic
278C22#57812yes—
279C4—22no—
280C4—22no—
281C4—41no—
282C4—42no—
283C4—42no—
284C4—42no—
285C4—41no—
286C4—82no—
287C4—82no—
288C4#52511yescharacteristic
289C4#53611yescharacteristic
290C4#54712yes—
291C2—24no—
292C2—42no—
293C2—82no—
294C2#240611yescharacteristic
295C2C2×#44711yescharacteristic
296C2#301111yescharacteristic
297C2#333412yes—
298C2#334312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#480C2semidirect
3. C22×#107C2semidirect
4. #2486C2semidirect
5. #2057C2semidirect
6. #2033C2semidirect
7. #1893C2semidirect
8. #1869C2semidirect
9. #1862C2semidirect
10. #1852C2semidirect
11. #1823C2semidirect
170. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4802
C22×#1072
#24862
#20572
#20332
#18932
#18692
#18622
#18522
#18232