Finite Group Databasebeta

← #24214

Subgroups — #24214

Order 256.

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

normal subgroups only

This group has 783 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×#532C211yesmaximal, characteristic, complemented
2C2×#463C212yesmaximal, complemented
3C2×#449C212yesmaximal, complemented
4C4×#121C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic, complemented
9#1983C212yesmaximal, complemented
10#1797C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×SD16C2212yes—
14C22×D8C2212yes—
15#532—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124C2212yes—
19C2×#121—21no—
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#463—28no—
28#449—28no—
29C2×C4⋊C8C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2212yes—
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#386—21no—
41#380C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4—21no—
44#336C2211yescharacteristic
45#336C2211yescharacteristic
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—24no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8C2312yes—
53C22×D4—21no—
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4C2311yescharacteristic
58C22×D4C2312yes—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4C2311yescharacteristic
64C2×SD16—24no—
65C2×SD16—24no—
66C2×SD16—42no—
67C2×D8—24no—
68C2×D8—24no—
69C2×D8—42no—
70C22×C8C2312yes—
71C22×C8C2312yes—
72C4⋊Q8—24no—
73#127—24no—
74#124—24no—
75#124D418yescomplemented
76#121—28no—
77#121—28no—
78#121—22no—
79#121D414yescomplemented
80C4×D4—28no—
81C4×D4—28no—
82C4×D4—24no—
83C4×D4—44no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—24no—
98C2×C42—21no—
99C2×C42D411yescharacteristic
100C2×C42D411yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C4⋊C8—24no—
104#103—24no—
105#103—24no—
106#103—42no—
107#102—24no—
108#102—24no—
109#102—42no—
110C22⋊C8—24no—
111#95—21no—
112#95—21no—
113#95—21no—
114C24—21no—
115C24—22no—
116C24C2×D411yescharacteristic
117#41—48no—
118#41—48no—
119#41—48no—
120#41—48no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8C2×D414yes—
127C2×D4—24no—
128C2×D4—24no—
129C2×D4—28no—
130C2×D4—24no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4C2×D414yes—
148C2×D4C2×D412yes—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—28no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4#4112yes—
174C22×C4C2411yescharacteristic
175SD16—48no—
176D8—48no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4C2×D412yes—
190C22⋊C4—28no—
191C22⋊C4—28no—
192C22⋊C4—24no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—42no—
201C42C2×D414yes—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—22no—
210C23—41no—
211C23—42no—
212C23—42no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—41no—
217C23—42no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23#12112yes—
223C23C22×D411yescharacteristic
224Q8—24no—
225Q8—24no—
226Q8—42no—
227Q8—42no—
228D4—24no—
229D4—24no—
230D4—24no—
231D4—48no—
232D4—48no—
233D4—48no—
234D4—416no—
235D4—42no—
236D4—42no—
237D4—42no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—416no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—84no—
266C2×C4#12012yes—
267C2×C4#12012yes—
268C2×C4#12112yes—
269C2×C4#13512yes—
270C2×C4AGL(1,ℤ/8)12yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D412yes—
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)12yes—
275C8—42no—
276C8—42no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—44no—
285C22—42no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—82no—
291C22—81no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—82no—
297C22—82no—
298C22C2×#12011yescharacteristic
299C22C2×#12111yescharacteristic
300C22#53511yescharacteristic
301C22C2×#13511yescharacteristic
302C22C2×AGL(1,ℤ/8)11yescharacteristic
303C22#57511yescharacteristic
304C22#57511yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—44no—
311C4—42no—
312C4—81no—
313C4—81no—
314C4D4212yes—
315C4#54612yes—
316C2—42no—
317C2—84no—
318C2—82no—
319C2#240811yescharacteristic
320C2#300211yescharacteristic
321C2#300611yescharacteristic
322C2#303511yescharacteristic
323C2#303611yescharacteristic
324C2#320711yescharacteristic
325C2#321911yescharacteristic
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×#463C2semidirect
3. C2×#449C2semidirect
4. C4×#121C2semidirect
5. #2043C2semidirect
6. #2041C2semidirect
7. #2013C2semidirect
8. #2011C2semidirect
9. #1983C2semidirect
10. #1797C2semidirect
11. #1788C2semidirect
75. #124D4semidirect
79. #121D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4632
C2×#4492
C4×#1212
#20432
#20412
#20132
#20112
#19832
#17972
#17882