Finite Group Databasebeta

← #23960

Subgroups — #23960

Order 256.

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

normal subgroups only

This group has 845 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, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3C22×#102C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic, complemented
5#2023C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1879C212yesmaximal, complemented
9#1874C212yesmaximal, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1823C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C23×D4C2211yescharacteristic
14C22×D8C2212yes—
15C23×C8C2211yescharacteristic
16#547—24no—
17#547—24no—
18C2×C4⋊Q8C2211yescharacteristic, complemented
19C2×#123—21no—
20C2×#123—21no—
21C2×C22⋊Q8—21no—
22C2×#121C2212yes—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic, complemented
25C2×#120—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic, complemented
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic, complemented
30#447—24no—
31C2×#102—24no—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yescomplemented
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yescomplemented
38#394—21no—
39#392—21no—
40#390C2212yes—
41#380C2211yescharacteristic
42#379—21no—
43#340C2212yescomplemented
44#340C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C2×#41—28no—
49C2×#41—44no—
50C2×#41—44no—
51C22×Q8—21no—
52C22×D4—24no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4—21no—
59C22×D4—42no—
60C22×D4D411yescharacteristic
61C22×D4C2311yescharacteristic
62C22×D4C2312yes—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C2×D8—24no—
69C2×D8—24no—
70C2×D8—42no—
71C22×C8—22no—
72C22×C8—42no—
73C22×C8C2312yes—
74C22×C8C2312yes—
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77#123—28no—
78#123—28no—
79C22⋊Q8—28no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121—24no—
84#121—42no—
85#121D414yescomplemented
86#120—44no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4D414yescomplemented
99C2×C4⋊C4C2312yes—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42C2311yescharacteristic
113#102—24no—
114#102—24no—
115#102—24no—
116#102—42no—
117C22⋊C8—22no—
118C22⋊C8—42no—
119#95—21no—
120C24—21no—
121C24—22no—
122C24—44no—
123C24—42no—
124C24—42no—
125C24—42no—
126C24—82no—
127C24C2×D411yescharacteristic
128C24C2×D411yescharacteristic
129#41—432no—
130C2×Q8—24no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—28no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—48no—
147C2×D4—44no—
148C2×D4—44no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—42no—
156C2×D4—82no—
157C2×D4D814yescomplemented
158C2×D4C2×D412yes—
159C2×D4#4114yescomplemented
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2×D411yescharacteristic
185C22×C4C2411yescharacteristic
186D8—48no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—44no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4C2×D412yes—
204C4⋊C4C2×D412yes—
205C4⋊C4C2×D412yes—
206C22⋊C4—24no—
207C22⋊C4—24no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—22no—
211C22⋊C4—44no—
212C22⋊C4—42no—
213C22⋊C4—44no—
214C22⋊C4—82no—
215C42—21no—
216C42—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—48no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—88no—
237C23—88no—
238C23—84no—
239C23—84no—
240C23#12012yes—
241C23C2×D812yes—
242C23C2×D812yes—
243C23AGL(1,ℤ/8)12yes—
244C23C22×D411yescharacteristic
245Q8—48no—
246D4—28no—
247D4—24no—
248D4—24no—
249D4—24no—
250D4—48no—
251D4—48no—
252D4—48no—
253D4—41no—
254D4—41no—
255D4—42no—
256D4—42no—
257D4—88no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—48no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4#12012yes—
287C2×C4C2×D812yes—
288C2×C4C22×D411yescharacteristic
289C2×C4C22×D411yescharacteristic
290C2×C4C2×#4112yes—
291C2×C4H(5,𝔽2)12yes—
292C8—42no—
293C8—42no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—44no—
308C22—44no—
309C22—816no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22#44714yes—
317C22C2×#12011yescharacteristic
318C22#54612yes—
319C22C22×D811yescharacteristic
320C22C2×AGL(1,ℤ/8)11yescharacteristic
321C22#57712yes—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—82no—
330C4#52511yescharacteristic
331C4#53511yescharacteristic
332C4#54612yes—
333C2—24no—
334C2—42no—
335C2—82no—
336C2—84no—
337C2#240411yescharacteristic
338C2C2×#44711yescharacteristic
339C2#301011yescharacteristic
340C2#329412yes—
341C2#330612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#547C2semidirect
2. C2×#447C2semidirect
3. C22×#102C2semidirect
4. #2408C2semidirect
5. #2023C2semidirect
6. #1999C2semidirect
7. #1893C2semidirect
8. #1879C2semidirect
9. #1874C2semidirect
10. #1869C2semidirect
11. #1823C2semidirect
18. C2×C4⋊Q8C22semidirect
24. C2×#121C22semidirect
27. C2×C4×D4C22semidirect
29. C22×C4⋊C4C22semidirect
35. C2×#102C22semidirect
37. C2×C22⋊C8C22semidirect
43. #340C22semidirect
44. #340C22semidirect
45. #340C22semidirect
46. #336C22semidirect
85. #121D4semidirect
98. C2×C4⋊C4D4semidirect
157. C2×D4D8semidirect
159. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4472
C22×#1022
#24082
#20232
#19992
#18932
#18792
#18742
#18692
#18232