Finite Group Databasebeta

← #24013

Subgroups — #24013

Order 256.

734 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 135 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×#535C211yesmaximal, characteristic
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#480C211yesmaximal, characteristic, complemented
4#2311C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1875C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1786C212yesmaximal, complemented
11#1780C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×D4C2211yescharacteristic
14#535—24no—
15#535—24no—
16C2×#127—21no—
17C2×#124—21no—
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#120—21no—
23C2×#120—21no—
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic, complemented
26C22×C4⋊C4C2211yescharacteristic, complemented
27#481—24no—
28#480—24no—
29C2×#107C2212yescomplemented
30C2×#106C2212yescomplemented
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2211yescharacteristic, complemented
37C2×C22⋊C8C2211yescharacteristic, complemented
38C2×C22⋊C8C2212yescomplemented
39#391—21no—
40#390—21no—
41#386—21no—
42#386—21no—
43#380C2212yes—
44#380C2212yescomplemented
45#340C2212yescomplemented
46#340C2212yescomplemented
47C25—21no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—24no—
51C2×#41—42no—
52C22×Q8D411yescharacteristic
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—21no—
60C22×D4—24no—
61C22×D4—42no—
62C22×D4D411yescharacteristic
63C22×D4D411yescharacteristic
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C22×C8C2312yes—
72C22×C8C2312yes—
73#127—28no—
74#124—28no—
75#121—22no—
76#121—22no—
77#121—22no—
78#121—22no—
79#121—22no—
80#121—22no—
81#121D414yes—
82#121D414yes—
83#120—28no—
84#120—28no—
85#117—22no—
86#117—22no—
87#117—42no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C42—21no—
104C2×C42—21no—
105#107—24no—
106#106—24no—
107#102—24no—
108#102—24no—
109#102—42no—
110#102—42no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115#95—22no—
116#95—22no—
117C24—24no—
118C24—21no—
119C24—21no—
120C24—42no—
121C24—42no—
122C24—82no—
123C24—82no—
124C24C2×D411yescharacteristic
125C24C2×D411yescharacteristic
126C24C2×D411yescharacteristic
127#41—48no—
128#41—48no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—41no—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—48no—
150C2×D4—48no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—82no—
157C2×D4—82no—
158C2×D4—82no—
159C2×D4—82no—
160C2×D4C2×D412yes—
161C2×D4C2×D412yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—82no—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D412yes—
180C22×C4C2×D412yes—
181C22×C4C2×D411yescharacteristic
182C22×C4#4114yes—
183C22×C4C2411yescharacteristic
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4C2×D412yes—
198C4⋊C4C2×D412yes—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C42—24no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—48no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—88no—
234C23—84no—
235C23—84no—
236C23—88no—
237C23#12012yes—
238C23#12312yes—
239C23C22×D411yescharacteristic
240Q8—44no—
241D4—24no—
242D4—24no—
243D4—44no—
244D4—44no—
245D4—48no—
246D4—44no—
247D4—42no—
248D4—42no—
249D4—84no—
250D4—84no—
251D4—84no—
252D4—84no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—84no—
277C2×C4#12012yes—
278C2×C4#12312yes—
279C2×C4C22×D411yescharacteristic
280C2×C4C22×D411yescharacteristic
281C2×C4C2×#4112yes—
282C2×C4H(5,𝔽2)12yes—
283C8—42no—
284C8—42no—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—816no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22C2×#12011yescharacteristic
312C22C2×#12311yescharacteristic
313C22#53411yescharacteristic
314C22#57612yes—
315C22#57712yes—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—42no—
320C4—42no—
321C4—82no—
322C4—82no—
323C4#54612yes—
324C4#54612yes—
325C2—42no—
326C2—42no—
327C2—81no—
328C2—82no—
329C2—82no—
330C2#240511yescharacteristic
331C2#300811yescharacteristic
332C2#300911yescharacteristic
333C2#309612yes—
334C2#319812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#481C2semidirect
3. C2×#480C2semidirect
4. #2311C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1875C2semidirect
8. #1875C2semidirect
9. #1854C2semidirect
10. #1786C2semidirect
11. #1780C2semidirect
25. C2×#117C22semidirect
26. C22×C4⋊C4C22semidirect
29. C2×#107C22semidirect
30. C2×#106C22semidirect
36. C2×C22⋊C8C22semidirect
37. C2×C22⋊C8C22semidirect
38. C2×C22⋊C8C22semidirect
44. #380C22semidirect
45. #340C22semidirect
46. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5352
C2×#4812
C2×#4802
#23112
#20342
#20332
#18752
#18542
#17862
#17802