Finite Group Databasebeta

← #24879

Subgroups — #24879

Order 256.

532 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#483C211yesmaximal, characteristic, complemented
2C2×#417C211yesmaximal, characteristic, complemented
3#2407C211yesmaximal, characteristic
4D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#483—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29#417—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#394—21no—
42#392—21no—
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#386—21no—
47#385—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4D411yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22⋊Q8—22no—
85C22⋊Q8—41no—
86#121—22no—
87#121—41no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4D411yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118#107—41no—
119#106—22no—
120#106—22no—
121#106—41no—
122#103—22no—
123#103—22no—
124#103—41no—
125#103D414yescomplemented
126#102—22no—
127#102—22no—
128#102—22no—
129#102—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142#41—28no—
143#41—44no—
144#41—44no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D412yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—81no—
166C2×D4#4112yes—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4#4112yes—
198C22×C4#4112yes—
199C22×C4C2411yescharacteristic
200M16—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8C2×D412yes—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4C2×D412yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—42no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#12312yes—
263C23AGL(1,ℤ/8)12yes—
264C23C22×D411yescharacteristic
265Q8—22no—
266Q8—22no—
267Q8—41no—
268Q8—41no—
269D4—28no—
270D4—22no—
271D4—22no—
272D4—48no—
273D4—41no—
274D4—41no—
275D4—84no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—48no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4#12312yes—
319C2×C4#13712yes—
320C2×C4C22×D411yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22C2×#12311yescharacteristic
352C22D4211yescharacteristic
353C22#54611yescharacteristic
354C22C2×AGL(1,ℤ/8)11yescharacteristic
355C22C2×#13711yescharacteristic
356C22#57711yescharacteristic
357C22#57711yescharacteristic
358C4—22no—
359C4—22no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4#52511yescharacteristic
370C4#53811yescharacteristic
371C4#54611yescharacteristic
372C4#55111yescharacteristic
373C2—42no—
374C2—42no—
375C2—82no—
376C2#241811yescharacteristic
377C2#309211yescharacteristic
378C2#309911yescharacteristic
379C2#326511yescharacteristic
380C2#327811yescharacteristic
381C2#330611yescharacteristic
382C2#333111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
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×#483C2semidirect
2. C2×#417C2semidirect
4. D4×C4⋊C4C2semidirect
5. #2094C2semidirect
6. #2072C2semidirect
7. #2035C2semidirect
8. #2033C2semidirect
9. #1928C2semidirect
10. #1926C2semidirect
11. #1874C2semidirect
12. #1870C2semidirect
13. #1853C2semidirect
14. #1809C2semidirect
15. #1788C2semidirect
125. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4172
#24072
D4×C4⋊C42
#20942
#20722
#20352
#20332
#19282
#19262
#18742
#18702
#18532
#18092
#17882