Finite Group Databasebeta

← #24538

Subgroups — #24538

Order 256.

578 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450C211yesmaximal, characteristic
2C2×#418C211yesmaximal, characteristic
3#2499C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic
5#2073C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1877C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C22⋊C4—21no—
25#450—24no—
26C2×C4⋊C8C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28#418—24no—
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#397—21no—
39#397—21no—
40#391C2211yescharacteristic, complemented
41#391C2211yescharacteristic, complemented
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#387—21no—
45#387—21no—
46#386—21no—
47#386—21no—
48#384C2211yescharacteristic, complemented
49#382C2211yescharacteristic, complemented
50#380C2211yescharacteristic
51#379—21no—
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—22no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4D414yes—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—41no—
93C4×D4—41no—
94#117—22no—
95#117—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
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—44no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C4⋊C8—41no—
120C4⋊C8—41no—
121#103—41no—
122#103—41no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129C22⋊C8—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—24no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×D4—28no—
156C2×D4—24no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4C2×D412yes—
171C2×D4C2×D412yes—
172C2×D4#4112yes—
173C2×D4#4112yes—
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—81no—
189C22×C4—81no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D412yes—
195C22×C4C2×D411yescharacteristic
196C22×C4#4112yes—
197C22×C4C2411yescharacteristic
198SD16—44no—
199SD16—44no—
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×C8—21no—
213C4⋊C4—21no—
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—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—22no—
227C22⋊C4—41no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—48no—
253C23—48no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—84no—
265C23—84no—
266C23—84no—
267C23—84no—
268C23—84no—
269C23#12112yes—
270C23C22×D411yescharacteristic
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275D4—22no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—48no—
281D4—41no—
282D4—41no—
283D4#12314yes—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
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—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4#12112yes—
322C2×C4AGL(1,ℤ/8)12yes—
323C2×C4C22×D411yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C8—41no—
330C8—41no—
331C8—41no—
332C8—41no—
333C22—22no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—816no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22C2×#12111yescharacteristic
356C22#54611yescharacteristic
357C22#54611yescharacteristic
358C22C2×AGL(1,ℤ/8)11yescharacteristic
359C22#57511yescharacteristic
360C22#57711yescharacteristic
361C22#57711yescharacteristic
362C4—22no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4C2×#12311yescharacteristic
374C4#53411yescharacteristic
375C4#55111yescharacteristic
376C4#55111yescharacteristic
377C2—42no—
378C2—44no—
379C2—82no—
380C2#241211yescharacteristic
381C2#303611yescharacteristic
382C2#304211yescharacteristic
383C2#331011yescharacteristic
384C2#331111yescharacteristic
385C2#332311yescharacteristic
386C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2499C2semidirect
5. #2073C2semidirect
6. #2071C2semidirect
7. #2055C2semidirect
8. #2055C2semidirect
9. #1894C2semidirect
10. #1877C2semidirect
11. #1877C2semidirect
12. #1851C2semidirect
13. #1825C2semidirect
14. #1804C2semidirect
15. #1802C2semidirect
26. C2×C4⋊C8C22semidirect
27. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
40. #391C22semidirect
41. #391C22semidirect
48. #384C22semidirect
49. #382C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×#4182
#24992
D4×C22⋊C42
#20732
#20712
#20552
#18942
#18772
#18512
#18252
#18042
#18022