Finite Group Databasebeta

← #25180

Subgroups — #25180

Order 256.

537 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#472C211yesmaximal, characteristic, complemented
2#3041C211yesmaximal, characteristic, complemented
3C2×#453C211yesmaximal, characteristic, complemented
4#3008C211yesmaximal, characteristic, complemented
5#2946C211yesmaximal, characteristic, complemented
6#2897C211yesmaximal, characteristic, complemented
7#2063C211yesmaximal, characteristic, complemented
8#2051C211yesmaximal, characteristic, complemented
9#2037C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1978C211yesmaximal, characteristic, complemented
12#1929C211yesmaximal, characteristic, complemented
13#1887C211yesmaximal, characteristic, complemented
14#1883C211yesmaximal, characteristic, complemented
15#1866C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C22×D8C2211yescharacteristic, complemented
22C2×#131C2211yescharacteristic
23#535C2211yescharacteristic
24C2×#127C2211yescharacteristic
25#520C2211yescharacteristic
26#518C2211yescharacteristic, complemented
27#472—24no—
28#463—21no—
29#463—21no—
30#459—21no—
31#459—21no—
32#453—24no—
33#449—21no—
34#449—21no—
35#447—21no—
36#447—21no—
37#442—21no—
38#440—21no—
39C4×SD16—21no—
40C4×D8—21no—
41C2×#108C2211yescharacteristic, complemented
42C2×#108C2211yescharacteristic, complemented
43#428C2211yescharacteristic, complemented
44#424C2211yescharacteristic, complemented
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47#418—21no—
48#418C2211yescharacteristic
49#418C2211yescharacteristic
50#418C2211yescharacteristic, complemented
51#417—21no—
52#417C2211yescharacteristic
53#416—21no—
54#416C2211yescharacteristic
55#416C2211yescharacteristic
56C2×#102—21no—
57C2×#102C2211yescharacteristic
58C2×#102C2211yescharacteristic, complemented
59C2×#100C2211yescharacteristic
60C2×#100C2211yescharacteristic, complemented
61C2×#100C2211yescharacteristic, complemented
62#408C2211yescharacteristic
63#405C2211yescharacteristic, complemented
64#390C2211yescharacteristic, complemented
65#343C2211yescharacteristic, complemented
66#339C2211yescharacteristic, complemented
67#339C2211yescharacteristic, complemented
68#336C2211yescharacteristic, complemented
69H(5,𝔽2)—22no—
70H(5,𝔽2)—22no—
71H(5,𝔽2)—22no—
72H(5,𝔽2)—41no—
73C2×#41—21no—
74C2×#41—21no—
75C2×#41C2311yescharacteristic
76C2×#41C2311yescharacteristic
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4C2311yescharacteristic
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84AGL(1,ℤ/8)—22no—
85AGL(1,ℤ/8)—22no—
86AGL(1,ℤ/8)—22no—
87AGL(1,ℤ/8)—22no—
88AGL(1,ℤ/8)—41no—
89AGL(1,ℤ/8)D414yescomplemented
90C2×SD16—21no—
91C2×SD16—21no—
92C2×SD16—21no—
93C2×D8—21no—
94C2×D8—22no—
95C2×D8—21no—
96C2×D8—21no—
97C2×D8—22no—
98C2×D8—21no—
99C2×D8—21no—
100C2×D8—21no—
101C2×D8—21no—
102C2×D8—41no—
103C2×D8—41no—
104#131—22no—
105#131—41no—
106C2×M16—21no—
107C2×M16C2311yescharacteristic
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C22×C8—21no—
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114#127—22no—
115#127—21no—
116#127—22no—
117#124—21no—
118#121—21no—
119#121—21no—
120#120—41no—
121C4×Q8—21no—
122C4×D4—21no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—41no—
128#117—21no—
129#117C2311yescharacteristic
130#117C2311yescharacteristic
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C42C2311yescharacteristic
136#108—22no—
137#108—41no—
138#107—21no—
139#106—21no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142#104—22no—
143#104—22no—
144#103—21no—
145#103D411yescharacteristic, complemented
146#103D411yescharacteristic, complemented
147#103D411yescharacteristic, complemented
148#103D411yescharacteristic, complemented
149#102—21no—
150#102—21no—
151#102—21no—
152#102—21no—
153#102—21no—
154#102—21no—
155#102—21no—
156#102—21no—
157#102—21no—
158#102—21no—
159#102—21no—
160#102D411yescharacteristic, complemented
161#102D411yescharacteristic, complemented
162#102D411yescharacteristic, complemented
163#102D411yescharacteristic, complemented
164#100—22no—
165#100—22no—
166#100—22no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169#97—21no—
170C4×C8—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—41no—
175#41—21no—
176#41—21no—
177#41—24no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—42no—
183#41—42no—
184#41—42no—
185#41C2×D412yes—
186#41#4112yes—
187C2×Q8C2×D411yescharacteristic
188C2×Q8C2×D411yescharacteristic
189C2×D4—21no—
190C2×D4—24no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C2×D412yes—
221C2×D4C2×D411yescharacteristic
222C2×D4C2×D411yescharacteristic
223C2×D4#4112yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4C2411yescharacteristic
233SD16—24no—
234SD16—42no—
235SD16—42no—
236D8—24no—
237D8—42no—
238D8—42no—
239D8—42no—
240D8—42no—
241D8—42no—
242D8—42no—
243M16—21no—
244M16—21no—
245M16—21no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—21no—
250M16—42no—
251M16C2×D412yes—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—42no—
261C2×C8C2×D411yescharacteristic
262C2×C8C2×D411yescharacteristic
263C2×C8C2×D411yescharacteristic
264C2×C8C2×D411yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4C2×D411yescharacteristic
269C4⋊C4C2×D411yescharacteristic
270C4⋊C4C2×D411yescharacteristic
271C4⋊C4C2×D411yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—82no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—41no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—44no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—161no—
304C23—161no—
305C23H(5,𝔽2)11yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8#12112yes—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—21no—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—22no—
319D4—44no—
320D4—44no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—88no—
332D4—161no—
333D4—161no—
334D4#12112yes—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—161no—
352C2×C4—161no—
353C2×C4—161no—
354C2×C4C22×D411yescharacteristic
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D411yescharacteristic
357C2×C4C2×#4111yescharacteristic
358C2×C4C2×#4111yescharacteristic
359C2×C4H(5,𝔽2)11yescharacteristic
360C8—22no—
361C8—21no—
362C8—21no—
363C8—21no—
364C8—21no—
365C8—41no—
366C8—41no—
367C8—41no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—161no—
389C22—161no—
390C22—161no—
391C22—164no—
392C22#53811yescharacteristic
393C22D4211yescharacteristic
394C22#54611yescharacteristic
395C4—22no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4C2×#12111yescharacteristic
404C4#52511yescharacteristic
405C4D4211yescharacteristic
406C4#54611yescharacteristic
407C2—21no—
408C2—21no—
409C2—42no—
410C2—81no—
411C2—82no—
412C2—82no—
413C2—161no—
414C2#242411yescharacteristic
415C2#328811yescharacteristic
416C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#472C2semidirect
2. #3041C2semidirect
3. C2×#453C2semidirect
4. #3008C2semidirect
5. #2946C2semidirect
6. #2897C2semidirect
7. #2063C2semidirect
8. #2051C2semidirect
9. #2037C2semidirect
10. #2011C2semidirect
11. #1978C2semidirect
12. #1929C2semidirect
13. #1887C2semidirect
14. #1883C2semidirect
15. #1866C2semidirect
21. C22×D8C22semidirect
26. #518C22semidirect
41. C2×#108C22semidirect
42. C2×#108C22semidirect
43. #428C22semidirect
44. #424C22semidirect
50. #418C22semidirect
58. C2×#102C22semidirect
60. C2×#100C22semidirect
61. C2×#100C22semidirect
63. #405C22semidirect
64. #390C22semidirect
65. #343C22semidirect
66. #339C22semidirect
67. #339C22semidirect
68. #336C22semidirect
89. AGL(1,ℤ/8)D4semidirect
145. #103D4semidirect
146. #103D4semidirect
147. #103D4semidirect
148. #103D4semidirect
160. #102D4semidirect
161. #102D4semidirect
162. #102D4semidirect
163. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4722
#30412
C2×#4532
#30082
#29462
#28972
#20632
#20512
#20372
#20112
#19782
#19292
#18872
#18832
#18662