Finite Group Databasebeta

← #24925

Subgroups — #24925

Order 256.

452 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3098C211yesmaximal, characteristic, complemented
2#3012C211yesmaximal, characteristic
3#2951C211yesmaximal, characteristic, complemented
4#2897C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic, complemented
6#2078C211yesmaximal, characteristic, complemented
7#2051C211yesmaximal, characteristic, complemented
8#2039C211yesmaximal, characteristic, complemented
9#1931C211yesmaximal, characteristic, complemented
10#1929C211yesmaximal, characteristic, complemented
11#1883C211yesmaximal, characteristic, complemented
12#1876C211yesmaximal, characteristic, complemented
13#1859C211yesmaximal, characteristic, complemented
14#1816C211yesmaximal, characteristic, complemented
15#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C2×#135C2211yescharacteristic
19C2×#131C2211yescharacteristic
20#537C2211yescharacteristic
21#520C2211yescharacteristic
22#520C2211yescharacteristic, complemented
23#485—21no—
24#484—21no—
25#482—21no—
26#480—21no—
27#452—21no—
28#450—21no—
29#449—21no—
30#447—21no—
31#440—21no—
32#440—21no—
33C4×Q16—21no—
34C4×D8—21no—
35C2×#108C2211yescharacteristic, complemented
36#428C2211yescharacteristic, complemented
37C2×#107C2211yescharacteristic, complemented
38C2×#107C2211yescharacteristic, complemented
39C2×#104C2211yescharacteristic
40C2×#104C2211yescharacteristic
41C2×#104C2211yescharacteristic, complemented
42#418—21no—
43#418C2211yescharacteristic
44#417—21no—
45#417C2211yescharacteristic
46#417C2211yescharacteristic
47#417C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic, complemented
50#416—21no—
51#416C2211yescharacteristic
52C2×#102—21no—
53C2×#102C2211yescharacteristic
54C2×#102C2211yescharacteristic
55C2×#100C2211yescharacteristic
56#408C2211yescharacteristic
57#408C2211yescharacteristic, complemented
58#405C2211yescharacteristic, complemented
59#390C2211yescharacteristic
60#384C2211yescharacteristic, complemented
61#343C2211yescharacteristic, complemented
62#343C2211yescharacteristic, complemented
63#343C2211yescharacteristic, complemented
64#341C2211yescharacteristic, complemented
65#339C2211yescharacteristic, complemented
66#336C2211yescharacteristic, complemented
67H(5,𝔽2)—22no—
68H(5,𝔽2)—22no—
69H(5,𝔽2)—22no—
70H(5,𝔽2)—41no—
71C2×#41—21no—
72C2×#41—21no—
73C2×#41C2311yescharacteristic
74C2×#41C2311yescharacteristic
75C2×#41C2311yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—41no—
79C22×D4C2311yescharacteristic
80AGL(1,ℤ/8)—22no—
81AGL(1,ℤ/8)—41no—
82#135—22no—
83#135—22no—
84C2×Q16D411yescharacteristic
85C2×SD16—21no—
86C2×SD16D411yescharacteristic
87C2×SD16D411yescharacteristic
88C2×D8—21no—
89C2×D8D411yescharacteristic
90#131—22no—
91#131—41no—
92C2×M16—21no—
93C2×M16C2311yescharacteristic
94C2×M16C2311yescharacteristic
95C2×M16C2311yescharacteristic
96C2×M16C2311yescharacteristic
97C22×C8—21no—
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22⋊Q8—21no—
101C22⋊Q8—21no—
102#121—21no—
103#121—21no—
104#121—41no—
105C4×Q8—21no—
106C4×Q8—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—41no—
110C4×D4—41no—
111#117—21no—
112#117—21no—
113#117C2311yescharacteristic
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C22⋊C4—41no—
121C2×C42C2311yescharacteristic
122#108—41no—
123#107—21no—
124#107—21no—
125#107—21no—
126#107—21no—
127#107—21no—
128#106—21no—
129#104—22no—
130#104—41no—
131#104—41no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103—21no—
137#103D411yescharacteristic
138#103D411yescharacteristic
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#102—21no—
148#102—21no—
149#102—21no—
150#102D411yescharacteristic
151#102D411yescharacteristic
152#100—22no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157#97—21no—
158C4×C8—21no—
159C24—21no—
160C24—41no—
161#41—21no—
162#41—21no—
163#41—24no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—42no—
173#41—42no—
174#41—42no—
175#41#4112yes—
176#41#4112yes—
177C2×Q8C2×D411yescharacteristic
178C2×Q8C2×D411yescharacteristic
179C2×Q8C2×D411yescharacteristic
180C2×D4—21no—
181C2×D4—24no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4C2×D411yescharacteristic
198C2×D4C2×D411yescharacteristic
199C2×D4C2×D411yescharacteristic
200C2×D4#4112yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4C2411yescharacteristic
212Q16—22no—
213SD16—22no—
214SD16—22no—
215SD16—42no—
216D8—22no—
217D8—42no—
218M16—21no—
219M16—21no—
220M16—21no—
221M16—21no—
222M16—21no—
223M16—21no—
224M16—21no—
225M16—21no—
226M16—42no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—42no—
236C2×C8C2×D411yescharacteristic
237C2×C8C2×D411yescharacteristic
238C2×C8C2×D411yescharacteristic
239C2×C8C2×D411yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—82no—
250C4⋊C4C2×D411yescharacteristic
251C4⋊C4C2×D411yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—82no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—41no—
262C42—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—44no—
268C23—41no—
269C23—41no—
270C23—81no—
271C23—81no—
272C23—161no—
273C23—161no—
274C23H(5,𝔽2)11yescharacteristic
275Q8—21no—
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8#12312yes—
281D4—22no—
282D4—22no—
283D4—21no—
284D4—21no—
285D4—22no—
286D4—21no—
287D4—21no—
288D4—22no—
289D4—21no—
290D4—44no—
291D4—44no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—161no—
299D4#12312yes—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
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—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—161no—
322C2×C4—161no—
323C2×C4C22×D411yescharacteristic
324C2×C4C22×D411yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C8—21no—
330C8—21no—
331C8—21no—
332C8—21no—
333C8—41no—
334C8—41no—
335C8—41no—
336C8—41no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—82no—
350C22—81no—
351C22—81no—
352C22—161no—
353C22—164no—
354C22#53811yescharacteristic
355C22D4211yescharacteristic
356C22#55111yescharacteristic
357C4—22no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4C2×#12311yescharacteristic
367C4#52511yescharacteristic
368C4#54611yescharacteristic
369C4#54611yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—81no—
374C2—81no—
375C2—82no—
376C2#241811yescharacteristic
377C2#328911yescharacteristic
378C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#3012C2
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. #3098C2semidirect
3. #2951C2semidirect
4. #2897C2semidirect
5. #2093C2semidirect
6. #2078C2semidirect
7. #2051C2semidirect
8. #2039C2semidirect
9. #1931C2semidirect
10. #1929C2semidirect
11. #1883C2semidirect
12. #1876C2semidirect
13. #1859C2semidirect
14. #1816C2semidirect
15. #1792C2semidirect
22. #520C22semidirect
35. C2×#108C22semidirect
36. #428C22semidirect
37. C2×#107C22semidirect
38. C2×#107C22semidirect
41. C2×#104C22semidirect
49. #417C22semidirect
57. #408C22semidirect
58. #405C22semidirect
60. #384C22semidirect
61. #343C22semidirect
62. #343C22semidirect
63. #343C22semidirect
64. #341C22semidirect
65. #339C22semidirect
66. #336C22semidirect

Maximal subgroups

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

subgroupindex
#30982
#30122
#29512
#28972
#20932
#20782
#20512
#20392
#19312
#19292
#18832
#18762
#18592
#18162
#17922