Finite Group Databasebeta

← #25183

Subgroups — #25183

Order 256.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#473C211yesmaximal, characteristic, complemented
2#3044C211yesmaximal, characteristic, complemented
3C2×#455C211yesmaximal, characteristic
4#3009C211yesmaximal, characteristic
5#2947C211yesmaximal, characteristic, complemented
6#2897C211yesmaximal, characteristic
7#2063C211yesmaximal, characteristic, complemented
8#2052C211yesmaximal, characteristic, complemented
9#2038C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#1977C211yesmaximal, characteristic, complemented
12#1929C211yesmaximal, characteristic, complemented
13#1888C211yesmaximal, characteristic, complemented
14#1882C211yesmaximal, characteristic, complemented
15#1866C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic
19C2×#137C2211yescharacteristic
20C22×Q16C2211yescharacteristic, complemented
21C22×SD16C2211yescharacteristic
22C2×#131C2211yescharacteristic
23#536C2211yescharacteristic
24C2×#124C2211yescharacteristic
25#520C2211yescharacteristic
26#519C2211yescharacteristic, complemented
27#473—24no—
28#464—21no—
29#464—21no—
30#462—21no—
31#462—21no—
32#455—24no—
33#452—21no—
34#452—21no—
35#450—21no—
36#450—21no—
37#441—21no—
38#440—21no—
39C4×Q16—21no—
40C4×SD16—21no—
41C2×#108C2211yescharacteristic, complemented
42C2×#108C2211yescharacteristic, complemented
43#428C2211yescharacteristic, complemented
44#424C2211yescharacteristic, complemented
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47#419C2211yescharacteristic, complemented
48#418—21no—
49#418C2211yescharacteristic
50#418C2211yescharacteristic
51#417—21no—
52#417C2211yescharacteristic
53#416—21no—
54#416C2211yescharacteristic
55#416C2211yescharacteristic
56C2×#103C2211yescharacteristic, complemented
57C2×#102—21no—
58C2×#102C2211yescharacteristic
59C2×#101C2211yescharacteristic, complemented
60C2×#101C2211yescharacteristic, complemented
61C2×#100C2211yescharacteristic
62#408C2211yescharacteristic
63#405C2211yescharacteristic, complemented
64#391C2211yescharacteristic, 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×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—41no—
82C22×D4C2311yescharacteristic
83#137—22no—
84#137—22no—
85#137—22no—
86#137—22no—
87#137—41no—
88#137D414yes—
89C2×Q16—21no—
90C2×Q16—22no—
91C2×Q16—21no—
92C2×Q16—21no—
93C2×Q16—21no—
94C2×Q16—21no—
95C2×Q16—41no—
96C2×SD16—21no—
97C2×SD16—21no—
98C2×SD16—21no—
99C2×SD16—22no—
100C2×SD16—21no—
101C2×SD16—21no—
102C2×SD16—41no—
103#131—22no—
104#131—41no—
105C2×M16—21no—
106C2×M16C2311yescharacteristic
107C2×M16C2311yescharacteristic
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C22×C8—21no—
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C4⋊Q8—21no—
114#124—22no—
115#124—21no—
116#124—22no—
117#123—41no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×D4—21no—
126C4×D4—41no—
127#117—21no—
128#117—41no—
129#117C2311yescharacteristic
130#117C2311yescharacteristic
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—41no—
134C2×C42C2311yescharacteristic
135#108—22no—
136#108—41no—
137#107—21no—
138#106—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141#104—22no—
142#104—22no—
143#103—21no—
144#103—21no—
145#103—21no—
146#103—21no—
147#103—21no—
148#103D411yescharacteristic
149#103D411yescharacteristic
150#103D411yescharacteristic
151#103D411yescharacteristic
152#102—21no—
153#102—21no—
154#102—21no—
155#102—21no—
156#102—21no—
157#102—21no—
158#102—21no—
159#102D411yescharacteristic
160#102D411yescharacteristic
161#102D411yescharacteristic
162#102D411yescharacteristic
163#101—22no—
164#101—22no—
165#100—22no—
166C22⋊C8—21no—
167C22⋊C8—21no—
168#97—21no—
169C4×C8—21no—
170#95—41no—
171C24—21no—
172C24—41no—
173#41—21no—
174#41—21no—
175#41—24no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—42no—
181#41—42no—
182#41—42no—
183#41C2×D412yes—
184#41#4112yes—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8C2×D412yes—
197C2×Q8C2×D411yescharacteristic
198C2×Q8C2×D411yescharacteristic
199C2×D4—21no—
200C2×D4—24no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4C2×D411yescharacteristic
215C2×D4C2×D411yescharacteristic
216C2×D4#4112yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4C2411yescharacteristic
226Q16—24no—
227Q16—42no—
228Q16—42no—
229Q16—42no—
230Q16—42no—
231SD16—24no—
232SD16—42no—
233SD16—42no—
234SD16—42no—
235SD16—42no—
236M16—21no—
237M16—21no—
238M16—21no—
239M16—21no—
240M16—21no—
241M16—21no—
242M16—21no—
243M16—42no—
244M16C2×D412yes—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—42no—
254C2×C8C2×D411yescharacteristic
255C2×C8C2×D411yescharacteristic
256C2×C8C2×D411yescharacteristic
257C2×C8C2×D411yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4C2×D411yescharacteristic
266C4⋊C4C2×D411yescharacteristic
267C4⋊C4C2×D411yescharacteristic
268C4⋊C4C2×D411yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—44no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—41no—
280C42—41no—
281C42—41no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—44no—
286C23—41no—
287C23—41no—
288C23—81no—
289C23—81no—
290C23—161no—
291C23—161no—
292C23H(5,𝔽2)11yescharacteristic
293Q8—21no—
294Q8—22no—
295Q8—21no—
296Q8—22no—
297Q8—21no—
298Q8—41no—
299Q8—41no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8#12112yes—
305D4—22no—
306D4—22no—
307D4—21no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—22no—
312D4—44no—
313D4—44no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4#12112yes—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—44no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—161no—
342C2×C4C22×D411yescharacteristic
343C2×C4C22×D411yescharacteristic
344C2×C4C22×D411yescharacteristic
345C2×C4C2×#4111yescharacteristic
346C2×C4C2×#4111yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C8—22no—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C8—41no—
354C8—41no—
355C8—41no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—82no—
366C22—81no—
367C22—81no—
368C22—161no—
369C22—164no—
370C22#53811yescharacteristic
371C22D4211yescharacteristic
372C22#54611yescharacteristic
373C4—22no—
374C4—41no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4C2×#12111yescharacteristic
384C4#52511yescharacteristic
385C4D4211yescharacteristic
386C4#54611yescharacteristic
387C2—21no—
388C2—21no—
389C2—42no—
390C2—81no—
391C2—82no—
392C2#242411yescharacteristic
393C2#328911yescharacteristic
394C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×H(5,𝔽2)C22
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×#473C2semidirect
2. #3044C2semidirect
5. #2947C2semidirect
7. #2063C2semidirect
8. #2052C2semidirect
9. #2038C2semidirect
10. #2015C2semidirect
11. #1977C2semidirect
12. #1929C2semidirect
13. #1888C2semidirect
14. #1882C2semidirect
15. #1866C2semidirect
20. C22×Q16C22semidirect
26. #519C22semidirect
41. C2×#108C22semidirect
42. C2×#108C22semidirect
43. #428C22semidirect
44. #424C22semidirect
47. #419C22semidirect
56. C2×#103C22semidirect
59. C2×#101C22semidirect
60. C2×#101C22semidirect
63. #405C22semidirect
64. #391C22semidirect
65. #343C22semidirect
66. #339C22semidirect
67. #339C22semidirect
68. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4732
#30442
C2×#4552
#30092
#29472
#28972
#20632
#20522
#20382
#20152
#19772
#19292
#18882
#18822
#18662