Finite Group Databasebeta

← #22467

Subgroups — #22467

Order 256.

675 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#577C211yesmaximal, characteristic
2C2×#443C211yesmaximal, characteristic, complemented
3#2952C211yesmaximal, characteristic, complemented
4#2898C211yesmaximal, characteristic, complemented
5#2897C211yesmaximal, characteristic, complemented
6#1977C211yesmaximal, characteristic, complemented
7#1968C211yesmaximal, characteristic, complemented
8#1908C211yesmaximal, characteristic, complemented
9#1907C211yesmaximal, characteristic, complemented
10#1883C211yesmaximal, characteristic, complemented
11#1882C211yesmaximal, characteristic, complemented
12#1815C211yesmaximal, characteristic, complemented
13#1793C211yesmaximal, characteristic, complemented
14#1792C211yesmaximal, characteristic, complemented
15#1764C211yesmaximal, characteristic, complemented
16C2×2−1+4C2211yescharacteristic
17C2×H(5,𝔽2)C2211yescharacteristic
18#577—24no—
19#577C418yescomplemented
20C2×#137—21no—
21C2×#137C2211yescharacteristic
22C2×AGL(1,ℤ/8)—21no—
23C2×AGL(1,ℤ/8)C2211yescharacteristic
24C2×#135—21no—
25C2×#135C2211yescharacteristic
26C22×SD16—21no—
27C22×SD16C2211yescharacteristic
28C2×#131C2211yescharacteristic
29C2×#131C2211yescharacteristic
30C2×#131C2211yescharacteristic, complemented
31#520C2211yescharacteristic
32#520C2211yescharacteristic, complemented
33#443—24no—
34#442—21no—
35#441—21no—
36C4×SD16—21no—
37C4×SD16—21no—
38C2×#108C2211yescharacteristic, complemented
39C2×#106C2211yescharacteristic, complemented
40C2×#104C2211yescharacteristic
41C2×#104C2211yescharacteristic
42C2×#104C2211yescharacteristic, complemented
43#419—21no—
44#419C2211yescharacteristic, complemented
45#418—21no—
46#418C2211yescharacteristic
47#417—21no—
48#417—21no—
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#417C2211yescharacteristic, complemented
53#416—21no—
54#416—21no—
55#416C2211yescharacteristic
56C2×#103—21no—
57C2×#103C2211yescharacteristic, complemented
58C2×#102—21no—
59C2×#102C2211yescharacteristic
60C2×#101C2211yescharacteristic, complemented
61C2×#100C2211yescharacteristic
62#405C2211yescharacteristic, complemented
63C2×C4×C8C2211yescharacteristic, complemented
64#391C2211yescharacteristic, complemented
65#390C2211yescharacteristic
66#343C2211yescharacteristic, complemented
67#343C2211yescharacteristic, complemented
68#341C2211yescharacteristic, complemented
69#339C2211yescharacteristic, complemented
702−1+4—22no—
712−1+4—22no—
722−1+4—41no—
732−1+4C2×C414yescomplemented
74H(5,𝔽2)—22no—
75H(5,𝔽2)—22no—
76H(5,𝔽2)—41no—
77H(5,𝔽2)C2×C414yescomplemented
78C2×#41—21no—
79C2×#41—21no—
80C2×#41—21no—
81C2×#41—21no—
82C2×#41—41no—
83C2×#41C2311yescharacteristic
84C2×#41C2311yescharacteristic
85C2×#41C2311yescharacteristic
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8C2311yescharacteristic
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—41no—
92C22×D4C2311yescharacteristic
93#137—22no—
94#137—28no—
95#137C2×C414yescomplemented
96AGL(1,ℤ/8)—22no—
97AGL(1,ℤ/8)—28no—
98AGL(1,ℤ/8)C2×C414yescomplemented
99#135—28no—
100#135C2×C414yes—
101#135D414yes—
102C2×Q16—21no—
103C2×Q16D411yescharacteristic
104C2×SD16—21no—
105C2×SD16—21no—
106C2×SD16—28no—
107C2×SD16—21no—
108C2×SD16—21no—
109C2×SD16—22no—
110C2×SD16C2×C414yescomplemented
111C2×SD16D411yescharacteristic
112C2×SD16D411yescharacteristic
113C2×D8—21no—
114C2×D8D411yescharacteristic
115#131—22no—
116#131—22no—
117#131—22no—
118#131—41no—
119#131—41no—
120#131C2×C414yescomplemented
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16C2311yescharacteristic
125C2×M16C2311yescharacteristic
126C2×M16C2311yescharacteristic
127C2×M16C2311yescharacteristic
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8C2311yescharacteristic
132C22×C8C2311yescharacteristic
133C4×Q8—21no—
134C4×Q8—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—41no—
138C4×D4—41no—
139#117—21no—
140#117—21no—
141#117C2311yescharacteristic
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C22⋊C4—41no—
147C2×C42C2311yescharacteristic
148#108—22no—
149#106—21no—
150#106—21no—
151#104—22no—
152#104—22no—
153#104—22no—
154#103—21no—
155#103—21no—
156#103—21no—
157#103—21no—
158#103—21no—
159#103—21no—
160#103—21no—
161#103—21no—
162#103D411yescharacteristic
163#103D411yescharacteristic
164#102—21no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—21no—
172#102D411yescharacteristic
173#102D411yescharacteristic
174#101—41no—
175#100—41no—
176#97—21no—
177C4×C8—22no—
178C4×C8—21no—
179#95—41no—
180C24—21no—
181C24—41no—
182#41—21no—
183#41—21no—
184#41—21no—
185#41—21no—
186#41—21no—
187#41—21no—
188#41—24no—
189#41—24no—
190#41—24no—
191#41—24no—
192#41—44no—
193#41—42no—
194#41—42no—
195#41—42no—
196#41—42no—
197#41—44no—
198#41C22×C412yes—
199#41C22×C412yes—
200#41C22×C412yes—
201#41C2×D412yes—
202#41C2×D412yes—
203#41#4112yes—
204C2×Q8—21no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8C22×C412yes—
215C2×Q8C2×D411yescharacteristic
216C2×Q8C2×D411yescharacteristic
217C2×Q8C2×D411yescharacteristic
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4C22×C412yes—
236C2×D4C2×D411yescharacteristic
237C2×D4C2×D411yescharacteristic
238C2×D4C2×D411yescharacteristic
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C2411yescharacteristic
252Q16—24no—
253Q16C22⋊C414yescomplemented
254SD16—24no—
255SD16—24no—
256SD16—24no—
257SD16—24no—
258SD16C22⋊C414yescomplemented
259SD16C22⋊C414yescomplemented
260D8—24no—
261D8C22⋊C414yescomplemented
262M16—21no—
263M16—24no—
264M16—21no—
265M16—21no—
266M16—21no—
267M16—21no—
268M16—21no—
269M16—21no—
270M16—42no—
271M16—42no—
272M16C22×C412yes—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—24no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8C22×C412yes—
283C2×C8C2×D411yescharacteristic
284C2×C8C2×D411yescharacteristic
285C2×C8C2×D411yescharacteristic
286C2×C8C2×D412yes—
287C2×C8C2×D411yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4C2×D411yescharacteristic
295C4⋊C4C2×D411yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—82no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—41no—
304C42—41no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23—44no—
310C23—41no—
311C23—41no—
312C23—81no—
313C23—81no—
314C23—161no—
315C23—161no—
316C23H(5,𝔽2)11yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—22no—
321Q8—22no—
322Q8—44no—
323Q8—41no—
324Q8—41no—
325Q8C2×C22⋊C412yes—
326Q8C2×C22⋊C412yes—
327Q8C4×D412yes—
328D4—22no—
329D4—22no—
330D4—22no—
331D4—22no—
332D4—22no—
333D4—22no—
334D4—22no—
335D4—22no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—44no—
340D4—44no—
341D4—44no—
342D4—41no—
343D4—41no—
344D4C2×C22⋊C412yes—
345D4C2×C22⋊C412yes—
346D4C4×D412yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—161no—
372C2×C4—161no—
373C2×C4C23×C411yescharacteristic
374C2×C4C22×D411yescharacteristic
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D411yescharacteristic
377C2×C4C2×#4111yescharacteristic
378C2×C4H(5,𝔽2)11yescharacteristic
379C8—22no—
380C8—21no—
381C8—21no—
382C8—41no—
383C8—41no—
384C8—41no—
385C8C2×C22⋊C412yes—
386C8C2×C22⋊C412yes—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22—82no—
400C22—81no—
401C22—81no—
402C22—161no—
403C22—164no—
404C22#51811yescharacteristic
405C22D4211yescharacteristic
406C22#54611yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—81no—
415C4—81no—
416C4C22×C22⋊C411yescharacteristic
417C4C2×C4×D411yescharacteristic
418C4D4211yescharacteristic
419C4#54611yescharacteristic
420C2—21no—
421C2—21no—
422C2—42no—
423C2—42no—
424C2—42no—
425C2—82no—
426C2D4×C22⋊C411yescharacteristic
427C2#329011yescharacteristic
428C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#577C2
LCSG ▷ C2×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
2. C2×#443C2semidirect
3. #2952C2semidirect
4. #2898C2semidirect
5. #2897C2semidirect
6. #1977C2semidirect
7. #1968C2semidirect
8. #1908C2semidirect
9. #1907C2semidirect
10. #1883C2semidirect
11. #1882C2semidirect
12. #1815C2semidirect
13. #1793C2semidirect
14. #1792C2semidirect
15. #1764C2semidirect
19. #577C4semidirect
30. C2×#131C22semidirect
32. #520C22semidirect
38. C2×#108C22semidirect
39. C2×#106C22semidirect
42. C2×#104C22semidirect
44. #419C22semidirect
52. #417C22semidirect
57. C2×#103C22semidirect
60. C2×#101C22semidirect
62. #405C22semidirect
63. C2×C4×C8C22semidirect
64. #391C22semidirect
66. #343C22semidirect
67. #343C22semidirect
68. #341C22semidirect
69. #339C22semidirect
73. 2−1+4C2×C4semidirect
77. H(5,𝔽2)C2×C4semidirect
95. #137C2×C4semidirect
98. AGL(1,ℤ/8)C2×C4semidirect
110. C2×SD16C2×C4semidirect
120. #131C2×C4semidirect
253. Q16C22⋊C4semidirect
258. SD16C22⋊C4semidirect
259. SD16C22⋊C4semidirect
261. D8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5772
C2×#4432
#29522
#28982
#28972
#19772
#19682
#19082
#19072
#18832
#18822
#18152
#17932
#17922
#17642