Finite Group Databasebeta

← #24230

Subgroups — #24230

Order 256.

812 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3498C211yesmaximal, characteristic, complemented
2#3036C211yesmaximal, characteristic, complemented
3C2×#453C212yesmaximal, complemented
4C4×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
5#2541C211yesmaximal, characteristic, complemented
6#2037C212yesmaximal, complemented
7#2002C212yesmaximal, complemented
8#1980C211yesmaximal, characteristic, complemented
9#1965C211yesmaximal, characteristic, complemented
10#1806C211yesmaximal, characteristic, complemented
11#1792C212yesmaximal, complemented
12C2×H(5,𝔽2)C2212yes—
13C2×AGL(1,ℤ/8)C2211yescharacteristic
14C2×AGL(1,ℤ/8)C2212yescomplemented
15#550—21no—
16#550—21no—
17D42—22no—
18D42—22no—
19D42—21no—
20D42—21no—
21#535—22no—
22#535C2212yes—
23#532—21no—
24#532C2211yescharacteristic
25C2×#127—21no—
26C2×#127C2211yescharacteristic
27C2×#127C2212yescomplemented
28C2×#121—22no—
29C4×#41C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic, complemented
31#463—22no—
32#461—21no—
33#459—21no—
34#453—28no—
35#442—21no—
36#440—21no—
37C4×SD16—21no—
38C4×D8—21no—
39#428C2211yescharacteristic, complemented
40#423C2211yescharacteristic, complemented
41C2×#104C2212yes—
42C2×#104C2212yescomplemented
43C2×#104C2212yescomplemented
44#418C2211yescharacteristic, complemented
45#417C2211yescharacteristic
46#417C2212yescomplemented
47C2×#100C2212yescomplemented
48C2×#100C2212yescomplemented
49C4×M16C2211yescharacteristic, complemented
50#392—22no—
51#390C2212yescomplemented
52C4×C4⋊C4C2211yescharacteristic, complemented
53#339C2212yescomplemented
54H(5,𝔽2)—24no—
55H(5,𝔽2)—24no—
56H(5,𝔽2)—42no—
57H(5,𝔽2)—42no—
58C2×#41—22no—
59C2×#41—22no—
60C2×#41C2312yes—
61C2×#41C2311yescharacteristic
62C22×D4—24no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—24no—
68C22×D4D412yes—
69C22×D4D412yes—
70C22×D4C2312yes—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73AGL(1,ℤ/8)—22no—
74AGL(1,ℤ/8)—24no—
75AGL(1,ℤ/8)—42no—
76AGL(1,ℤ/8)D414yescomplemented
77C2×SD16—21no—
78C2×SD16—22no—
79C2×D8—21no—
80C2×D8—22no—
81C2×M16C2312yes—
82C2×M16C2311yescharacteristic
83C2×M16C2311yescharacteristic
84C4⋊Q8D411yescharacteristic, complemented
85#127—22no—
86#127—22no—
87#127—22no—
88#127—24no—
89#127—21no—
90#127—21no—
91#127—21no—
92#127—21no—
93#127—42no—
94#127D411yescharacteristic, complemented
95#124—22no—
96#124D412yescomplemented
97#121—22no—
98#121—22no—
99#121—24no—
100#121—24no—
101#121—21no—
102#121—21no—
103#121—22no—
104#121—22no—
105#121—48no—
106#120—24no—
107#120—24no—
108#120—24no—
109C4×Q8D411yescharacteristic, complemented
110C4×Q8D411yescharacteristic, complemented
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4D411yescharacteristic, complemented
122C4×D4D411yescharacteristic, complemented
123#117—21no—
124#117C2311yescharacteristic
125C2×C4⋊C4—21no—
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4—24no—
128C2×C22⋊C4—21no—
129C2×C42—21no—
130C2×C42C2312yes—
131C2×C42C2311yescharacteristic
132#107—21no—
133#106—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136#104—24no—
137#104—24no—
138#104—42no—
139#103—21no—
140#103—22no—
141#102—21no—
142#102—22no—
143#102—21no—
144#102—21no—
145#100—24no—
146#100—42no—
147#97—21no—
148C4×C8—21no—
149#95—21no—
150C24—24no—
151C24—21no—
152C24C2×D414yes—
153#41—21no—
154#41—21no—
155#41—22no—
156#41—22no—
157#41—22no—
158#41—22no—
159#41—44no—
160#41—44no—
161#41—44no—
162#41—44no—
163#41C2×D412yes—
164#41#4112yes—
165C2×Q8C2×D412yes—
166C2×Q8C2×D411yescharacteristic
167C2×D4—21no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—22no—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—48no—
194C2×D4—48no—
195C2×D4—48no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—48no—
199C2×D4—84no—
200C2×D4—84no—
201C2×D4—82no—
202C2×D4—84no—
203C2×D4C2×D412yes—
204C2×D4C2×D411yescharacteristic
205C2×D4C2×D412yescomplemented
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—44no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2411yescharacteristic
219SD16—24no—
220SD16—44no—
221D8—24no—
222D8—44no—
223M16—21no—
224M16—22no—
225M16—21no—
226M16—22no—
227M16—21no—
228M16C2×D412yescomplemented
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—22no—
232C4⋊C4—24no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4C2×D411yescharacteristic
240C4⋊C4C2×D411yescharacteristic
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—21no—
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—48no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—22no—
255C42—22no—
256C42C2×D411yescharacteristic
257C42C2×D411yescharacteristic
258C42C2×D411yescharacteristic
259C42C2×D411yescharacteristic
260C23—21no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—24no—
266C23—42no—
267C23—42no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23—41no—
272C23—41no—
273C23—84no—
274C23—84no—
275C23C22×D411yescharacteristic
276Q8—21no—
277Q8—22no—
278Q8—22no—
279Q8#12112yescomplemented
280D4—22no—
281D4—22no—
282D4—22no—
283D4—21no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—48no—
288D4—44no—
289D4—42no—
290D4—48no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—41no—
295D4—42no—
296D4—42no—
297D4—41no—
298D4—42no—
299D4—44no—
300D4—42no—
301D4—82no—
302D4—82no—
303D4—84no—
304D4—84no—
305D4—816no—
306D4#12112yescomplemented
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#12011yescharacteristic
328C2×C4#12011yescharacteristic
329C2×C4#12011yescharacteristic
330C2×C4#12011yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C22×D411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C8—22no—
338C8—41no—
339C8—42no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—44no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—44no—
348C22—42no—
349C22—44no—
350C22—42no—
351C22—44no—
352C22—42no—
353C22—84no—
354C22—84no—
355C22—82no—
356C22—81no—
357C22—81no—
358C22—168no—
359C22C2×#12011yescharacteristic
360C22D4211yescharacteristic
361C22#54611yescharacteristic
362C4—22no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—44no—
367C4—42no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4#45312yescomplemented
372C4C2×#12111yescharacteristic
373C4#53511yescharacteristic
374C4D4211yescharacteristic
375C4#54611yescharacteristic
376C2—21no—
377C2—21no—
378C2—42no—
379C2—82no—
380C2—84no—
381C2—82no—
382C2#240811yescharacteristic
383C2C2×#45311yescharacteristic
384C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3498C2semidirect
2. #3036C2semidirect
3. C2×#453C2semidirect
4. C4×AGL(1,ℤ/8)C2semidirect
5. #2541C2semidirect
6. #2037C2semidirect
7. #2002C2semidirect
8. #1980C2semidirect
9. #1965C2semidirect
10. #1806C2semidirect
11. #1792C2semidirect
14. C2×AGL(1,ℤ/8)C22semidirect
27. C2×#127C22semidirect
30. C2×C4×D4C22semidirect
39. #428C22semidirect
40. #423C22semidirect
42. C2×#104C22semidirect
43. C2×#104C22semidirect
44. #418C22semidirect
46. #417C22semidirect
47. C2×#100C22semidirect
48. C2×#100C22semidirect
49. C4×M16C22semidirect
51. #390C22semidirect
52. C4×C4⋊C4C22semidirect
53. #339C22semidirect
76. AGL(1,ℤ/8)D4semidirect
84. C4⋊Q8D4semidirect
94. #127D4semidirect
96. #124D4semidirect
109. C4×Q8D4semidirect
110. C4×Q8D4semidirect
121. C4×D4D4semidirect
122. C4×D4D4semidirect
205. C2×D4C2×D4semidirect
228. M16C2×D4semidirect
279. Q8#121semidirect
306. D4#121semidirect
371. C4#453semidirect

Maximal subgroups

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

subgroupindex
#34982
#30362
C2×#4532
C4×AGL(1,ℤ/8)2
#25412
#20372
#20022
#19802
#19652
#18062
#17922