Finite Group Databasebeta

← #24294

Subgroups — #24294

Order 256.

679 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 679 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#576C211yesmaximal, characteristic, complemented
2C2×#472C211yesmaximal, characteristic, complemented
3#3060C211yesmaximal, characteristic, complemented
4#3012C212yesmaximal, complemented
5C2×#444C211yesmaximal, characteristic, complemented
6#2051C212yesmaximal, complemented
7#2006C212yesmaximal, complemented
8#1975C211yesmaximal, characteristic, complemented
9#1972C211yesmaximal, characteristic, complemented
10#1815C211yesmaximal, characteristic, complemented
11#1792C212yesmaximal, complemented
12C2×H(5,𝔽2)C2212yes—
13#576—24no—
14#576—24no—
15C2×AGL(1,ℤ/8)—22no—
16C2×AGL(1,ℤ/8)C2212yes—
17C2×AGL(1,ℤ/8)C2212yescomplemented
18C2×#135—21no—
19C2×#135C2211yescharacteristic
20C2×#135C2211yescharacteristic
21C22×D8—21no—
22C22×D8C2211yescharacteristic
23C2×#131C2211yescharacteristic
24C2×#131C2211yescharacteristic, complemented
25#537C2212yescomplemented
26#472—24no—
27#470—21no—
28#469—21no—
29#465—22no—
30#452—22no—
31#450—22no—
32#449—22no—
33#447—22no—
34#444—24no—
35C2×#108C2211yescharacteristic, complemented
36C2×#108C2211yescharacteristic, complemented
37C2×#106C2211yescharacteristic, complemented
38C2×#104C2211yescharacteristic
39C2×#104C2211yescharacteristic, complemented
40C2×#104C2212yescomplemented
41#417C2212yes—
42#417C2212yescomplemented
43C2×#102C2212yescomplemented
44C2×#100C2212yescomplemented
45#408C2212yescomplemented
46C2×#97C2211yescharacteristic, complemented
47#390C2212yescomplemented
48#343C2212yescomplemented
49H(5,𝔽2)—24no—
50H(5,𝔽2)—24no—
51H(5,𝔽2)—24no—
52H(5,𝔽2)—42no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41C2311yescharacteristic
56C2×#41C2312yes—
57C2×#41C2311yescharacteristic
58C22×D4—22no—
59C22×D4—22no—
60C22×D4—42no—
61C22×D4C2312yes—
62AGL(1,ℤ/8)—216no—
63AGL(1,ℤ/8)—24no—
64AGL(1,ℤ/8)—24no—
65AGL(1,ℤ/8)—24no—
66AGL(1,ℤ/8)—42no—
67#135—28no—
68#135—22no—
69#135—22no—
70#135—22no—
71#135—22no—
72C2×Q16D411yescharacteristic
73C2×Q16D411yescharacteristic
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16D412yes—
77C2×SD16D411yescharacteristic
78C2×SD16D411yescharacteristic
79C2×D8—28no—
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—21no—
83C2×D8—21no—
84C2×D8—22no—
85C2×D8D411yescharacteristic
86C2×D8D411yescharacteristic
87C2×D8D414yescomplemented
88#131—22no—
89#131—22no—
90#131—22no—
91#131D414yescomplemented
92C2×M16—21no—
93C2×M16—21no—
94C2×M16C2311yescharacteristic
95C2×M16C2312yes—
96C2×M16C2311yescharacteristic
97C22×C8—21no—
98C22×C8—21no—
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103#121—22no—
104#121—22no—
105#121—42no—
106C2×C4⋊C4C2312yes—
107C2×C22⋊C4—42no—
108C2×C42C2311yescharacteristic
109#108—22no—
110#108—22no—
111#106—21no—
112#106—21no—
113#104—22no—
114#104—22no—
115#104—42no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#100—24no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125#97—22no—
126C24—22no—
127C24—42no—
128#41—21no—
129#41—21no—
130#41—28no—
131#41—21no—
132#41—21no—
133#41—22no—
134#41—22no—
135#41—21no—
136#41—22no—
137#41—22no—
138#41—21no—
139#41—44no—
140#41—44no—
141#41—44no—
142#41C2×D412yes—
143#41#4112yes—
144C2×Q8C2×D412yes—
145C2×Q8C2×D411yescharacteristic
146C2×Q8C2×D411yescharacteristic
147C2×D4—21no—
148C2×D4—28no—
149C2×D4—21no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—44no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—48no—
161C2×D4—48no—
162C2×D4—44no—
163C2×D4—42no—
164C2×D4C2×D414yes—
165C2×D4C2×D412yes—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4C2411yescharacteristic
175Q16—22no—
176Q16—22no—
177SD16—28no—
178SD16—22no—
179SD16—22no—
180SD16—24no—
181SD16—44no—
182D8—28no—
183D8—22no—
184D8—22no—
185D8—24no—
186D8—24no—
187D8—44no—
188M16—21no—
189M16—24no—
190M16—21no—
191M16—22no—
192M16—21no—
193M16—22no—
194M16—42no—
195M16C2×D412yes—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—24no—
200C2×C8—22no—
201C2×C8—21no—
202C2×C8—42no—
203C2×C8C2×D411yescharacteristic
204C2×C8C2×D411yescharacteristic
205C2×C8C2×D412yes—
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C2×C8C2×D412yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—84no—
215C42—21no—
216C42—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—24no—
221C23—48no—
222C23—42no—
223C23—42no—
224C23—82no—
225C23—82no—
226C23—162no—
227C23—162no—
228C23H(5,𝔽2)11yescharacteristic
229Q8—21no—
230Q8—22no—
231Q8—21no—
232Q8—21no—
233Q8—22no—
234Q8#12112yes—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—24no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—24no—
243D4—22no—
244D4—48no—
245D4—48no—
246D4—42no—
247D4—41no—
248D4—42no—
249D4—41no—
250D4—42no—
251D4—42no—
252D4—162no—
253D4#12112yes—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—48no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4H(5,𝔽2)11yescharacteristic
274C8—22no—
275C8—21no—
276C8—21no—
277C8—41no—
278C8—42no—
279C8#12012yes—
280C8#12012yes—
281C22—22no—
282C22—22no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—84no—
291C22—82no—
292C22—82no—
293C22—162no—
294C22—168no—
295C22#53511yescharacteristic
296C22D4211yescharacteristic
297C22#54611yescharacteristic
298C4—22no—
299C4—42no—
300C4—41no—
301C4—81no—
302C4—82no—
303C4C2×#12011yescharacteristic
304C4C2×#12111yescharacteristic
305C4D4211yescharacteristic
306C4#54611yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2—81no—
311C2—82no—
312C2—84no—
313C2#240811yescharacteristic
314C2#328812yes—
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×#576C2semidirect
2. C2×#472C2semidirect
3. #3060C2semidirect
4. #3012C2semidirect
5. C2×#444C2semidirect
6. #2051C2semidirect
7. #2006C2semidirect
8. #1975C2semidirect
9. #1972C2semidirect
10. #1815C2semidirect
11. #1792C2semidirect
17. C2×AGL(1,ℤ/8)C22semidirect
24. C2×#131C22semidirect
25. #537C22semidirect
35. C2×#108C22semidirect
36. C2×#108C22semidirect
37. C2×#106C22semidirect
39. C2×#104C22semidirect
40. C2×#104C22semidirect
42. #417C22semidirect
43. C2×#102C22semidirect
44. C2×#100C22semidirect
45. #408C22semidirect
46. C2×#97C22semidirect
47. #390C22semidirect
48. #343C22semidirect
87. C2×D8D4semidirect
91. #131D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5762
C2×#4722
#30602
#30122
C2×#4442
#20512
#20062
#19752
#19722
#18152
#17922