Finite Group Databasebeta

← #29718

Subgroups — #29718

Order 256.

627 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 129 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×#450C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3#2850C211yesmaximal, characteristic, complemented
4C2×#390C211yesmaximal, characteristic, complemented
5#2073C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#2054C212yesmaximal, complemented
8#2012C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1872C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2212yes—
15C2×#124C2211yescharacteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20C22×C22⋊C4—21no—
21C22×C42C2211yescharacteristic
22#450—24no—
23#447—24no—
24C2×C4⋊C8C2212yescomplemented
25C2×#103C2212yescomplemented
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yescomplemented
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31#391C2211yescharacteristic, complemented
32#390—21no—
33#390—21no—
34#390C2212yes—
35#390C2212yes—
36#390C2211yescharacteristic, complemented
37#388—21no—
38#388—21no—
39#386—21no—
40#386—21no—
41#382C2211yescharacteristic, complemented
42#340C2212yescomplemented
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—42no—
51C22×D4—42no—
52C22×D4D414yes—
53C22×D4C2311yescharacteristic
54C22×D4C2312yes—
55C23×C4—21no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4C2311yescharacteristic
59C2×SD16—24no—
60C2×SD16—42no—
61C2×SD16—42no—
62C2×D8—24no—
63C2×D8—42no—
64C2×D8—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67#124—42no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70#121—22no—
71#121—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—44no—
89C2×C22⋊C4—44no—
90C2×C42—42no—
91C2×C42D411yescharacteristic
92C2×C42D411yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C4⋊C8—42no—
96#103—42no—
97#102—24no—
98#102—24no—
99#102—42no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—24no—
107C24—22no—
108C24—21no—
109C24—42no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24—42no—
114C24—42no—
115C24C2×D411yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×D4—24no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—48no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—48no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4C2×D414yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4#4112yes—
160C22×C4C2411yescharacteristic
161SD16—48no—
162D8—48no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—44no—
187C42—22no—
188C42—22no—
189C42—41no—
190C42—41no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—24no—
196C23—48no—
197C23—48no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—48no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—84no—
213C23—84no—
214C23—84no—
215C23—84no—
216C23#12112yes—
217C23#13512yes—
218C23AGL(1,ℤ/8)12yes—
219C23C22×D411yescharacteristic
220Q8—42no—
221Q8—42no—
222D4—24no—
223D4—24no—
224D4—48no—
225D4—42no—
226D4—42no—
227D4—42no—
228D4—42no—
229D4—84no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—84no—
260C2×C4#12112yes—
261C2×C4AGL(1,ℤ/8)12yes—
262C2×C4C22×D411yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4H(5,𝔽2)12yes—
265C2×C4H(5,𝔽2)12yes—
266C8—42no—
267C8—42no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—816no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22#46314yes—
290C22C2×#12111yescharacteristic
291C22#53411yescharacteristic
292C22#53511yescharacteristic
293C22C2×#13511yescharacteristic
294C22C2×AGL(1,ℤ/8)11yescharacteristic
295C22C2×AGL(1,ℤ/8)11yescharacteristic
296C22#57511yescharacteristic
297C4—22no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—81no—
303C4—82no—
304C4—81no—
305C4—81no—
306C4#53412yes—
307C4#53812yes—
308C2—24no—
309C2—84no—
310C2—161no—
311C2#261711yescharacteristic
312C2C2×#46311yescharacteristic
313C2#303611yescharacteristic
314C2#318811yescharacteristic
315C2#319411yescharacteristic
316C2#321111yescharacteristic
317C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#450C2semidirect
3. #2850C2semidirect
4. C2×#390C2semidirect
5. #2073C2semidirect
6. #2055C2semidirect
7. #2054C2semidirect
8. #2012C2semidirect
9. #1874C2semidirect
10. #1872C2semidirect
11. #1797C2semidirect
15. C2×#124C22semidirect
24. C2×C4⋊C8C22semidirect
25. C2×#103C22semidirect
28. C2×#102C22semidirect
31. #391C22semidirect
36. #390C22semidirect
41. #382C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×#4472
#28502
C2×#3902
#20732
#20552
#20542
#20122
#18742
#18722
#17972