Finite Group Databasebeta

← #23642

Subgroups — #23642

Order 256.

583 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#428C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic
3#2280C211yesmaximal, characteristic
4C2×#380C211yesmaximal, characteristic
5#2077C212yesmaximal, complemented
6#2077C212yesmaximal, complemented
7#2076C212yesmaximal, complemented
8#2075C212yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10#1736C211yesmaximal, characteristic, complemented
11#1736C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20#428—24no—
21C2×#107C2212yescomplemented
22C2×#106C2212yescomplemented
23#407—24no—
24C2×C22⋊C8C2212yes—
25C2×C22⋊C8C2212yes—
26#381—21no—
27#380—21no—
28#380—21no—
29#380C2211yescharacteristic
30#380C2212yes—
31#380C2211yescharacteristic
32#380C2212yes—
33#380C2211yescharacteristic
34#380C2211yescharacteristic
35#379—21no—
36C4×C22⋊C4—21no—
37C4×C22⋊C4—21no—
38C2×#95—21no—
39#340C2212yescomplemented
40#340C2212yescomplemented
41#340C2212yescomplemented
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—22no—
65C2×M16D414yes—
66C22×C8C2312yes—
67C22×C8C2312yes—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—44no—
93C2×C42—21no—
94C2×C42—21no—
95#107—24no—
96#106—24no—
97C22⋊C8—24no—
98C22⋊C8—24no—
99#95—22no—
100#95—22no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—41no—
111C24—41no—
112C24C2×D411yescharacteristic
113C24C2×D411yescharacteristic
114C24C2×D411yescharacteristic
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D412yes—
156C22×C4#4112yes—
157C22×C4#4112yes—
158C22×C4#4112yes—
159C22×C4#4112yes—
160C22×C4#4112yes—
161C22×C4C2411yescharacteristic
162M16—28no—
163M16—44no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8C2×D414yes—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—44no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—44no—
191C22⋊C4—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—28no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23#12112yes—
218C23#12312yes—
219C23#12312yes—
220C23AGL(1,ℤ/8)12yes—
221C23#13712yes—
222C23C22×D411yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—44no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4#12112yes—
261C2×C4#12314yes—
262C2×C4#12312yes—
263C2×C4#12312yes—
264C2×C4#12414yes—
265C2×C4#12614yes—
266C2×C4#12614yes—
267C2×C4C22×D411yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C8—24no—
274C8—42no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—416no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22#39714yes—
297C22C2×#12111yescharacteristic
298C22C2×#12311yescharacteristic
299C22C2×#12311yescharacteristic
300C22C2×AGL(1,ℤ/8)11yescharacteristic
301C22C2×#13711yescharacteristic
302C22#57511yescharacteristic
303C22#57511yescharacteristic
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—82no—
309C4—81no—
310C4—82no—
311C4—81no—
312C4#39714yes—
313C4C2×#12311yescharacteristic
314C4C2×#12411yescharacteristic
315C4C2×#12611yescharacteristic
316C4C2×#12611yescharacteristic
317C2—24no—
318C2—42no—
319C2—42no—
320C2C2×#39711yescharacteristic
321C2#305511yescharacteristic
322C2#305611yescharacteristic
323C2#309211yescharacteristic
324C2#309311yescharacteristic
325C2#309411yescharacteristic
326C2#309511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#428C2semidirect
5. #2077C2semidirect
6. #2077C2semidirect
7. #2076C2semidirect
8. #2075C2semidirect
9. #1736C2semidirect
10. #1736C2semidirect
11. #1736C2semidirect
21. C2×#107C22semidirect
22. C2×#106C22semidirect
39. #340C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
C2×#4072
#22802
C2×#3802
#20772
#20762
#20752
#17362