Finite Group Databasebeta

← #24640

Subgroups — #24640

Order 256.

459 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3#2545C211yesmaximal, characteristic, complemented
4#2354C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic
7#2013C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic, complemented
10#1981C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1928C211yesmaximal, characteristic, complemented
14#1925C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25#463—24no—
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#396—21no—
43#396—21no—
44#391C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#388—21no—
49#388—21no—
50#388—21no—
51#388—21no—
52#384C2211yescharacteristic
53#380—21no—
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×D8—22no—
73C2×D8—41no—
74C2×D8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C4⋊Q8—22no—
80C4⋊Q8—41no—
81#124—22no—
82#124—22no—
83#124—22no—
84#124—41no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×D4—22no—
88C4×D4D414yes—
89#117—44no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#107—22no—
119C4⋊C8—22no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
127#103—41no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#97—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8#4112yes—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4C2×D412yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181SD16—28no—
182SD16—44no—
183D8—44no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8#4112yes—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4C2×D412yes—
211C22⋊C4—24no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42C2×D412yes—
230C42#4112yes—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23C22×D411yescharacteristic
246Q8—22no—
247Q8—22no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254D4—22no—
255D4—22no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—48no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4#12112yes—
297C2×C4#12112yes—
298C2×C4AGL(1,ℤ/8)12yes—
299C2×C4C22×D411yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C8—22no—
306C8—41no—
307C8—41no—
308C8—41no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22C2×#12111yescharacteristic
324C22#54611yescharacteristic
325C22#55011yescharacteristic
326C22C2×AGL(1,ℤ/8)11yescharacteristic
327C22#57511yescharacteristic
328C22#57711yescharacteristic
329C22#57811yescharacteristic
330C4—22no—
331C4—22no—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#52511yescharacteristic
343C4#53511yescharacteristic
344C4#55311yescharacteristic
345C4#55511yescharacteristic
346C2—82no—
347C2—161no—
348C2#241411yescharacteristic
349C2#303611yescharacteristic
350C2#304611yescharacteristic
351C2#330811yescharacteristic
352C2#332111yescharacteristic
353C2#339711yescharacteristic
354C2#340211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#440C2semidirect
3. #2545C2semidirect
4. #2354C2semidirect
5. #2055C2semidirect
7. #2013C2semidirect
9. #1983C2semidirect
10. #1981C2semidirect
11. #1959C2semidirect
12. #1940C2semidirect
13. #1928C2semidirect
14. #1925C2semidirect
15. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4402
#25452
#23542
#20552
#20542
#20132
#20122
#19832
#19812
#19592
#19402
#19282
#19252
#18032