Finite Group Databasebeta

← #14300

Subgroups — #14300

Order 256.

1121 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2C2×#394C211yesmaximal, characteristic
3C2×#387C211yesmaximal, characteristic, complemented
4C2×#385C212yesmaximal, complemented
5#1444C211yesmaximal, characteristic, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1440C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic
10#1434C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2212yes—
14C2×C4×D4—28no—
15C2×C4×D4—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C42C2211yescharacteristic
26#394C418yescomplemented
27#387—24no—
28#385—28no—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2312yes—
41C22×D4—24no—
42C22×D4—24no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4C2×C414yes—
46C23×C4—24no—
47C23×C4—24no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4—42no—
52C23×C4D418yes—
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2312yes—
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C4×D4—232no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4C2×C414yescomplemented
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4C2×C418yescomplemented
88C2×C22⋊C4C2×C418yescomplemented
89C2×C42—24no—
90C2×C42D414yes—
91#95—24no—
92#95—24no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95—42no—
99#95—42no—
100#95—42no—
101#95C2×C414yescomplemented
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24C22×C414yes—
115C24C2×D414yes—
116C24#4114yes—
117C24C2411yescharacteristic
118C2×D4—28no—
119C2×D4—28no—
120C2×D4—216no—
121C2×D4—216no—
122C2×D4—28no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—216no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4C22×C414yes—
179C22×C4C22×C412yes—
180C22×C4C22×C412yes—
181C22×C4C22×C412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D414yes—
184C22×C4C2×D412yes—
185C22×C4C2×D414yes—
186C22×C4C2×D412yes—
187C22×C4#4112yes—
188C4⋊C4—28no—
189C4⋊C4—28no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C22⋊C4—216no—
193C22⋊C4—216no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C42—28no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—24no—
202C23—24no—
203C23—24no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23C4×D418yes—
233C23C23×C411yescharacteristic
234C23C22×D412yes—
235C23C22×D411yescharacteristic
236C23C2×#4112yes—
237C23C2×#4111yescharacteristic
238C23H(5,𝔽2)11yescharacteristic
239C23H(5,𝔽2)11yescharacteristic
240C23H(5,𝔽2)12yes—
241C23H(5,𝔽2)11yescharacteristic
242C232−1+411yescharacteristic
243C232−1+412yes—
244D4—232no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—432no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4C4×D414yes—
303C2×C4#12014yes—
304C22—28no—
305C22—28no—
306C22—28no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22C2×C4×D412yes—
320C22C2×C4×D411yescharacteristic
321C22#51811yescharacteristic
322C22#52011yescharacteristic
323C22#52012yes—
324C22C2×#12011yescharacteristic
325C22#52512yes—
326C22#52511yescharacteristic
327C22#53412yes—
328C22#53412yes—
329C22#53511yescharacteristic
330C22#53511yescharacteristic
331C22#53612yes—
332C22#53611yescharacteristic
333C22#53711yescharacteristic
334C22#53712yes—
335C22#53812yes—
336C22#53912yes—
337C22#53912yes—
338C22#53912yes—
339C22#54011yescharacteristic
340C22#54211yescharacteristic
341C22#54212yes—
342C4—24no—
343C4—24no—
344C4—42no—
345C4—44no—
346C4—44no—
347C4—42no—
348C4—42no—
349C4—81no—
350C4—82no—
351C4—81no—
352C2—48no—
353C2C4×#12011yescharacteristic
354C2#232212yes—
355C2#232411yescharacteristic
356C2#232512yes—
357C2#232611yescharacteristic
358C2#261311yescharacteristic
359C2#261611yescharacteristic
360C2#263012yes—
361C2#263311yescharacteristic
362C2#263611yescharacteristic
363C2#263812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#387C2semidirect
4. C2×#385C2semidirect
5. #1444C2semidirect
6. #1441C2semidirect
7. #1440C2semidirect
8. #1439C2semidirect
10. #1434C2semidirect
11. C22⋊#95C2semidirect
26. #394C4semidirect
73. C2×C4⋊C4C2×C4semidirect
87. C2×C22⋊C4C2×C4semidirect
88. C2×C22⋊C4C2×C4semidirect
101. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#3942
C2×#3872
C2×#3852
#14442
#14412
#14402
#14392
#14382
#14342
C22⋊#952