Finite Group Databasebeta

← #14932

Subgroups — #14932

Order 256.

969 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic
2C2×#388C212yesmaximal, complemented
3C2×#387C211yesmaximal, characteristic, complemented
4#1446C211yesmaximal, characteristic, complemented
5#1446C211yesmaximal, characteristic, complemented
6#1440C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C211yesmaximal, characteristic
9#1438C211yesmaximal, characteristic
10#1437C212yesmaximal, complemented
11C4×#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C2×#124—28no—
15C2×#124—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C42C2212yes—
25C22×C42C2211yescharacteristic
26#388—28no—
27#387—24no—
28C2×#95C2211yescharacteristic
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C25C2312yes—
40C22×Q8—24no—
41C22×Q8—22no—
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—42no—
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—42no—
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60#124—232no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4D418yes—
84C2×C42—24no—
85C2×C42—24no—
86C2×C42—24no—
87C2×C42—42no—
88C2×C42D414yes—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95—22no—
97#95—22no—
98#95—42no—
99#95—42no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24C2×D414yes—
115C24C2411yescharacteristic
116C2×Q8—28no—
117C2×Q8—48no—
118C2×Q8—48no—
119C2×Q8—44no—
120C2×Q8—44no—
121C2×D4—28no—
122C2×D4—48no—
123C2×D4—48no—
124C2×D4—44no—
125C2×D4—44no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4C2×D414yes—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D414yes—
175C22×C4C2×D412yes—
176C22×C4#4114yes—
177C22×C4#4114yes—
178C22×C4#4112yes—
179C22×C4#4112yes—
180C22×C4#4112yes—
181C22×C4#4112yes—
182C4⋊C4—28no—
183C4⋊C4—44no—
184C22⋊C4—216no—
185C22⋊C4—216no—
186C22⋊C4—216no—
187C22⋊C4—48no—
188C22⋊C4—48no—
189C22⋊C4—48no—
190C42—28no—
191C42—48no—
192C23—24no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—24no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23C22×D411yescharacteristic
228C23C22×D412yes—
229C23C2×#4111yescharacteristic
230C23C2×#4112yes—
231C23C2×#4111yescharacteristic
232C23C2×#4111yescharacteristic
233C23C2×#4111yescharacteristic
234C23C2×#4112yes—
235C23H(5,𝔽2)12yes—
236C23H(5,𝔽2)11yescharacteristic
237C232−1+411yescharacteristic
238Q8—416no—
239D4—416no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4#12014yes—
295C2×C4#12414yes—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22C2×#12011yescharacteristic
312C22#52511yescharacteristic
313C22#52512yes—
314C22C2×#12411yescharacteristic
315C22#52912yes—
316C22#52912yes—
317C22#53211yescharacteristic
318C22#53211yescharacteristic
319C22#53212yes—
320C22#53511yescharacteristic
321C22#53711yescharacteristic
322C22#53812yes—
323C22#54212yes—
324C22#54612yes—
325C22#54612yes—
326C22#54612yes—
327C22#54712yes—
328C22#55112yes—
329C22#55512yes—
330C22#55611yescharacteristic
331C22#55611yescharacteristic
332C22#55911yescharacteristic
333C22#55911yescharacteristic
334C4—24no—
335C4—24no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—44no—
340C4—44no—
341C4—44no—
342C4—81no—
343C4—81no—
344C2—84no—
345C2#238811yescharacteristic
346C2#243011yescharacteristic
347C2#243511yescharacteristic
348C2#245912yes—
349C2#247412yes—
350C2#252112yes—
351C2#255712yes—
352C2#259211yescharacteristic
353C2#261011yescharacteristic
354C2#266211yescharacteristic
355C2#286111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#388C2semidirect
3. C2×#387C2semidirect
4. #1446C2semidirect
5. #1446C2semidirect
6. #1440C2semidirect
7. #1439C2semidirect
10. #1437C2semidirect
11. C4×#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3882
C2×#3872
#14462
#14402
#14392
#14382
#14372
C4×#952