Finite Group Databasebeta

← #15113

Subgroups — #15113

Order 256.

1031 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117C211yesmaximal, characteristic, complemented
2C2×#388C212yesmaximal, complemented
3C2×#386C211yesmaximal, characteristic
4#1444C211yesmaximal, characteristic, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1439C211yesmaximal, characteristic, complemented
8#1439C211yesmaximal, characteristic, complemented
9#1438C212yesmaximal
10#1434C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×#117—24no—
15C2×#117—28no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C42C2212yes—
27#388—28no—
28#386—24no—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C25C2311yescharacteristic
40C25C2311yescharacteristic
41C22×D4—22no—
42C22×D4—22no—
43C22×D4—24no—
44C22×D4—22no—
45C22×D4—42no—
46C23×C4—24no—
47C23×C4—24no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4D414yes—
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60#117—232no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
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—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C42—24no—
88C2×C42—24no—
89C2×C42D418yescomplemented
90#95—22no—
91#95—22no—
92#95—24no—
93#95—24no—
94#95—22no—
95#95—24no—
96#95—24no—
97#95—22no—
98#95—42no—
99#95—42no—
100C24—22no—
101C24—22no—
102C24—21no—
103C24—21no—
104C24—21no—
105C24—21no—
106C24—21no—
107C24—22no—
108C24—21no—
109C24—22no—
110C24—21no—
111C24—22no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—21no—
117C24—22no—
118C24—21no—
119C24C2×D412yes—
120C24#4112yes—
121C24#4114yes—
122C24C2411yescharacteristic
123C2×D4—28no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—44no—
127C2×D4—44no—
128C22×C4—216no—
129C22×C4—216no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—28no—
137C22×C4—28no—
138C22×C4—21no—
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—22no—
149C22×C4—22no—
150C22×C4—22no—
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—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C2×D414yes—
178C22×C4C2×D414yes—
179C22×C4C2×D412yes—
180C22×C4C2×D414yes—
181C22×C4C2×D412yes—
182C22×C4#4112yes—
183C22×C4#4114yes—
184C22×C4#4112yes—
185C22×C4#4112yes—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—48no—
189C22⋊C4—216no—
190C22⋊C4—216no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C42—216no—
197C23—22no—
198C23—22no—
199C23—24no—
200C23—24no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—22no—
211C23—24no—
212C23—22no—
213C23—22no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—42no—
232C23—41no—
233C23—42no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23#12314yes—
243C23C22×D411yescharacteristic
244C23C22×D412yes—
245C23C2×#4112yes—
246C23C2×#4111yescharacteristic
247C23C2×#4111yescharacteristic
248C23C2×#4111yescharacteristic
249C23C2×#4111yescharacteristic
250C23C2×#4112yes—
251C23H(5,𝔽2)11yescharacteristic
252C23H(5,𝔽2)12yes—
253C232−1+411yescharacteristic
254D4—416no—
255C2×C4—232no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—41no—
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—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4#12014yes—
308C2×C4#12314yes—
309C22—24no—
310C22—24no—
311C22—24no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—81no—
326C22—82no—
327C22—82no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22C2×#12011yescharacteristic
332C22C2×#12311yescharacteristic
333C22C2×#12311yescharacteristic
334C22#52512yes—
335C22#52512yes—
336C22#52512yes—
337C22#52511yescharacteristic
338C22#53212yes—
339C22#53211yescharacteristic
340C22#53512yes—
341C22#53511yescharacteristic
342C22#53712yes—
343C22#53711yescharacteristic
344C22#55112yes—
345C22#55112yes—
346C22#55112yes—
347C22#55212yes—
348C22#55211yescharacteristic
349C22#55311yescharacteristic
350C22#55312yes—
351C22#55512yes—
352C22#55511yescharacteristic
353C22#55611yescharacteristic
354C4—24no—
355C4—24no—
356C4—44no—
357C4—44no—
358C4—44no—
359C4—42no—
360C4—82no—
361C4—81no—
362C4—81no—
363C2—44no—
364C2—82no—
365C2#238811yescharacteristic
366C2#248211yescharacteristic
367C2#248511yescharacteristic
368C2#249811yescharacteristic
369C2#250011yescharacteristic
370C2#259812yes—
371C2#260112yes—
372C2#260912yes—
373C2#261012yes—
374C2#261511yescharacteristic
375C2#284911yescharacteristic
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
1. C22×#117C2semidirect
2. C2×#388C2semidirect
4. #1444C2semidirect
5. #1442C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
10. #1434C2semidirect
11. C22⋊#95C2semidirect
89. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#3882
C2×#3862
#14442
#14422
#14392
#14382
#14342
C22⋊#952