Finite Group Databasebeta

← #16884

Subgroups — #16884

Order 256.

811 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386C211yesmaximal, characteristic
2C2×#380C211yesmaximal, characteristic, complemented
3#1443C212yesmaximal, complemented
4#1442C211yesmaximal, characteristic, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1440C212yesmaximal, complemented
7#1440C212yesmaximal, complemented
8#1439C211yesmaximal, characteristic, complemented
9#1439C211yesmaximal, characteristic, complemented
10#1438C212yesmaximal
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23#386—24no—
24#380—24no—
25C2×#95C2211yescharacteristic
26C2×#95C2211yescharacteristic
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C25C2311yescharacteristic
38C25C2311yescharacteristic
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—24no—
42C22×D4—22no—
43C22×D4—42no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—42no—
47C23×C4—42no—
48C23×C4D414yes—
49C23×C4C2312yes—
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81#95—24no—
82#95—22no—
83#95—22no—
84#95—24no—
85#95—22no—
86#95—22no—
87#95—42no—
88#95—42no—
89#95—42no—
90#95—42no—
91#95—42no—
92#95—42no—
93C24—21no—
94C24—21no—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—21no—
99C24—22no—
100C24—22no—
101C24—21no—
102C24—21no—
103C24—21no—
104C24—21no—
105C24—21no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—21no—
110C24—22no—
111C24—21no—
112C24C2×D412yes—
113C24#4112yes—
114C24#4114yes—
115C24C2411yescharacteristic
116C2×D4—28no—
117C2×D4—48no—
118C2×D4—48no—
119C2×D4—44no—
120C2×D4—44no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—28no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
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—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4#4112yes—
177C22×C4#4112yes—
178C22×C4#4112yes—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C22⋊C4—44no—
182C22⋊C4—48no—
183C22⋊C4—48no—
184C22⋊C4—48no—
185C22⋊C4—48no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—24no—
199C23—24no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23#12414yes—
234C23C22×D411yescharacteristic
235C23C2×#4111yescharacteristic
236C23C2×#4111yescharacteristic
237C23C2×#4111yescharacteristic
238C23C2×#4111yescharacteristic
239C23C2×#4112yes—
240C23H(5,𝔽2)12yes—
241C23H(5,𝔽2)12yes—
242C23H(5,𝔽2)12yes—
243C23H(5,𝔽2)11yescharacteristic
244C232−1+411yescharacteristic
245D4—416no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—416no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
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—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4#12314yes—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—44no—
316C22—81no—
317C22—81no—
318C22—82no—
319C22—81no—
320C22—82no—
321C22—81no—
322C22C2×#12311yescharacteristic
323C22#52511yescharacteristic
324C22C2×#12411yescharacteristic
325C22#53411yescharacteristic
326C22#53411yescharacteristic
327C22#53511yescharacteristic
328C22#53711yescharacteristic
329C22#53812yes—
330C22#53812yes—
331C22#53812yes—
332C22#53812yes—
333C22#53912yes—
334C22#54012yes—
335C22#55112yes—
336C22#55112yes—
337C22#55211yescharacteristic
338C22#55311yescharacteristic
339C22#55312yes—
340C22#55312yes—
341C22#55511yescharacteristic
342C22#55611yescharacteristic
343C22#56012yes—
344C22#56212yes—
345C4—24no—
346C4—42no—
347C4—42no—
348C4—82no—
349C4—81no—
350C4—82no—
351C4—81no—
352C4—82no—
353C4—82no—
354C2—44no—
355C2—82no—
356C2#248511yescharacteristic
357C2#250411yescharacteristic
358C2#256111yescharacteristic
359C2#262211yescharacteristic
360C2#263511yescharacteristic
361C2#263911yescharacteristic
362C2#275012yes—
363C2#275612yes—
364C2#277812yes—
365C2#278612yes—
366C2#281311yescharacteristic
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×#380C2semidirect
3. #1443C2semidirect
4. #1442C2semidirect
5. #1442C2semidirect
6. #1440C2semidirect
7. #1440C2semidirect
8. #1439C2semidirect
9. #1439C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3862
C2×#3802
#14432
#14422
#14402
#14392
#14382
C22⋊#952