Finite Group Databasebeta

← #15384

Subgroups — #15384

Order 256.

1059 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C2×#399C211yesmaximal, characteristic, complemented
3C2×#397C211yesmaximal, characteristic, complemented
4C2×#394C212yesmaximal
5C2×#385C212yesmaximal, complemented
6C2×#384C211yesmaximal, characteristic, complemented
7#1445C211yesmaximal, characteristic, complemented
8#1444C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10#1437C211yesmaximal, characteristic, complemented
11#1437C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×#123—24no—
15C2×#123—24no—
16C2×#123—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#399—24no—
32#397—24no—
33#394—28no—
34#385—28no—
35#384—24no—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C25C2311yescharacteristic
44C25C2311yescharacteristic
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—24no—
48C22×D4—22no—
49C22×D4—42no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—24no—
54C23×C4—42no—
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2312yes—
64#123—232no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4D414yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4D414yes—
102C2×C22⋊C4D414yes—
103C2×C22⋊C4D418yes—
104C2×C42—22no—
105C2×C42—24no—
106#95—24no—
107#95—24no—
108#95—24no—
109#95—22no—
110#95—24no—
111#95—22no—
112#95—24no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—21no—
121C24—21no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24C2×D412yes—
129C24C2×D412yes—
130C24C2×D414yes—
131C24C2411yescharacteristic
132C2×D4—28no—
133C2×D4—48no—
134C2×D4—48no—
135C2×D4—44no—
136C2×D4—44no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—28no—
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—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
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—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4C2×D412yes—
185C22×C4C2×D414yes—
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4C2×D414yes—
193C22×C4#4114yes—
194C22×C4#4114yes—
195C4⋊C4—28no—
196C4⋊C4—28no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—48no—
200C4⋊C4—44no—
201C22⋊C4—28no—
202C22⋊C4—28no—
203C22⋊C4—28no—
204C22⋊C4—216no—
205C22⋊C4—48no—
206C22⋊C4—48no—
207C22⋊C4—48no—
208C42—28no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—24no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23#12714yes—
251C23C22×D411yescharacteristic
252C23C22×D411yescharacteristic
253C23C22×D411yescharacteristic
254C23C22×D412yes—
255C23C2×#4112yes—
256C23C2×#4112yes—
257C23H(5,𝔽2)12yes—
258C232−1+412yes—
259C232−1+412yes—
260D4—416no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—416no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4#12014yes—
314C2×C4#12318yescomplemented
315C22—24no—
316C22—28no—
317C22—42no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—42no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—44no—
326C22—44no—
327C22—42no—
328C22—42no—
329C22—82no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22C2×#12011yescharacteristic
334C22C2×#12312yes—
335C22#52511yescharacteristic
336C22C2×#12711yescharacteristic
337C22#53211yescharacteristic
338C22#53511yescharacteristic
339C22#53611yescharacteristic
340C22#53612yes—
341C22#53611yescharacteristic
342C22#53912yes—
343C22#54212yes—
344C22D4212yes—
345C22D4212yes—
346C22#54712yes—
347C22#54712yes—
348C22#54712yes—
349C22#54712yes—
350C22#54712yes—
351C22#54812yes—
352C22#55312yes—
353C22#56312yes—
354C4—24no—
355C4—28no—
356C4—44no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—82no—
362C4—82no—
363C2—44no—
364C2—82no—
365C2#240611yescharacteristic
366C2#242212yes—
367C2#243211yescharacteristic
368C2#243311yescharacteristic
369C2#244312yes—
370C2#250112yes—
371C2#267412yes—
372C2#268012yes—
373C2#271812yes—
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×#399C2semidirect
3. C2×#397C2semidirect
5. C2×#385C2semidirect
6. C2×#384C2semidirect
7. #1445C2semidirect
8. #1444C2semidirect
9. #1439C2semidirect
10. #1437C2semidirect
11. #1437C2semidirect
314. C2×C4#123semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#3992
C2×#3972
C2×#3942
C2×#3852
C2×#3842
#14452
#14442
#14392
#14372