Finite Group Databasebeta

← #14801

Subgroups — #14801

Order 256.

1127 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2C2×#398C211yesmaximal, characteristic, complemented
3C2×#394C211yesmaximal, characteristic
4C2×#387C211yesmaximal, characteristic, complemented
5C2×#385C212yesmaximal, complemented
6#1445C211yesmaximal, 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—24no—
15C2×C4×D4—24no—
16C2×C4×D4—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C42C2211yescharacteristic
27#398—24no—
28#394—24no—
29#387—24no—
30#385—28no—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C2×#95C2212yes—
42C25C2312yes—
43C22×D4—24no—
44C22×D4—24no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4D414yes—
48C23×C4—24no—
49C23×C4—24no—
50C23×C4—24no—
51C23×C4—24no—
52C23×C4—24no—
53C23×C4—42no—
54C23×C4—42no—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C4×D4—232no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4D414yes—
77C2×C4⋊C4D414yescomplemented
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42D414yes—
96#95—24no—
97#95—24no—
98#95—22no—
99#95—22no—
100#95—22no—
101#95—22no—
102#95—22no—
103#95—22no—
104#95—24no—
105#95—24no—
106#95—42no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24C2×D414yes—
120C24#4114yes—
121C24#4114yes—
122C24C2411yescharacteristic
123C2×D4—28no—
124C2×D4—28no—
125C2×D4—216no—
126C2×D4—216no—
127C2×D4—28no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—216no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—48no—
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D414yes—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4C2×D412yes—
193C22×C4#4114yes—
194C22×C4#4112yes—
195C4⋊C4—28no—
196C4⋊C4—28no—
197C4⋊C4—48no—
198C4⋊C4—44no—
199C22⋊C4—216no—
200C22⋊C4—216no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C22⋊C4—48no—
204C42—28no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23#12318yes—
240C23C22×D411yescharacteristic
241C23C22×D411yescharacteristic
242C23C22×D411yescharacteristic
243C23C22×D411yescharacteristic
244C23C2×#4111yescharacteristic
245C23C2×#4112yes—
246C23C2×#4112yes—
247C23C2×#4112yes—
248C23H(5,𝔽2)11yescharacteristic
249C23H(5,𝔽2)11yescharacteristic
250C232−1+412yes—
251D4—232no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—432no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4#12014yes—
312C2×C4#12114yes—
313C22—28no—
314C22—28no—
315C22—28no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22C2×#12011yescharacteristic
329C22C2×#12111yescharacteristic
330C22C2×#12312yes—
331C22#52511yescharacteristic
332C22#52511yescharacteristic
333C22#52512yes—
334C22#52512yes—
335C22#52912yes—
336C22#53211yescharacteristic
337C22#53511yescharacteristic
338C22#53611yescharacteristic
339C22#53912yes—
340C22#53912yes—
341C22D4211yescharacteristic
342C22D4211yescharacteristic
343C22#54611yescharacteristic
344C22#54611yescharacteristic
345C22#54712yes—
346C22#54812yes—
347C22#54812yes—
348C22#55112yes—
349C22#55212yes—
350C22#55312yes—
351C4—24no—
352C4—24no—
353C4—42no—
354C4—42no—
355C4—44no—
356C4—42no—
357C4—44no—
358C4—81no—
359C4—81no—
360C4—82no—
361C2—48no—
362C2#239211yescharacteristic
363C2#240811yescharacteristic
364C2#241912yes—
365C2#243211yescharacteristic
366C2#244312yes—
367C2#246012yes—
368C2#246312yes—
369C2#253611yescharacteristic
370C2#260311yescharacteristic
371C2#263311yescharacteristic
372C2#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
2. C2×#398C2semidirect
4. C2×#387C2semidirect
5. C2×#385C2semidirect
6. #1445C2semidirect
7. #1440C2semidirect
8. #1439C2semidirect
10. #1434C2semidirect
11. C22⋊#95C2semidirect
77. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#3982
C2×#3942
C2×#3872
C2×#3852
#14452
#14402
#14392
#14382
#14342
C22⋊#952