Finite Group Databasebeta

← #24210

Subgroups — #24210

Order 256.

783 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#532C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic, complemented
4C2×#449C212yesmaximal, complemented
5C4×#121C211yesmaximal, characteristic, complemented
6#2042C212yesmaximal, complemented
7#2012C212yesmaximal, complemented
8#1982C211yesmaximal, characteristic, complemented
9#1981C211yesmaximal, characteristic, complemented
10#1797C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×SD16C2212yes—
14C22×D8C2212yes—
15#532—24no—
16#532—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#124C2212yes—
20C2×#121—21no—
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4—21no—
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C22×C42C2211yescharacteristic
28#461—24no—
29#459—24no—
30#449—28no—
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#102C2212yes—
37C2×#102C2212yes—
38C2×#102C2212yes—
39C2×C22⋊C8C2212yes—
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#386—21no—
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4—21no—
47#336C2212yes—
48C2×#41—24no—
49C2×#41—24no—
50C2×#41—24no—
51C2×#41—24no—
52C2×#41—42no—
53C2×#41—42no—
54C22×Q8C2312yes—
55C22×D4—21no—
56C22×D4—22no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4C2311yescharacteristic
60C22×D4C2312yes—
61C23×C4—22no—
62C23×C4—21no—
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4C2311yescharacteristic
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—42no—
69C2×D8—24no—
70C2×D8—24no—
71C2×D8—42no—
72C22×C8C2312yes—
73C22×C8C2312yes—
74C4⋊Q8—22no—
75C4⋊Q8D414yescomplemented
76#127—22no—
77#127D414yescomplemented
78#124—24no—
79#124—24no—
80#121—28no—
81#121—28no—
82#121—22no—
83#121D414yescomplemented
84C4×D4—28no—
85C4×D4—28no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—44no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42D411yescharacteristic
106C2×C42D411yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C4⋊C8—22no—
110C4⋊C8—22no—
111#103—24no—
112#103—24no—
113#103—42no—
114#102—24no—
115#102—24no—
116#102—42no—
117C22⋊C8—24no—
118#95—21no—
119#95—21no—
120#95—21no—
121C24—21no—
122C24—22no—
123C24C2×D411yescharacteristic
124#41—48no—
125#41—48no—
126#41—48no—
127#41—48no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8C2×D414yes—
134C2×D4—28no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4C2×D414yes—
157C2×D4C2×D412yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—28no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D412yes—
183C22×C4C2×D411yescharacteristic
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186SD16—48no—
187D8—48no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4C2×D412yes—
203C22⋊C4—28no—
204C22⋊C4—28no—
205C22⋊C4—24no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C42—22no—
211C42—22no—
212C42—21no—
213C42—21no—
214C42—42no—
215C42C2×D412yes—
216C42C2×D412yes—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—24no—
224C23—22no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—42no—
231C23—41no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—41no—
237C23#12112yes—
238C23C22×D411yescharacteristic
239Q8—24no—
240Q8—24no—
241Q8—42no—
242Q8—42no—
243D4—22no—
244D4—24no—
245D4—22no—
246D4—24no—
247D4—48no—
248D4—48no—
249D4—416no—
250D4—42no—
251D4—41no—
252D4—41no—
253D4—42no—
254D4—48no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—44no—
272C2×C4—416no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—84no—
285C2×C4#12012yes—
286C2×C4#12012yes—
287C2×C4#12112yes—
288C2×C4#13512yes—
289C2×C4AGL(1,ℤ/8)12yes—
290C2×C4C22×D411yescharacteristic
291C2×C4C22×D411yescharacteristic
292C2×C4C22×D411yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C8—42no—
297C8—42no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—44no—
306C22—42no—
307C22—44no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—81no—
313C22—82no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—82no—
318C22—82no—
319C22C2×#12011yescharacteristic
320C22C2×#12111yescharacteristic
321C22#53511yescharacteristic
322C22C2×#13511yescharacteristic
323C22C2×AGL(1,ℤ/8)11yescharacteristic
324C22#57511yescharacteristic
325C22#57511yescharacteristic
326C4—22no—
327C4—22no—
328C4—44no—
329C4—41no—
330C4—41no—
331C4—42no—
332C4—81no—
333C4—81no—
334C4#44914yescomplemented
335C4D4211yescharacteristic
336C4D4211yescharacteristic
337C4#54611yescharacteristic
338C4#54611yescharacteristic
339C2—42no—
340C2—84no—
341C2—82no—
342C2#240811yescharacteristic
343C2C2×#44911yescharacteristic
344C2#300511yescharacteristic
345C2#303511yescharacteristic
346C2#303611yescharacteristic
347C2#321211yescharacteristic
348C2#321411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#532C2semidirect
2. C2×#461C2semidirect
3. C2×#459C2semidirect
4. C2×#449C2semidirect
5. C4×#121C2semidirect
6. #2042C2semidirect
7. #2012C2semidirect
8. #1982C2semidirect
9. #1981C2semidirect
10. #1797C2semidirect
11. #1788C2semidirect
75. C4⋊Q8D4semidirect
77. #127D4semidirect
83. #121D4semidirect
334. C4#449semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4612
C2×#4592
C2×#4492
C4×#1212
#20422
#20122
#19822
#19812
#17972
#17882