Finite Group Databasebeta

← #27278

Subgroups — #27278

Order 256.

499 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#417C211yesmaximal, characteristic
3#2440C211yesmaximal, characteristic, complemented
4#2432C211yesmaximal, characteristic
5#2071C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#1956C211yesmaximal, characteristic, complemented
10#1935C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1894C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#489—24no—
25#417—24no—
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#398—21no—
42#395—21no—
43#394—21no—
44#394—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#385—21no—
50#385—21no—
51#385—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8—21no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86#124—22no—
87#124—41no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—41no—
123#103—41no—
124#103—41no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102D414yes—
129C22⋊C8—22no—
130#97—22no—
131C4×C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24C2×D411yescharacteristic
141#41—28no—
142#41—44no—
143#41—44no—
144#41—44no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—81no—
159C2×Q8#4112yes—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4C2×D412yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4#4112yes—
192C22×C4#4112yes—
193C22×C4C2411yescharacteristic
194M16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8C2×D412yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—81no—
225C4⋊C4C2×D412yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#12412yes—
252C23C22×D411yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—84no—
262D4—22no—
263D4—22no—
264D4—48no—
265D4—41no—
266D4—41no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—48no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—84no—
306C2×C4#12412yes—
307C2×C4AGL(1,ℤ/8)12yes—
308C2×C4#13712yes—
309C2×C4C22×D411yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C8—22no—
316C8—41no—
317C8—41no—
318C8—41no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C2×#12411yescharacteristic
334C22D4211yescharacteristic
335C22#54611yescharacteristic
336C22C2×AGL(1,ℤ/8)11yescharacteristic
337C22C2×#13711yescharacteristic
338C22#57711yescharacteristic
339C22#57811yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—41no—
345C4—42no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4#52511yescharacteristic
353C4#53811yescharacteristic
354C4#54611yescharacteristic
355C4#55111yescharacteristic
356C2—42no—
357C2—82no—
358C2#247211yescharacteristic
359C2#313711yescharacteristic
360C2#314111yescharacteristic
361C2#327211yescharacteristic
362C2#327611yescharacteristic
363C2#330811yescharacteristic
364C2#331611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#489C2semidirect
3. #2440C2semidirect
5. #2071C2semidirect
6. #2043C2semidirect
7. #2042C2semidirect
8. #2024C2semidirect
9. #1956C2semidirect
10. #1935C2semidirect
11. #1934C2semidirect
12. #1894C2semidirect
13. #1872C2semidirect
15. #1753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4172
#24402
#24322
#20712
#20432
#20422
#20242
#19562
#19352
#19342
#18942
#18722
#17882
#17532