Finite Group Databasebeta

← #31307

Subgroups — #31307

Order 256.

452 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic, complemented
2#2431C211yesmaximal, characteristic
3#2094C211yesmaximal, characteristic, complemented
4#2066C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1924C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#126—21no—
19C2×#124C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#400—21no—
40#397—21no—
41#392—21no—
42#392—21no—
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#387—21no—
48#386—21no—
49#386—21no—
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—41no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#127—22no—
79#127—41no—
80#126—44no—
81#124—22no—
82#124—41no—
83#121—41no—
84#121—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#106—41no—
107#106—41no—
108C4⋊C8—41no—
109#103—22no—
110#103—41no—
111#103—41no—
112#103—41no—
113#102—22no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120#102—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123#97—41no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24—21no—
135C24C2×D411yescharacteristic
136#41—44no—
137#41—44no—
138#41—44no—
139#41—44no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8#4112yes—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4#4112yes—
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—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2411yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C22⋊C4—22no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—42no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C22×D411yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—41no—
256Q8—41no—
257D4—22no—
258D4—22no—
259D4—48no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—84no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—48no—
285C2×C4—41no—
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—41no—
306C2×C4AGL(1,ℤ/8)12yes—
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—41no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22#53511yescharacteristic
339C22#54611yescharacteristic
340C22#54611yescharacteristic
341C22C2×AGL(1,ℤ/8)11yescharacteristic
342C22#57511yescharacteristic
343C22#57711yescharacteristic
344C22#57711yescharacteristic
345C4—22no—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4#53811yescharacteristic
357C4#53811yescharacteristic
358C4#55111yescharacteristic
359C4#56011yescharacteristic
360C2—81no—
361C2—82no—
362C2—161no—
363C2#268511yescharacteristic
364C2#321411yescharacteristic
365C2#322211yescharacteristic
366C2#330611yescharacteristic
367C2#331111yescharacteristic
368C2#331711yescharacteristic
369C2#333211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2431C2
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. #2496C2semidirect
3. #2094C2semidirect
4. #2066C2semidirect
5. #2042C2semidirect
6. #2041C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #1982C2semidirect
10. #1959C2semidirect
11. #1936C2semidirect
12. #1927C2semidirect
13. #1924C2semidirect
14. #1788C2semidirect
15. #1788C2semidirect

Maximal subgroups

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

subgroupindex
#24962
#24312
#20942
#20662
#20422
#20412
#20342
#19822
#19592
#19362
#19272
#19242
#17882