Finite Group Databasebeta

← #31839

Subgroups — #31839

Order 256.

437 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2499C211yesmaximal, characteristic, complemented
2#2428C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2073C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1957C211yesmaximal, characteristic, complemented
10#1933C211yesmaximal, characteristic, complemented
11#1923C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#397—21no—
38#397—21no—
39#394—21no—
40#392—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#387—21no—
46#387—21no—
47#386—21no—
48#384C2211yescharacteristic
49#382C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×SD16—41no—
73C2×SD16—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#127—22no—
79#127—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82#121—22no—
83#121—22no—
84#121—44no—
85#121—41no—
86#121—41no—
87C4×D4—22no—
88C4×D4—41no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C4⋊C8—41no—
113#103—22no—
114#103—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—
123C4×C8—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
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4#4112yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
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—42no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2411yescharacteristic
187SD16—44no—
188SD16—44no—
189C2×C8—21no—
190C2×C8—21no—
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—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4#4112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—42no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23C22×D411yescharacteristic
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255D4—22no—
256D4—22no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—84no—
264D4—84no—
265D4—84no—
266D4—84no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
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—84no—
303C2×C4#13512yes—
304C2×C4AGL(1,ℤ/8)12yes—
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C8—41no—
312C8—41no—
313C8—41no—
314C8—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22#53511yescharacteristic
337C22#54611yescharacteristic
338C22#54611yescharacteristic
339C22C2×#13511yescharacteristic
340C22C2×AGL(1,ℤ/8)11yescharacteristic
341C22#57711yescharacteristic
342C22#57711yescharacteristic
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4#53811yescharacteristic
355C4#54011yescharacteristic
356C4#55311yescharacteristic
357C4#56011yescharacteristic
358C2—81no—
359C2—82no—
360C2—161no—
361C2#270311yescharacteristic
362C2#320711yescharacteristic
363C2#322111yescharacteristic
364C2#329611yescharacteristic
365C2#330411yescharacteristic
366C2#331011yescharacteristic
367C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2428C2
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. #2499C2semidirect
3. #2099C2semidirect
4. #2073C2semidirect
5. #2055C2semidirect
6. #2014C2semidirect
7. #2000C2semidirect
8. #2000C2semidirect
9. #1957C2semidirect
10. #1933C2semidirect
11. #1923C2semidirect
12. #1877C2semidirect
13. #1875C2semidirect
14. #1875C2semidirect
15. #1804C2semidirect

Maximal subgroups

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

subgroupindex
#24992
#24282
#20992
#20732
#20552
#20142
#20002
#19572
#19332
#19232
#18772
#18752
#18042