Finite Group Databasebeta

← #31287

Subgroups — #31287

Order 256.

426 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic, complemented
2#2432C211yesmaximal, characteristic
3#2093C211yesmaximal, characteristic, complemented
4#2068C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#1983C211yesmaximal, characteristic, complemented
10#1956C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#126—21no—
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#400—21no—
40#397—21no—
41#394—21no—
42#394—21no—
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#387—21no—
48#385—21no—
49#385—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×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77#126—44no—
78#124—22no—
79#124—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
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#107—41no—
107#107—41no—
108C4⋊C8—41no—
109#103—22no—
110#103—41no—
111#103—41no—
112#103—41no—
113#103—41no—
114#103—41no—
115#103—41no—
116#103—41no—
117#102—22no—
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—
134C24C2×D411yescharacteristic
135#41—44no—
136#41—44no—
137#41—44no—
138#41—44no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8#4112yes—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4#4112yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2411yescharacteristic
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—42no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23C22×D411yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248D4—22no—
249D4—22no—
250D4—48no—
251D4—41no—
252D4—41no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—48no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
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#13512yes—
293C2×C4AGL(1,ℤ/8)12yes—
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C22—41no—
305C22—42no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—42no—
313C22—41no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#53511yescharacteristic
319C22#54611yescharacteristic
320C22#54611yescharacteristic
321C22C2×#13511yescharacteristic
322C22C2×AGL(1,ℤ/8)11yescharacteristic
323C22#57811yescharacteristic
324C22#57811yescharacteristic
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—42no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4#53811yescharacteristic
338C4#53811yescharacteristic
339C4#55111yescharacteristic
340C4#56011yescharacteristic
341C2—81no—
342C2—82no—
343C2#268511yescharacteristic
344C2#320711yescharacteristic
345C2#322311yescharacteristic
346C2#330011yescharacteristic
347C2#330211yescharacteristic
348C2#330811yescharacteristic
349C2#333011yescharacteristic
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. #2496C2semidirect
3. #2093C2semidirect
4. #2068C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #2036C2semidirect
8. #2036C2semidirect
9. #1983C2semidirect
10. #1956C2semidirect
11. #1934C2semidirect
12. #1928C2semidirect
13. #1923C2semidirect

Maximal subgroups

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

subgroupindex
#24962
#24322
#20932
#20682
#20432
#20422
#20362
#19832
#19562
#19342
#19282
#19232
#17882