Finite Group Databasebeta

← #29557

Subgroups — #29557

Order 256.

371 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2861C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic, complemented
3#2059C211yesmaximal, characteristic, complemented
4#2042C211yesmaximal, characteristic
5#2026C211yesmaximal, characteristic, complemented
6#2015C211yesmaximal, characteristic
7#1969C211yesmaximal, characteristic, complemented
8#1968C211yesmaximal, characteristic
9#1934C211yesmaximal, characteristic
10#1933C211yesmaximal, characteristic, complemented
11#1927C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic
13#1880C211yesmaximal, characteristic
14#1878C211yesmaximal, characteristic, complemented
15#1744C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#391C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#388—21no—
44#388—21no—
45#387—21no—
46#387—21no—
47C4×C4⋊C4—21no—
48C4×C4⋊C4C2211yescharacteristic
49#376—21no—
50C43C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C2×Q16—22no—
63C2×Q16—22no—
64C2×Q16—41no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#125—41no—
73#125—41no—
74#124—41no—
75#124—41no—
76C4×Q8—22no—
77C4×Q8—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42D411yescharacteristic
97C2×C42D411yescharacteristic
98C2×C42D411yescharacteristic
99C2×C42D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—41no—
104#106—41no—
105#103—22no—
106#103—22no—
107#103—41no—
108#103—41no—
109#103—41no—
110#102—22no—
111#102—22no—
112#102—41no—
113C4×C8—22no—
114C4×C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8#4112yes—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159Q16—44no—
160SD16—44no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8#4112yes—
172C2×C8#4112yes—
173C4⋊C4—24no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4#4112yes—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C42—24no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—42no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23C22×D411yescharacteristic
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—41no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4#13512yes—
263C2×C4#13512yes—
264C2×C4#13712yes—
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C8—22no—
272C8—22no—
273C8—41no—
274C8—41no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22#52511yescharacteristic
283C22#53211yescharacteristic
284C22#53511yescharacteristic
285C22C2×#13511yescharacteristic
286C22C2×#13511yescharacteristic
287C22C2×#13711yescharacteristic
288C22#57511yescharacteristic
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#55111yescharacteristic
303C4#55111yescharacteristic
304C4#55511yescharacteristic
305C4#55511yescharacteristic
306C2—161no—
307C2#261011yescharacteristic
308C2#310211yescharacteristic
309C2#310811yescharacteristic
310C2#316811yescharacteristic
311C2#318111yescharacteristic
312C2#321011yescharacteristic
313C2#321211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2295C2semidirect
3. #2059C2semidirect
5. #2026C2semidirect
7. #1969C2semidirect
10. #1933C2semidirect
11. #1927C2semidirect
14. #1878C2semidirect
15. #1744C2semidirect

Maximal subgroups

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

subgroupindex
#28612
#22952
#20592
#20422
#20262
#20152
#19692
#19682
#19342
#19332
#19272
#19252
#18802
#18782
#17442