Finite Group Databasebeta

← #27508

Subgroups — #27508

Order 256.

338 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2513C211yesmaximal, characteristic
2#2355C211yesmaximal, characteristic, complemented
3#2097C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2083C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic
7#1973C211yesmaximal, characteristic
8#1949C211yesmaximal, characteristic, complemented
9#1927C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic
11#1925C211yesmaximal, characteristic
12#1917C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic
14#1863C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#97C2211yescharacteristic
31#397—21no—
32#397—21no—
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#388—21no—
41#387—21no—
42#387—21no—
43#384C2211yescharacteristic
44#382—21no—
45#382C2211yescharacteristic
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C2×SD16—22no—
62C2×SD16—22no—
63C2×SD16—41no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C4×Q8—22no—
69C4×Q8—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#107—22no—
96#107—41no—
97#107—41no—
98#106—22no—
99#106—41no—
100C4⋊C8—22no—
101#103—22no—
102#103—22no—
103#103—41no—
104#102—22no—
105#102—22no—
106#102—41no—
107#97—22no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8#4112yes—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147SD16—44no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8#4112yes—
159C2×C8#4112yes—
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4#4112yes—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C22⋊C4—81no—
184C42—24no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—81no—
192C42—81no—
193C42—81no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23C22×D411yescharacteristic
202Q8—22no—
203Q8—22no—
204Q8—41no—
205Q8—41no—
206D4—41no—
207D4—41no—
208D4—41no—
209D4—41no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4#13712yes—
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4C2×#4111yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C2×C42−1+411yescharacteristic
255C8—22no—
256C8—22no—
257C8—41no—
258C8—41no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22#52511yescharacteristic
267C22#54711yescharacteristic
268C22#55011yescharacteristic
269C22C2×#13711yescharacteristic
270C22#57511yescharacteristic
271C22#57611yescharacteristic
272C22#57711yescharacteristic
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—42no—
280C4—42no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4#52911yescharacteristic
287C4#54211yescharacteristic
288C4#55111yescharacteristic
289C4#55211yescharacteristic
290C2—161no—
291C2#247911yescharacteristic
292C2#311711yescharacteristic
293C2#311911yescharacteristic
294C2#335011yescharacteristic
295C2#335711yescharacteristic
296C2#340011yescharacteristic
297C2#340311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2355C2semidirect
3. #2097C2semidirect
5. #2083C2semidirect
8. #1949C2semidirect
9. #1927C2semidirect
12. #1917C2semidirect
14. #1863C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
#25132
#23552
#20972
#20942
#20832
#20552
#19732
#19492
#19272
#19252
#19172
#18802
#18632
#17762