Finite Group Databasebeta

← #27565

Subgroups — #27565

Order 256.

376 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2511C211yesmaximal, characteristic, complemented
2#2349C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#1933C211yesmaximal, characteristic, complemented
8#1921C211yesmaximal, characteristic, complemented
9#1916C211yesmaximal, characteristic, complemented
10#1877C211yesmaximal, characteristic
11#1877C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic
13#1862C211yesmaximal, characteristic
14#1802C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#400—21no—
30#400—21no—
31#391C2211yescharacteristic
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#385—21no—
40#385—21no—
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43#380—21no—
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C2×#95—21no—
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—41no—
76C4×D4—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—22no—
98C4⋊C8—41no—
99#103—22no—
100#103—41no—
101#103—41no—
102#102—22no—
103#102—22no—
104#102—22no—
105C4×C8—41no—
106#95—21no—
107#95—21no—
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—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4#4112yes—
131C2×D4#4112yes—
132C2×D4#4112yes—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156SD16—44no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
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—42no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4#4112yes—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—81no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23C22×D411yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210D4—22no—
211D4—22no—
212D4—22no—
213D4#12614yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
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—21no—
230C2×C4—21no—
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—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4#13512yes—
255C2×C4#13712yes—
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C42−1+411yescharacteristic
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22#52511yescharacteristic
274C22#54611yescharacteristic
275C22#54811yescharacteristic
276C22C2×#13511yescharacteristic
277C22C2×#13711yescharacteristic
278C22#57611yescharacteristic
279C22#57711yescharacteristic
280C4—22no—
281C4—22no—
282C4—41no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4C2×#12611yescharacteristic
294C4#53911yescharacteristic
295C4#55211yescharacteristic
296C4#55311yescharacteristic
297C2—44no—
298C2#248111yescharacteristic
299C2#310711yescharacteristic
300C2#311811yescharacteristic
301C2#329611yescharacteristic
302C2#331411yescharacteristic
303C2#336111yescharacteristic
304C2#337211yescharacteristic
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
1. #2511C2semidirect
3. #2101C2semidirect
4. #2099C2semidirect
5. #2074C2semidirect
6. #2068C2semidirect
7. #1933C2semidirect
8. #1921C2semidirect
9. #1916C2semidirect

Maximal subgroups

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

subgroupindex
#25112
#23492
#21012
#20992
#20742
#20682
#19332
#19212
#19162
#18772
#18622
#18022
C4×#1022