Finite Group Databasebeta

← #27941

Subgroups — #27941

Order 256.

493 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#469C211yesmaximal, characteristic
4C2×#428C211yesmaximal, characteristic, complemented
5#2501C211yesmaximal, characteristic, complemented
6#2500C211yesmaximal, characteristic, complemented
7#2094C211yesmaximal, characteristic, complemented
8#2094C211yesmaximal, characteristic, complemented
9#2093C211yesmaximal, characteristic, complemented
10#2092C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1970C211yesmaximal, characteristic, complemented
13#1847C211yesmaximal, characteristic, complemented
14#1844C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#123—22no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#482—24no—
24#481—24no—
25#469—24no—
26#428—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#399—22no—
46#394—22no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#388—22no—
52#384C2211yescharacteristic
53#384C2211yescharacteristic
54#384C2211yescharacteristic
55#382C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×D8—22no—
66C2×D8—22no—
67C2×D8—22no—
68C2×M16—22no—
69C2×M16—22no—
70C2×M16D414yes—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#123—48no—
76#121—22no—
77#121—22no—
78#121—22no—
79#121—22no—
80#117—22no—
81#117—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C2311yescharacteristic
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—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4D412yes—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—22no—
107#107—22no—
108#106—22no—
109#106—22no—
110#106—22no—
111#106—22no—
112#106—22no—
113#106—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120C22⋊C8D414yes—
121#97—22no—
122C4×C8—22no—
123#95—22no—
124#95D412yes—
125C24—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—82no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4C2×D412yes—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2411yescharacteristic
162D8—44no—
163M16—28no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8C2×D412yes—
171C2×C8C2×D412yes—
172C2×C8C2×D412yes—
173C2×C8C2×D412yes—
174C2×C8#4112yes—
175C2×C8#4112yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—82no—
192C22⋊C4—24no—
193C22⋊C4—44no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—82no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23#12712yes—
220C23C22×D411yescharacteristic
221D4—41no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—84no—
230D4—84no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4#12712yes—
256C2×C4AGL(1,ℤ/8)14yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C42−1+411yescharacteristic
262C2×C42−1+411yescharacteristic
263C8—22no—
264C8—22no—
265C8—22no—
266C8#12314yescomplemented
267C22—22no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—42no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22C2×#12711yescharacteristic
283C22#54711yescharacteristic
284C22#54711yescharacteristic
285C22C2×AGL(1,ℤ/8)12yes—
286C22#57712yes—
287C4—24no—
288C4—22no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4C2×#12311yescharacteristic
297C4#53611yescharacteristic
298C4#54711yescharacteristic
299C4#54711yescharacteristic
300C2—42no—
301C2—161no—
302C2—161no—
303C2#250111yescharacteristic
304C2#315111yescharacteristic
305C2#315411yescharacteristic
306C2#334512yes—
307C2#335412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
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. C2×#481C2semidirect
2. C2×#482C2semidirect
4. C2×#428C2semidirect
5. #2501C2semidirect
6. #2500C2semidirect
7. #2094C2semidirect
8. #2094C2semidirect
9. #2093C2semidirect
10. #2092C2semidirect
11. #1972C2semidirect
12. #1970C2semidirect
13. #1847C2semidirect
14. #1844C2semidirect
15. #1753C2semidirect
266. C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4822
C2×#4692
C2×#4282
#25012
#25002
#20942
#20932
#20922
#19722
#19702
#18472
#18442
#17532