Finite Group Databasebeta

← #21125

Subgroups — #21125

Order 256.

445 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#440C211yesmaximal, characteristic
2#2320C211yesmaximal, characteristic, complemented
3#2275C211yesmaximal, characteristic
4#1928C211yesmaximal, characteristic, complemented
5#1927C211yesmaximal, characteristic, complemented
6#1921C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1877C211yesmaximal, characteristic
9#1877C211yesmaximal, characteristic
10#1864C211yesmaximal, characteristic
11#1862C211yesmaximal, characteristic
12#1776C211yesmaximal, characteristic, complemented
13C4×#103C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic
15#1746C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#440C418yescomplemented
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#383—21no—
40#382—21no—
41C4×C4⋊C4C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48#376C2211yescharacteristic
49C2×#95—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16C2×C414yes—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4×Q8—22no—
73C4×Q8C2×C414yescomplemented
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4C2×C414yes—
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×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#107C2×C414yescomplemented
102#106—22no—
103#103—22no—
104#103—22no—
105#103C2×C414yescomplemented
106#102—22no—
107#102—22no—
108#102C2×C414yes—
109#97C2×C414yescomplemented
110C4×C8—22no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95D411yescharacteristic
122#95D411yescharacteristic
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8C22×C412yes—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4C22×C412yes—
136C2×D4#4112yes—
137C2×D4#4112yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161SD16—28no—
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—21no—
172C2×C8C22×C412yes—
173C2×C8C22×C412yes—
174C4⋊C4—24no—
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—81no—
184C4⋊C4—81no—
185C4⋊C4C22×C412yes—
186C4⋊C4C22×C412yes—
187C4⋊C4#4112yes—
188C4⋊C4#4112yes—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C42—24no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—81no—
198C42—81no—
199C42—81no—
200C42—81no—
201C42C22×C412yes—
202C42#4112yes—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23C22×D411yescharacteristic
211Q8—22no—
212Q8—22no—
213Q8—41no—
214Q8—41no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4#11714yes—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4C4×D412yes—
260C2×C4C4×D412yes—
261C2×C4C23×C411yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C8—22no—
268C8—22no—
269C8—41no—
270C8—41no—
271C22—24no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22C2×C4×D411yescharacteristic
279C22#52511yescharacteristic
280C22#52511yescharacteristic
281C22#57611yescharacteristic
282C22#57711yescharacteristic
283C22#57711yescharacteristic
284C22#57811yescharacteristic
285C4—22no—
286C4—22no—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—41no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—81no—
297C4—81no—
298C4C2×#11711yescharacteristic
299C4C4×#4111yescharacteristic
300C4#52911yescharacteristic
301C4#52911yescharacteristic
302C2—44no—
303C2#229011yescharacteristic
304C2#294611yescharacteristic
305C2#294711yescharacteristic
306C2#311811yescharacteristic
307C2#312011yescharacteristic
308C2#312311yescharacteristic
309C2#312411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2320C2semidirect
4. #1928C2semidirect
5. #1927C2semidirect
6. #1921C2semidirect
7. #1918C2semidirect
12. #1776C2semidirect
13. C4×#103C2semidirect
15. #1746C2semidirect
21. #440C4semidirect
73. C4×Q8C2×C4semidirect
101. #107C2×C4semidirect
105. #103C2×C4semidirect
109. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
#23202
#22752
#19282
#19272
#19212
#19182
#18772
#18642
#18622
#17762
C4×#1032
C4×#1022
#17462