Finite Group Databasebeta

← #27943

Subgroups — #27943

Order 256.

423 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic
3C2×#470C211yesmaximal, characteristic
4C2×#428C211yesmaximal, characteristic
5#2503C211yesmaximal, characteristic
6#2499C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2097C211yesmaximal, characteristic, complemented
9#2097C211yesmaximal, characteristic, complemented
10#2095C211yesmaximal, characteristic, complemented
11#1971C211yesmaximal, characteristic, complemented
12#1969C211yesmaximal, characteristic, complemented
13#1847C211yesmaximal, characteristic, complemented
14#1846C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#485—24no—
23#484—24no—
24#470—24no—
25#428—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#400—22no—
45#397—22no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#387—22no—
51#384C2211yescharacteristic
52#382C2211yescharacteristic
53#382C2211yescharacteristic
54#382C2211yescharacteristic
55#381—22no—
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C2×Q16—22no—
64C2×Q16—22no—
65C2×Q16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16D414yes—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#117—22no—
78#117—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D412yes—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107—22no—
100#107—22no—
101#107—22no—
102#107—22no—
103#106—22no—
104#106—22no—
105#106—22no—
106#106—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113C22⋊C8D414yes—
114#97—22no—
115C4×C8—22no—
116#95—22no—
117#95—22no—
118#95—22no—
119#95—22no—
120#95—22no—
121#95—22no—
122#95—22no—
123#95D412yes—
124C24C2×D411yescharacteristic
125C2×Q8—21no—
126C2×Q8—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—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D412yes—
155C22×C4C2×D412yes—
156C22×C4C2411yescharacteristic
157Q16—44no—
158M16—28no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8C2×D412yes—
166C2×C8C2×D412yes—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C2×C8#4112yes—
170C2×C8#4112yes—
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—21no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23#12712yes—
201C23C22×D411yescharacteristic
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—41no—
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—84no—
241C2×C4#12712yes—
242C2×C4C22×D411yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4C2×#4111yescharacteristic
246C2×C42−1+411yescharacteristic
247C2×C42−1+411yescharacteristic
248C8—22no—
249C8—22no—
250C8—22no—
251C8#12314yes—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22C2×#12711yescharacteristic
258C22#54711yescharacteristic
259C22#54711yescharacteristic
260C22#57512yes—
261C22#57812yes—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4C2×#12311yescharacteristic
274C4#53611yescharacteristic
275C4#54711yescharacteristic
276C4#54711yescharacteristic
277C2—42no—
278C2#250111yescharacteristic
279C2#315311yescharacteristic
280C2#315611yescharacteristic
281C2#334912yes—
282C2#335212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
7. #2097C2semidirect
8. #2097C2semidirect
9. #2097C2semidirect
10. #2095C2semidirect
11. #1971C2semidirect
12. #1969C2semidirect
13. #1847C2semidirect
14. #1846C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4852
C2×#4702
C2×#4282
#25032
#24992
#20972
#20952
#19712
#19692
#18472
#18462
#17532