Finite Group Databasebeta

← #27947

Subgroups — #27947

Order 256.

444 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic
4C2×#428C211yesmaximal, characteristic, complemented
5#2504C211yesmaximal, characteristic
6#2499C211yesmaximal, characteristic, complemented
7#2097C211yesmaximal, characteristic, complemented
8#2097C211yesmaximal, characteristic, complemented
9#2094C211yesmaximal, characteristic, complemented
10#2092C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1847C211yesmaximal, characteristic, complemented
14#1846C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#483—24no—
23#482—24no—
24#468—24no—
25#428—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#397—22no—
45#397—22no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#387—22no—
51#386—22no—
52#384C2211yescharacteristic
53#382C2211yescharacteristic
54#382C2211yescharacteristic
55#382C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×M16—22no—
68C2×M16—22no—
69C2×M16D414yes—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76#121—22no—
77#121—22no—
78#117—22no—
79#117—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
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⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4D412yes—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#107—22no—
102#107—22no—
103#107—22no—
104#106—22no—
105#106—22no—
106#106—22no—
107#106—22no—
108#106—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115C22⋊C8D414yes—
116#97—22no—
117C4×C8—22no—
118#95—22no—
119#95—22no—
120#95—22no—
121#95—22no—
122#95—22no—
123#95D412yes—
124C24—21no—
125C24C2×D411yescharacteristic
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—44no—
139C2×D4—42no—
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—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—82no—
153C22×C4C2×D412yes—
154C22×C4C2×D412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159SD16—44no—
160M16—28no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C2×C8#4112yes—
172C2×C8#4112yes—
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—21no—
184C4⋊C4—21no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—42no—
199C23—21no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23#12712yes—
209C23C22×D411yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—84no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
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—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4#12712yes—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C42−1+411yescharacteristic
253C2×C42−1+411yescharacteristic
254C8—22no—
255C8—22no—
256C8—22no—
257C8#12314yes—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22—82no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22C2×#12711yescharacteristic
269C22#54711yescharacteristic
270C22#54711yescharacteristic
271C22#57512yes—
272C22#57712yes—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4C2×#12311yescharacteristic
284C4#53611yescharacteristic
285C4#54711yescharacteristic
286C4#54711yescharacteristic
287C2—42no—
288C2—161no—
289C2#250111yescharacteristic
290C2#315311yescharacteristic
291C2#315411yescharacteristic
292C2#335012yes—
293C2#335112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#483C2semidirect
4. C2×#428C2semidirect
6. #2499C2semidirect
7. #2097C2semidirect
8. #2097C2semidirect
9. #2094C2semidirect
10. #2092C2semidirect
11. #1973C2semidirect
12. #1968C2semidirect
13. #1847C2semidirect
14. #1846C2semidirect
15. #1753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4832
C2×#4682
C2×#4282
#25042
#24992
#20972
#20942
#20922
#19732
#19682
#18472
#18462
#17532