Finite Group Databasebeta

← #20943

Subgroups — #20943

Order 256.

889 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575C211yesmaximal, characteristic
2C2×#520C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic, complemented
4#2896C211yesmaximal, characteristic, complemented
5#2311C211yesmaximal, characteristic, complemented
6#1883C214yesmaximal, complemented
7#1882C214yesmaximal, complemented
8C2×#343C211yesmaximal, characteristic, complemented
9#1733C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×#41C2211yescharacteristic
12#575—24no—
13#575C418yescomplemented
14C2×#137—21no—
15C2×#137C2212yes—
16C2×AGL(1,ℤ/8)—21no—
17C2×AGL(1,ℤ/8)C2212yes—
18C2×#135—22no—
19C22×M16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21#520—22no—
22#520—22no—
23#520—22no—
24#520C2214yes—
25C2×C4×Q8—21no—
26C2×C4×D4—21no—
27C2×C4×D4—22no—
28C2×#117—21no—
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2211yescharacteristic
33#421—24no—
34C2×#104C2214yes—
35#419C2212yescomplemented
36#418C2212yescomplemented
37#417—21no—
38#417—21no—
39#416—22no—
40#391C2212yescomplemented
41#390C2212yescomplemented
42#386—22no—
43#380—22no—
44#343C2214yescomplemented
45#339C2214yescomplemented
46C2×#41—21no—
47C2×#41—21no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—21no—
52C2×#41—21no—
53C2×#41—22no—
54C2×#41—21no—
55C2×#41—21no—
56C2×#41—22no—
57C2×#41C2×C414yes—
58C2×#41C2×C414yescomplemented
59C2×#41D414yes—
60C2×#41D412yes—
61C2×#41C2312yes—
62C22×Q8D411yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4D411yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4D412yes—
72C23×C4C2311yescharacteristic
73#137—24no—
74#137—28no—
75#137C2×C418yescomplemented
76AGL(1,ℤ/8)—24no—
77AGL(1,ℤ/8)—28no—
78AGL(1,ℤ/8)C2×C418yescomplemented
79#135—216no—
80C2×Q16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×D8—22no—
84C2×M16—21no—
85C2×M16—22no—
86C2×M16—21no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16C2×C414yescomplemented
90C2×M16C2312yes—
91C2×M16C2312yes—
92C22×C8—21no—
93C22×C8—21no—
94C4×Q8—24no—
95C4×Q8—24no—
96C4×D4—28no—
97C4×D4—24no—
98C4×D4—24no—
99C4×D4—28no—
100#117—24no—
101#117—21no—
102#117—21no—
103#117—24no—
104#117—22no—
105#117—22no—
106#117D412yes—
107#117C2312yes—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4D412yes—
117C2×C4⋊C4D411yescharacteristic
118C2×C4⋊C4D411yescharacteristic
119C2×C4⋊C4C2312yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—24no—
124C2×C42—22no—
125C2×C42—21no—
126C2×C42C2312yes—
127#104—28no—
128#103—22no—
129#103—22no—
130#102—22no—
131#102—22no—
132#95—24no—
133C24—21no—
134C24—21no—
135C24C2×D411yescharacteristic
136#41—24no—
137#41—24no—
138#41—24no—
139#41—24no—
140#41—24no—
141#41—22no—
142#41—22no—
143#41—24no—
144#41—24no—
145#41—24no—
146#41—24no—
147#41—42no—
148#41—42no—
149#41—42no—
150#41—42no—
151#41C22⋊C418yescomplemented
152#41C22×C414yes—
153#41C2×D414yes—
154C2×Q8—21no—
155C2×Q8—22no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊C414yescomplemented
162C2×Q8C22×C412yescomplemented
163C2×Q8C2×D412yes—
164C2×D4—24no—
165C2×D4—22no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4C22⋊C414yescomplemented
180C2×D4C22×C412yescomplemented
181C2×D4C2×D412yes—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—28no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—82no—
199C22×C4—84no—
200C22×C4C22×C412yes—
201C22×C4C2×D414yes—
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D412yes—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4#4112yes—
209C22×C4#4112yes—
210C22×C4C2411yescharacteristic
211Q16—28no—
212SD16—28no—
213SD16—28no—
214D8—28no—
215M16—24no—
216M16—22no—
217M16—22no—
218M16—22no—
219M16—42no—
220M16C22×C414yescomplemented
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—42no—
225C4⋊C4—28no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—42no—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—28no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—42no—
239C22⋊C4—84no—
240C42—28no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C23—22no—
246C23—24no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23C4×D412yes—
257C23#12011yescharacteristic
258C23#12011yescharacteristic
259C23#12112yes—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—24no—
263Q8—22no—
264Q8—22no—
265Q8—41no—
266Q8—41no—
267Q8C2×C22⋊C414yes—
268D4—24no—
269D4—24no—
270D4—28no—
271D4—22no—
272D4—22no—
273D4—22no—
274D4—24no—
275D4—44no—
276D4—44no—
277D4—41no—
278D4—41no—
279D4C2×C22⋊C414yes—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—88no—
302C2×C4C2×C22⋊C412yes—
303C2×C4C2×C22⋊C412yes—
304C2×C4C4×D412yes—
305C2×C4C4×D412yes—
306C2×C4C4×D412yes—
307C2×C4#12011yescharacteristic
308C2×C4#12011yescharacteristic
309C2×C4#12112yes—
310C2×C4#12112yes—
311C2×C4#12112yes—
312C2×C4#12312yes—
313C2×C4#12312yes—
314C2×C4C23×C411yescharacteristic
315C2×C4C22×D411yescharacteristic
316C2×C4C22×D411yescharacteristic
317C2×C4C22×D411yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C8—24no—
321C8—42no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—24no—
329C22—42no—
330C22—42no—
331C22—48no—
332C22—44no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—82no—
337C22—81no—
338C22—81no—
339C22#38612yes—
340C22#38612yes—
341C22C2×C4×D411yescharacteristic
342C22C2×#12011yescharacteristic
343C22C2×#12111yescharacteristic
344C4—24no—
345C4—44no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—82no—
350C4#38612yes—
351C4#38612yes—
352C4C22×C22⋊C411yescharacteristic
353C4C2×C4×D411yescharacteristic
354C4C2×#12111yescharacteristic
355C4C2×#12311yescharacteristic
356C2—21no—
357C2—21no—
358C2—22no—
359C2—22no—
360C2—44no—
361C2—82no—
362C2C2×#38611yescharacteristic
363C2#301911yescharacteristic
364C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#575C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#520C2semidirect
3. C2×#421C2semidirect
4. #2896C2semidirect
5. #2311C2semidirect
6. #1883C2semidirect
7. #1882C2semidirect
8. C2×#343C2semidirect
9. #1733C2semidirect
13. #575C4semidirect
35. #419C22semidirect
36. #418C22semidirect
40. #391C22semidirect
41. #390C22semidirect
44. #343C22semidirect
45. #339C22semidirect
58. C2×#41C2×C4semidirect
75. #137C2×C4semidirect
78. AGL(1,ℤ/8)C2×C4semidirect
89. C2×M16C2×C4semidirect
151. #41C22⋊C4semidirect
161. C2×Q8C22⋊C4semidirect
162. C2×Q8C22×C4semidirect
179. C2×D4C22⋊C4semidirect
180. C2×D4C22×C4semidirect
220. M16C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5752
C2×#5202
C2×#4212
#28962
#23112
#18832
#18822
C2×#3432
#17332