Finite Group Databasebeta

← C4×#471 #21552

Subgroups — C4×#471 #21552

Order 256.

457 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#471C211yesmaximal, characteristic
2C4×#137C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5C4×#131C211yesmaximal, characteristic, complemented
6#1977C211yesmaximal, characteristic, complemented
7#1951C211yesmaximal, characteristic, complemented
8#1931C211yesmaximal, characteristic, complemented
9#1930C211yesmaximal, characteristic, complemented
10#1908C211yesmaximal, characteristic, complemented
11#1907C211yesmaximal, characteristic
12#1815C211yesmaximal, characteristic, complemented
13C4×#108C211yesmaximal, characteristic, complemented
14C4×#101C211yesmaximal, characteristic, complemented
15C4×#100C211yesmaximal, characteristic, complemented
16C2×#137C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×#131C2211yescharacteristic
20C2×#131C2211yescharacteristic
21C4×#41C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#471C418yesdirect product factor
25#442—21no—
26#441—21no—
27#440—21no—
28#440—21no—
29C4×Q16—21no—
30C4×SD16—21no—
31C4×SD16—21no—
32C4×SD16—21no—
33C4×SD16—21no—
34C4×SD16—21no—
35C4×SD16—21no—
36C4×D8—21no—
37C2×#108C2211yescharacteristic
38C2×#108C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41C2×#106C2211yescharacteristic
42#419C2211yescharacteristic
43#418C2211yescharacteristic
44#417C2211yescharacteristic
45#417C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#101C2211yescharacteristic
49C2×#101C2211yescharacteristic
50C2×#100C2211yescharacteristic
51C2×#100C2211yescharacteristic
52#405—21no—
53#405—21no—
54#405C2211yescharacteristic
55#405C2211yescharacteristic
56C4×M16—21no—
57C4×M16C2211yescharacteristic
58C4×M16C2211yescharacteristic
59C4×M16C2211yescharacteristic
60C2×C4×C8—21no—
61C2×C4×C8C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#343C2211yescharacteristic
65#343C2211yescharacteristic
66#341C2211yescharacteristic
67#341C2211yescharacteristic
68C2×#41C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72#137—22no—
73#137C2×C414yescomplemented
74AGL(1,ℤ/8)—22no—
75AGL(1,ℤ/8)C2×C414yes—
76C2×Q16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16—21no—
81C2×SD16—22no—
82C2×SD16C2×C414yescomplemented
83C2×D8—21no—
84#131—22no—
85#131—22no—
86#131—22no—
87#131C2×C414yescomplemented
88C2×M16—21no—
89C2×M16—21no—
90C2×M16C2311yescharacteristic
91C2×M16C2311yescharacteristic
92C2×M16C2311yescharacteristic
93C2×M16C2311yescharacteristic
94C2×M16C2311yescharacteristic
95C2×M16C2311yescharacteristic
96C22×C8—21no—
97C22×C8—21no—
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8D411yescharacteristic
105C4×Q8D411yescharacteristic
106C4×D4—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4D411yescharacteristic
113C4×D4D411yescharacteristic
114#117—21no—
115#117C2311yescharacteristic
116C2×C4⋊C4—21no—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42C2311yescharacteristic
123#108—22no—
124#108C2×C414yescomplemented
125#107—21no—
126#107—21no—
127#106—21no—
128#106—21no—
129#106—21no—
130#106—21no—
131#103—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103—21no—
137#102—21no—
138#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#101—22no—
144#101C2×C414yescomplemented
145#100—22no—
146#100C2×C414yescomplemented
147#97—21no—
148#97—21no—
149#97—21no—
150#97—21no—
151#97—21no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8D411yescharacteristic
158C4×C8D411yescharacteristic
159C4×C8D411yescharacteristic
160C4×C8D411yescharacteristic
161C24—21no—
162#41—21no—
163#41—21no—
164#41C22×C412yes—
165#41#4112yes—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22×C412yes—
172C2×Q8C2×D411yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4C22×C412yes—
180C2×D4C2×D411yescharacteristic
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—82no—
190C22×C4#4111yescharacteristic
191C22×C4#4111yescharacteristic
192C22×C4C2411yescharacteristic
193Q16—24no—
194SD16—24no—
195SD16—24no—
196SD16—24no—
197SD16—24no—
198D8—24no—
199M16—24no—
200M16—21no—
201M16—21no—
202M16—21no—
203M16—21no—
204M16—24no—
205M16—21no—
206M16—21no—
207M16—21no—
208M16—21no—
209M16C22×C412yes—
210M16C22×C412yes—
211M16C22×C412yes—
212M16#4112yes—
213C2×C8—24no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—24no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8C22×C412yes—
224C2×C8C2×D411yescharacteristic
225C2×C8C2×D411yescharacteristic
226C2×C8C2×D411yescharacteristic
227C2×C8C2×D411yescharacteristic
228C2×C8#4112yes—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4C2×D411yescharacteristic
238C4⋊C4C2×D411yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—21no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—41no—
248C42—41no—
249C42C2×D411yescharacteristic
250C42C2×D411yescharacteristic
251C42C2×D411yescharacteristic
252C42C2×D411yescharacteristic
253C23—21no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—81no—
258C23—81no—
259C23C2×#4111yescharacteristic
260Q8—21no—
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8C4×D412yes—
266D4—22no—
267D4—22no—
268D4—22no—
269D4—22no—
270D4—21no—
271D4—41no—
272D4—41no—
273D4C4×D412yes—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4#12111yescharacteristic
293C2×C4#12111yescharacteristic
294C2×C4#12111yescharacteristic
295C2×C4#12111yescharacteristic
296C2×C4C23×C411yescharacteristic
297C2×C4C22×D411yescharacteristic
298C2×C4C22×D411yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C8—22no—
303C8—22no—
304C8—22no—
305C8—21no—
306C8—22no—
307C8—21no—
308C8—41no—
309C8—41no—
310C8C4×D412yes—
311C8C4×D412yes—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—81no—
321C22—81no—
322C22C4×#4111yescharacteristic
323C22C2×#12111yescharacteristic
324C22#52911yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—42no—
332C4—81no—
333C4—81no—
334C4#47112yesdirect product factor
335C4C2×C4×D411yescharacteristic
336C4C2×C4×D411yescharacteristic
337C4#52511yescharacteristic
338C4#53211yescharacteristic
339C2—21no—
340C2—21no—
341C2—42no—
342C2—82no—
343C2C4×#12111yescharacteristic
344C2C2×#47111yescharacteristic
345C2#306711yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×#121
prime radicalC2×AGL(1,ℤ/8)C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#137C2semidirect
4. C2×C4×SD16C2semidirect
5. C4×#131C2semidirect
6. #1977C2semidirect
7. #1951C2semidirect
8. #1931C2semidirect
9. #1930C2semidirect
10. #1908C2semidirect
12. #1815C2semidirect
13. C4×#108C2semidirect
14. C4×#101C2semidirect
15. C4×#100C2semidirect
24. #471C4direct (N × H)
73. #137C2×C4semidirect
82. C2×SD16C2×C4semidirect
87. #131C2×C4semidirect
124. #108C2×C4semidirect
144. #101C2×C4semidirect
146. #100C2×C4semidirect
334. C4#471direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#4712
C4×#1372
C4×AGL(1,ℤ/8)2
C2×C4×SD162
C4×#1312
#19772
#19512
#19312
#19302
#19082
#19072
#18152
C4×#1082
C4×#1012
C4×#1002