Finite Group Databasebeta

← C2×#2048 #23523

Subgroups — C2×#2048 #23523

Order 256.

725 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#430C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic, complemented
4C2×#418C211yesmaximal, characteristic, complemented
5C2×#413C211yesmaximal, characteristic, complemented
6C22×#100C211yesmaximal, characteristic, complemented
7#2048C218yesmaximal, direct product factor
8C2×#344C211yesmaximal, characteristic, complemented
9C22×#41C2211yescharacteristic
10C23×D4C2211yescharacteristic
11C2×AGL(1,ℤ/8)—22no—
12C2×AGL(1,ℤ/8)—22no—
13C2×AGL(1,ℤ/8)—22no—
14C2×AGL(1,ℤ/8)—22no—
15C2×AGL(1,ℤ/8)—22no—
16C2×AGL(1,ℤ/8)C2214yes—
17C22×SD16—21no—
18C22×D8—21no—
19C22×M16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22C22×M16C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#430—22no—
25#430C2214yescomplemented
26C2×#108—22no—
27#421—22no—
28#421C2214yescomplemented
29C2×#104—22no—
30#418—22no—
31#418C2214yescomplemented
32C2×#102—22no—
33#413—22no—
34#413C2214yescomplemented
35C2×#101—21no—
36C2×#100—21no—
37C2×#100—22no—
38C2×#100—22no—
39C2×#100C2214yescomplemented
40#344C2214yescomplemented
41C25—21no—
42C2×#41—21no—
43C2×#41—21no—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—21no—
47C2×#41—21no—
48C2×#41—21no—
49C2×#41—21no—
50C2×#41—21no—
51C2×#41—21no—
52C2×#41D412yes—
53C2×#41C2312yes—
54C22×Q8D411yescharacteristic
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4D412yes—
69C22×D4D412yes—
70C22×D4C2312yes—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73AGL(1,ℤ/8)—28no—
74AGL(1,ℤ/8)—28no—
75AGL(1,ℤ/8)—28no—
76AGL(1,ℤ/8)—28no—
77C2×SD16—24no—
78C2×SD16—24no—
79C2×SD16—24no—
80C2×D8—24no—
81C2×D8—24no—
82C2×D8—24no—
83C2×M16—21no—
84C2×M16—21no—
85C2×M16—21no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16D412yes—
100C2×M16D412yes—
101C2×M16D412yes—
102C2×M16D412yes—
103C2×M16C2312yes—
104C2×M16C2312yes—
105C2×M16C2312yes—
106C2×M16C2312yes—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111#117—21no—
112#117—21no—
113#117—21no—
114#117C2312yes—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42—21no—
118#108—28no—
119#104—28no—
120#102—28no—
121#101—24no—
122#100—24no—
123#100—28no—
124C24—22no—
125C24—22no—
126C24—21no—
127C24—22no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24—84no—
133C24#4111yescharacteristic
134#41—22no—
135#41—22no—
136#41—22no—
137#41—22no—
138#41—22no—
139#41—22no—
140#41—22no—
141#41—22no—
142#41—41no—
143#41—41no—
144#41—41no—
145#41—41no—
146#41—41no—
147#41—41no—
148#41—41no—
149#41—41no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C2×D412yes—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C2×D412yes—
195C2×D4C2×D414yes—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4#4111yescharacteristic
213C22×C4#4111yescharacteristic
214C22×C4#4111yescharacteristic
215C22×C4#4111yescharacteristic
216C22×C4#4111yescharacteristic
217C22×C4C2411yescharacteristic
218SD16—216no—
219D8—216no—
220M16—22no—
221M16—22no—
222M16—22no—
223M16—22no—
224M16—22no—
225M16—22no—
226M16—22no—
227M16—22no—
228M16—22no—
229M16—22no—
230M16—41no—
231M16—41no—
232M16—41no—
233M16—41no—
234M16C2×D414yes—
235M16C2×D414yes—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C22⋊C4—22no—
251C22⋊C4—41no—
252C42—22no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—42no—
265C23—82no—
266C23—84no—
267C23—84no—
268C23—84no—
269C23—84no—
270C23—84no—
271C23—84no—
272C23—84no—
273C23#12111yescharacteristic
274C23#12111yescharacteristic
275C23#12311yescharacteristic
276C23#12311yescharacteristic
277C23#12411yescharacteristic
278C23#12411yescharacteristic
279C23C2×#4111yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284D4—22no—
285D4—24no—
286D4—22no—
287D4—24no—
288D4—42no—
289D4—42no—
290D4—42no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—41no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4#12011yescharacteristic
310C2×C4#12011yescharacteristic
311C2×C4#12011yescharacteristic
312C2×C4#12011yescharacteristic
313C2×C4#12111yescharacteristic
314C2×C4#12111yescharacteristic
315C2×C4#12111yescharacteristic
316C2×C4#12111yescharacteristic
317C2×C4#12111yescharacteristic
318C2×C4#12111yescharacteristic
319C2×C4#12111yescharacteristic
320C2×C4#12111yescharacteristic
321C2×C4#12111yescharacteristic
322C2×C4#12111yescharacteristic
323C2×C4#12311yescharacteristic
324C2×C4#12311yescharacteristic
325C2×C4#12311yescharacteristic
326C2×C4#12311yescharacteristic
327C2×C4#12311yescharacteristic
328C2×C4#12311yescharacteristic
329C2×C4#12411yescharacteristic
330C2×C4#12411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C22×D411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C8—24no—
338C8—24no—
339C8—42no—
340C8—42no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—44no—
350C22—82no—
351C22—84no—
352C22—82no—
353C22—84no—
354C22—84no—
355C22—84no—
356C22—84no—
357C22—84no—
358C22—82no—
359C22—84no—
360C22#39411yescharacteristic
361C22#39411yescharacteristic
362C22#39411yescharacteristic
363C22#39411yescharacteristic
364C22C2×#12111yescharacteristic
365C22C2×#12311yescharacteristic
366C22C2×#12411yescharacteristic
367C4—42no—
368C4—82no—
369C4#39411yescharacteristic
370C4#39411yescharacteristic
371C4#39411yescharacteristic
372C4#39411yescharacteristic
373C4C2×#12011yescharacteristic
374C4C2×#12111yescharacteristic
375C4C2×#12111yescharacteristic
376C4C2×#12311yescharacteristic
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—82no—
384C2—84no—
385C2#204812yesdirect product factor
386C2C2×#39411yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×AGL(1,ℤ/8)C2
LCSG ▷ C22×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×#430C2semidirect
3. C2×#421C2semidirect
4. C2×#418C2semidirect
5. C2×#413C2semidirect
6. C22×#100C2semidirect
7. #2048C2direct (N × H)
8. C2×#344C2semidirect
25. #430C22semidirect
28. #421C22semidirect
31. #418C22semidirect
34. #413C22semidirect
39. C2×#100C22semidirect
40. #344C22semidirect
385. C2#2048direct (N × H)

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)2
C2×#4302
C2×#4212
C2×#4182
C2×#4132
C22×#1002
#20482
C2×#3442