Finite Group Databasebeta

← #24256

Subgroups — #24256

Order 256.

755 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic, complemented
5C2×#440C211yesmaximal, characteristic, complemented
6#2537C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1955C211yesmaximal, characteristic, complemented
13#1803C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21#548—24no—
22#548—24no—
23C2×#124—21no—
24C2×#124C2211yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121—21no—
28C2×#121—21no—
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33C2×C4×D4—21no—
34C2×C4×D4C2211yescharacteristic
35C2×C4×D4C2211yescharacteristic
36#463—24no—
37#449—24no—
38#448—24no—
39#440—24no—
40C2×#107C2211yescharacteristic
41C2×C4⋊C8C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55C2×C22⋊C8C2211yescharacteristic
56C2×#97C2211yescharacteristic
57#394—21no—
58#394—21no—
59#384C2211yescharacteristic
60#380—21no—
61#380C2211yescharacteristic
62#380C2211yescharacteristic
63#340C2211yescharacteristic
64#340C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—41no—
70C2×#41—41no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—41no—
97C2×SD16D414yescomplemented
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—41no—
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105#124—28no—
106#124—22no—
107#124D414yescomplemented
108C22⋊Q8—28no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111#121—28no—
112#121—22no—
113#121—22no—
114#121—22no—
115#121—22no—
116#121—41no—
117#121—41no—
118#121—44no—
119C4×Q8—22no—
120C4×Q8D414yescomplemented
121C4×D4—28no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4D411yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4D411yescharacteristic
145C2×C22⋊C4D411yescharacteristic
146C2×C42—21no—
147C2×C42C2311yescharacteristic
148#107—22no—
149C4⋊C8—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—41no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—41no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—22no—
165#95—21no—
166#95—21no—
167C24—21no—
168C24—21no—
169C24C2×D411yescharacteristic
170C24C2×D411yescharacteristic
171#41—44no—
172#41—44no—
173#41—44no—
174#41—44no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8C2×D412yes—
191C2×Q8C2×D412yes—
192C2×Q8C2×D412yes—
193C2×Q8#4112yes—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—81no—
218C2×D4—81no—
219C2×D4C2×D412yes—
220C2×D4C2×D412yes—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2411yescharacteristic
253SD16—28no—
254SD16—44no—
255D8—44no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8C2×D412yes—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—42no—
279C4⋊C4—81no—
280C4⋊C4C2×D412yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—24no—
293C42C2×D412yes—
294C42C2×D412yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23AGL(1,ℤ/8)12yes—
323C23C22×D411yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—22no—
327Q8—22no—
328Q8—22no—
329Q8—48no—
330Q8—41no—
331Q8—41no—
332Q8#12114yescomplemented
333D4—22no—
334D4—22no—
335D4—22no—
336D4—22no—
337D4—44no—
338D4—48no—
339D4—48no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—84no—
345D4—84no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—84no—
385C2×C4#12012yes—
386C2×C4#12012yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C2×#4111yescharacteristic
391C2×C4H(5,𝔽2)11yescharacteristic
392C2×C4H(5,𝔽2)11yescharacteristic
393C8—22no—
394C8—41no—
395C8—41no—
396C8—41no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—42no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22C2×#12011yescharacteristic
426C22D4211yescharacteristic
427C22#54611yescharacteristic
428C22C2×AGL(1,ℤ/8)11yescharacteristic
429C22#57511yescharacteristic
430C22#57711yescharacteristic
431C22#57811yescharacteristic
432C4—24no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—42no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4C2×#12111yescharacteristic
443C4#53511yescharacteristic
444C4D4211yescharacteristic
445C4#54611yescharacteristic
446C2—42no—
447C2—81no—
448C2—82no—
449C2—82no—
450C2#240811yescharacteristic
451C2#300611yescharacteristic
452C2#301311yescharacteristic
453C2#326511yescharacteristic
454C2#326911yescharacteristic
455C2#330811yescharacteristic
456C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#548C2semidirect
2. C2×#463C2semidirect
3. C2×#449C2semidirect
4. C2×#448C2semidirect
5. C2×#440C2semidirect
6. #2537C2semidirect
7. #2036C2semidirect
8. #2033C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1959C2semidirect
12. #1955C2semidirect
13. #1803C2semidirect
14. #1788C2semidirect
15. #1787C2semidirect
97. C2×SD16D4semidirect
107. #124D4semidirect
120. C4×Q8D4semidirect
332. Q8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4632
C2×#4492
C2×#4482
C2×#4402
#25372
#20362
#20332
#20002
#19992
#19592
#19552
#18032
#17882
#17872