Finite Group Databasebeta

← #24367

Subgroups — #24367

Order 256.

802 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
1C22×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5#2432C211yesmaximal, characteristic, complemented
6D4×C22⋊C4C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1851C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C2×AGL(1,ℤ/8)—24no—
19C2×AGL(1,ℤ/8)—24no—
20C2×AGL(1,ℤ/8)—24no—
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×SD16C2211yescharacteristic
25C22×D8C2211yescharacteristic
26C22×D8C2211yescharacteristic
27C22×M16C2211yescharacteristic
28C2×C4⋊Q8C2211yescharacteristic, complemented
29C2×#124C2211yescharacteristic, complemented
30C2×C22⋊Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C22×C22⋊C4—21no—
34#463—24no—
35#461—24no—
36#450—24no—
37C2×C4⋊C8C2211yescharacteristic, complemented
38C2×C4⋊C8C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#103C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#394—21no—
50#391C2211yescharacteristic, complemented
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#390C2211yescharacteristic, complemented
54#386—21no—
55#386—21no—
56#385—21no—
57#385—21no—
58#380C2211yescharacteristic
59#379—21no—
60C4×C22⋊C4C2211yescharacteristic
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—22no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4D414yes—
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89AGL(1,ℤ/8)—232no—
90C2×SD16—22no—
91C2×SD16—22no—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—41no—
100C2×SD16—41no—
101C2×SD16D414yes—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8D414yes—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C4⋊Q8—22no—
116C4⋊Q8—41no—
117#124—22no—
118#124—41no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—41no—
124C4×D4—41no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
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—44no—
144C2×C22⋊C4D411yescharacteristic
145C2×C22⋊C4D411yescharacteristic
146C2×C42C2311yescharacteristic
147C2×C42C2311yescharacteristic
148C4⋊C8—22no—
149C4⋊C8—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—41no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—41no—
158C22⋊C8—22no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95D411yescharacteristic
163#95D411yescharacteristic
164C24—21no—
165C24—21no—
166C24—21no—
167C24—24no—
168C24—42no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—81no—
173C24—81no—
174C24C2×D411yescharacteristic
175#41—28no—
176#41—44no—
177#41—44no—
178#41—44no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8C2×D412yes—
191C2×D4—28no—
192C2×D4—24no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
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—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4C2×D412yes—
215C2×D4C2×D412yes—
216C2×D4C2×D412yes—
217C2×D4C2×D412yes—
218C2×D4C2×D412yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4C2×D411yescharacteristic
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D412yes—
242C22×C4C2×D411yescharacteristic
243C22×C4#4112yes—
244C22×C4C2411yescharacteristic
245SD16—28no—
246SD16—28no—
247SD16—44no—
248SD16—44no—
249D8—28no—
250D8—28no—
251M16—28no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8C2×D412yes—
263C2×C8C2×D412yes—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—81no—
286C22⋊C4—81no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—22no—
301C23—22no—
302C23—48no—
303C23—48no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—84no—
321C23—84no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23#12112yes—
326C23C22×D411yescharacteristic
327Q8—22no—
328Q8—22no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335D4—22no—
336D4—22no—
337D4—22no—
338D4—22no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—48no—
343D4—48no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4#12014yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—48no—
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—84no—
383C2×C4—84no—
384C2×C4#12112yes—
385C2×C4AGL(1,ℤ/8)12yescomplemented
386C2×C4C22×D411yescharacteristic
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C2×#4111yescharacteristic
390C2×C4H(5,𝔽2)11yescharacteristic
391C2×C4H(5,𝔽2)11yescharacteristic
392C8—22no—
393C8—22no—
394C8—41no—
395C8—41no—
396C22—22no—
397C22—24no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—816no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22C2×#12111yescharacteristic
426C22D4211yescharacteristic
427C22D4211yescharacteristic
428C22C2×AGL(1,ℤ/8)11yescharacteristic
429C22#57511yescharacteristic
430C22#57711yescharacteristic
431C22#57711yescharacteristic
432C4—22no—
433C4—22no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4C2×#12011yescharacteristic
443C4#53411yescharacteristic
444C4#54611yescharacteristic
445C4#54611yescharacteristic
446C2—42no—
447C2—44no—
448C2—82no—
449C2—82no—
450C2#241011yescharacteristic
451C2#303611yescharacteristic
452C2#304211yescharacteristic
453C2#326511yescharacteristic
454C2#326711yescharacteristic
455C2#327211yescharacteristic
456C2#327511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×AGL(1,ℤ/8)C2
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×#463C2semidirect
3. C2×#461C2semidirect
4. C2×#450C2semidirect
5. #2432C2semidirect
6. D4×C22⋊C4C2semidirect
7. #2055C2semidirect
8. #2054C2semidirect
9. #2013C2semidirect
10. #2012C2semidirect
11. #1894C2semidirect
12. #1879C2semidirect
13. #1877C2semidirect
14. #1851C2semidirect
15. #1788C2semidirect
28. C2×C4⋊Q8C22semidirect
29. C2×#124C22semidirect
37. C2×C4⋊C8C22semidirect
38. C2×C4⋊C8C22semidirect
39. C2×#103C22semidirect
40. C2×#103C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
50. #391C22semidirect
53. #390C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
385. C2×C4AGL(1,ℤ/8)semidirect

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×#4632
C2×#4612
C2×#4502
#24322
D4×C22⋊C42
#20552
#20542
#20132
#20122
#18942
#18792
#18772
#18512
#17882