Finite Group Databasebeta

← #24434

Subgroups — #24434

Order 256.

837 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5C22×#102C211yesmaximal, characteristic, complemented
6#2416C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C23×C8C2211yescharacteristic
22#546—24no—
23#546—24no—
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121—21no—
28C2×#121—21no—
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic, complemented
32C2×#120—21no—
33C2×C4×D4C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic
35C22×C4⋊C4C2211yescharacteristic
36C22×C22⋊C4—21no—
37#465—24no—
38#449—24no—
39#447—24no—
40C2×#106C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#102—24no—
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic, complemented
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic, complemented
54C2×C22⋊C8C2211yescharacteristic, complemented
55#396—21no—
56#394—21no—
57#390C2211yescharacteristic, complemented
58#389C2211yescharacteristic, complemented
59#386—21no—
60#380C2211yescharacteristic, complemented
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×D4—22no—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—22no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4D411yescharacteristic
84C22×D4D414yescomplemented
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C22×D4C2311yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—41no—
96C2×D8—22no—
97C2×D8—22no—
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—41no—
101C2×D8—41no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109#124—22no—
110#124—22no—
111#123—28no—
112C22⋊Q8—22no—
113C22⋊Q8D414yescomplemented
114#121—28no—
115#121—22no—
116#121—22no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—41no—
121#121—44no—
122#120—28no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4D411yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C22⋊C4—28no—
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⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4D411yescharacteristic
149C2×C22⋊C4D411yescharacteristic
150C2×C42C2311yescharacteristic
151#106—22no—
152#103—22no—
153#103—22no—
154#103—41no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—22no—
160#102—22no—
161#102D414yescomplemented
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—41no—
165#95—21no—
166#95—21no—
167C24—21no—
168C24—21no—
169C24—24no—
170C24—21no—
171C24—42no—
172C24—42no—
173C24—42no—
174C24—42no—
175C24—81no—
176C24—81no—
177C24C2×D411yescharacteristic
178C24C2×D411yescharacteristic
179#41—44no—
180#41—44no—
181#41—44no—
182#41—44no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8C2×D412yes—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—24no—
195C2×D4—24no—
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—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—81no—
225C2×D4C2×D412yes—
226C2×D4C2×D412yes—
227C2×D4C2×D412yes—
228C2×D4#4112yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D412yes—
258C22×C4C2×D412yes—
259C22×C4C2411yescharacteristic
260SD16—44no—
261D8—44no—
262D8—44no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—44no—
275C2×C8C2×D412yes—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4C2×D412yes—
287C4⋊C4C2×D412yes—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—22no—
293C22⋊C4—24no—
294C22⋊C4—22no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—81no—
300C42—21no—
301C42—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—48no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—48no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—84no—
337C23—84no—
338C23—84no—
339C23—84no—
340C23—84no—
341C23#12012yes—
342C23#13512yes—
343C23AGL(1,ℤ/8)12yes—
344C23C22×D411yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
349D4—22no—
350D4—22no—
351D4—22no—
352D4—22no—
353D4—22no—
354D4—48no—
355D4—48no—
356D4—44no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—84no—
364D4—84no—
365D4—84no—
366D4#12114yescomplemented
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—48no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4#12012yes—
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D411yescharacteristic
407C2×C4C22×D411yescharacteristic
408C2×C4C2×#4111yescharacteristic
409C2×C4H(5,𝔽2)11yescharacteristic
410C2×C4H(5,𝔽2)11yescharacteristic
411C8—22no—
412C8—41no—
413C8—41no—
414C8—41no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—24no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—816no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22#44914yescomplemented
451C22C2×#12011yescharacteristic
452C22D4211yescharacteristic
453C22#54611yescharacteristic
454C22C2×#13511yescharacteristic
455C22C2×AGL(1,ℤ/8)11yescharacteristic
456C22#57611yescharacteristic
457C22#57711yescharacteristic
458C4—22no—
459C4—41no—
460C4—42no—
461C4—41no—
462C4—42no—
463C4—41no—
464C4—42no—
465C4—81no—
466C4—81no—
467C4C2×#12111yescharacteristic
468C4#53411yescharacteristic
469C4D4211yescharacteristic
470C4#54611yescharacteristic
471C2—24no—
472C2—44no—
473C2—81no—
474C2—82no—
475C2—161no—
476C2#241011yescharacteristic
477C2C2×#44911yescharacteristic
478C2#301211yescharacteristic
479C2#326411yescharacteristic
480C2#328211yescharacteristic
481C2#329711yescharacteristic
482C2#330611yescharacteristic
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×#546C2semidirect
2. C2×#465C2semidirect
3. C2×#449C2semidirect
4. C2×#447C2semidirect
5. C22×#102C2semidirect
6. #2416C2semidirect
7. #2042C2semidirect
8. #2034C2semidirect
9. #2012C2semidirect
10. #1999C2semidirect
11. #1879C2semidirect
12. #1872C2semidirect
13. #1869C2semidirect
14. #1864C2semidirect
15. #1788C2semidirect
18. C22×SD16C22semidirect
31. C2×#121C22semidirect
40. C2×#106C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
51. C2×#102C22semidirect
53. C2×C22⋊C8C22semidirect
54. C2×C22⋊C8C22semidirect
57. #390C22semidirect
58. #389C22semidirect
60. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
84. C22×D4D4semidirect
113. C22⋊Q8D4semidirect
161. #102D4semidirect
366. D4#121semidirect
450. C22#449semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4652
C2×#4492
C2×#4472
C22×#1022
#24162
#20422
#20342
#20122
#19992
#18792
#18722
#18692
#18642
#17882