Finite Group Databasebeta

← #29600

Subgroups — #29600

Order 256.

770 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4#2847C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1999C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1779C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2211yescharacteristic, complemented
20#525—24no—
21#525—24no—
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#120—21no—
31C2×#120—21no—
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34C2×#117C2211yescharacteristic
35#482—24no—
36#449—24no—
37C2×#107C2211yescharacteristic, complemented
38C2×#106C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#103C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53#386—21no—
54#386—21no—
55#386—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25D411yescharacteristic
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41D414yescomplemented
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4D411yescharacteristic
87C23×C4D411yescharacteristic
88C23×C4D411yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C2×SD16—22no—
93C2×SD16—41no—
94C2×SD16—41no—
95C2×D8—22no—
96C2×D8—41no—
97C2×D8—41no—
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102#123—28no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107#121—22no—
108#121—22no—
109#121—22no—
110#121—22no—
111#121—22no—
112#121—22no—
113#121—41no—
114#121—41no—
115#121—41no—
116#121—41no—
117#120—28no—
118#120—44no—
119C4×D4—28no—
120C4×D4—28no—
121#117—22no—
122#117D414yescomplemented
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C22⋊C4—21no—
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×C42—21no—
142#107—22no—
143#106—22no—
144#103—22no—
145#103—41no—
146#103—41no—
147#102—22no—
148#102—22no—
149#102—22no—
150#102—41no—
151#102—41no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158#95—21no—
159#95—21no—
160#95—21no—
161C24—22no—
162C24—22no—
163C24—22no—
164C24—21no—
165C24—21no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24C2×D411yescharacteristic
171C24C2×D411yescharacteristic
172C24C2×D411yescharacteristic
173#41—44no—
174#41—44no—
175#41—44no—
176#41—44no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8C2×D412yes—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4C2×D412yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
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—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D412yes—
255C22×C4C2×D412yes—
256C22×C4C2411yescharacteristic
257SD16—44no—
258D8—44no—
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×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4C2×D412yes—
282C4⋊C4C2×D412yes—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—24no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#12012yes—
335C23#13512yes—
336C23AGL(1,ℤ/8)12yes—
337C23C22×D411yescharacteristic
338Q8—22no—
339Q8—22no—
340Q8—41no—
341Q8—41no—
342D4—22no—
343D4—22no—
344D4—48no—
345D4—48no—
346D4—48no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—84no—
354D4—84no—
355D4—84no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—44no—
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—41no—
385C2×C4—41no—
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#12012yes—
399C2×C4C22×D411yescharacteristic
400C2×C4C22×D411yescharacteristic
401C2×C4H(5,𝔽2)11yescharacteristic
402C2×C4H(5,𝔽2)11yescharacteristic
403C2×C4H(5,𝔽2)11yescharacteristic
404C2×C4H(5,𝔽2)11yescharacteristic
405C8—41no—
406C8—41no—
407C8—41no—
408C8—41no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—416no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22C2×#12011yescharacteristic
446C22#53411yescharacteristic
447C22#53411yescharacteristic
448C22C2×#13511yescharacteristic
449C22C2×AGL(1,ℤ/8)11yescharacteristic
450C22#57511yescharacteristic
451C22#57511yescharacteristic
452C4—22no—
453C4—41no—
454C4—41no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4#53411yescharacteristic
462C4#53411yescharacteristic
463C4#53511yescharacteristic
464C4#53511yescharacteristic
465C2—42no—
466C2—42no—
467C2—42no—
468C2—82no—
469C2—161no—
470C2#261311yescharacteristic
471C2#300211yescharacteristic
472C2#300611yescharacteristic
473C2#318811yescharacteristic
474C2#319111yescharacteristic
475C2#319411yescharacteristic
476C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#525C2semidirect
2. C2×#482C2semidirect
3. C2×#449C2semidirect
4. #2847C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1999C2semidirect
12. #1999C2semidirect
13. #1825C2semidirect
14. #1788C2semidirect
15. #1779C2semidirect
18. C22×SD16C22semidirect
19. C22×D8C22semidirect
37. C2×#107C22semidirect
38. C2×#106C22semidirect
39. C2×#103C22semidirect
40. C2×#103C22semidirect
41. C2×#103C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect
69. C2×#41D4semidirect
122. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4822
C2×#4492
#28472
#20362
#20352
#20342
#20332
#20002
#19992
#18252
#17882
#17792