Finite Group Databasebeta

← #24382

Subgroups — #24382

Order 256.

839 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×#449C211yesmaximal, characteristic, complemented
5C22×C22⋊C8C211yesmaximal, characteristic, complemented
6#2847C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2211yescharacteristic, complemented
20C23×C8C2211yescharacteristic
21C23×C8C2211yescharacteristic
22#525—24no—
23#525—24no—
24C2×#123—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×#120—21no—
33C2×#120—21no—
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36C2×#117C2211yescharacteristic
37#466—24no—
38#465—24no—
39#449—24no—
40C2×#107C2211yescharacteristic, complemented
41C2×#106C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#103C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×#102C2211yescharacteristic, complemented
48C2×#102C2211yescharacteristic, complemented
49C2×#102C2211yescharacteristic, complemented
50C2×C22⋊C8—24no—
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55#386—21no—
56#386—21no—
57#386—21no—
58#380C2211yescharacteristic
59#380C2211yescharacteristic
60#380C2211yescharacteristic
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64#340C2211yescharacteristic, complemented
65C25D411yescharacteristic
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4C2311yescharacteristic
80C22×D4C2311yescharacteristic
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4D411yescharacteristic
89C23×C4D411yescharacteristic
90C23×C4D411yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—41no—
97C2×D8—22no—
98C2×D8—22no—
99C2×D8—41no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109C22×C8C2311yescharacteristic
110#123—28no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8D414yescomplemented
115#121—22no—
116#121—22no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#121—41no—
123#121—41no—
124#121D414yescomplemented
125#120—28no—
126#120—44no—
127C4×D4—28no—
128C4×D4—28no—
129#117—22no—
130#117—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
149C2×C42—21no—
150#107—22no—
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—41no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8D414yescomplemented
164#95—21no—
165#95—21no—
166#95—21no—
167C24—22no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—21no—
173C24—21no—
174C24—41no—
175C24—41no—
176C24C2×D411yescharacteristic
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—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—21no—
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—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×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
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—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
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—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D412yes—
258C22×C4C2×D412yes—
259C22×C4C2×D411yescharacteristic
260C22×C4C2×D412yes—
261C22×C4#4112yes—
262C22×C4C2411yescharacteristic
263SD16—44no—
264D8—44no—
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—21no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8C2×D412yes—
278C2×C8C2×D412yes—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4C2×D412yes—
290C4⋊C4C2×D412yes—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C42—24no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—28no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23#12012yes—
341C23#12112yes—
342C23#13512yes—
343C23#13512yes—
344C23AGL(1,ℤ/8)12yes—
345C23C22×D411yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350D4—22no—
351D4—22no—
352D4—48no—
353D4—48no—
354D4—48no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—84no—
362D4—84no—
363D4—84no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
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—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4#12012yes—
406C2×C4#12112yes—
407C2×C4C22×D411yescharacteristic
408C2×C4C22×D411yescharacteristic
409C2×C4C22×D411yescharacteristic
410C2×C4C2×#4111yescharacteristic
411C2×C4H(5,𝔽2)11yescharacteristic
412C2×C4H(5,𝔽2)11yescharacteristic
413C8—22no—
414C8—22no—
415C8—41no—
416C8—41no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—416no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#44914yescomplemented
453C22C2×#12011yescharacteristic
454C22C2×#12111yescharacteristic
455C22#53411yescharacteristic
456C22C2×#13511yescharacteristic
457C22C2×#13511yescharacteristic
458C22C2×AGL(1,ℤ/8)11yescharacteristic
459C22#57511yescharacteristic
460C4—22no—
461C4—22no—
462C4—41no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4D4211yescharacteristic
470C4D4211yescharacteristic
471C4#54611yescharacteristic
472C4#54611yescharacteristic
473C2—24no—
474C2—42no—
475C2—42no—
476C2—82no—
477C2—161no—
478C2#241011yescharacteristic
479C2C2×#44911yescharacteristic
480C2#300211yescharacteristic
481C2#305011yescharacteristic
482C2#305511yescharacteristic
483C2#318811yescharacteristic
484C2#319111yescharacteristic
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×#466C2semidirect
3. C2×#465C2semidirect
4. C2×#449C2semidirect
5. C22×C22⋊C8C2semidirect
6. #2847C2semidirect
7. #2036C2semidirect
8. #2034C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1893C2semidirect
12. #1892C2semidirect
13. #1875C2semidirect
14. #1854C2semidirect
15. #1788C2semidirect
18. C22×SD16C22semidirect
19. C22×D8C22semidirect
40. C2×#107C22semidirect
41. C2×#106C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
44. C2×#103C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#102C22semidirect
49. C2×#102C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #340C22semidirect
64. #340C22semidirect
114. C22⋊Q8D4semidirect
124. #121D4semidirect
163. C22⋊C8D4semidirect
452. C22#449semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4662
C2×#4652
C2×#4492
C22×C22⋊C82
#28472
#20362
#20342
#20002
#19992
#18932
#18922
#18752
#18542
#17882