Finite Group Databasebeta

← #24245

Subgroups — #24245

Order 256.

814 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5C2×C4×D8C211yesmaximal, characteristic, complemented
6#2535C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic, complemented
13#1804C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21#546—24no—
22#546—24no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic, complemented
30C2×#120—21no—
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic, complemented
34C22×C22⋊C4—21no—
35#463—24no—
36#449—24no—
37#447—24no—
38C4×D8—24no—
39C2×#107C2211yescharacteristic, complemented
40C2×C4⋊C8C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic, complemented
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic, complemented
54C2×C22⋊C8C2211yescharacteristic, complemented
55C2×C4×C8C2211yescharacteristic, complemented
56#397—21no—
57#392—21no—
58#386—21no—
59#386—21no—
60#382C2211yescharacteristic, complemented
61#380C2211yescharacteristic
62#380C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64#340C2211yescharacteristic, 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—21no—
74C22×D4—21no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4D411yescharacteristic
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×D8D414yescomplemented
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106#124—22no—
107#124D414yescomplemented
108#123—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#120—28no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4D414yescomplemented
125C2×C4⋊C4—21no—
126C2×C4⋊C4D411yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C22⋊C4—28no—
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⋊C4D411yescharacteristic
143C2×C22⋊C4D411yescharacteristic
144C2×C42C2311yescharacteristic
145#107—22no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—41no—
150#102—22no—
151#102—22no—
152#102—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—41no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C4×C8—22no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24—24no—
168C24—21no—
169C24—21no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—42no—
174C24—81no—
175C24—81no—
176C24C2×D411yescharacteristic
177C24C2×D411yescharacteristic
178#41—44no—
179#41—44no—
180#41—44no—
181#41—44no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8C2×D412yes—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—24no—
192C2×D4—22no—
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—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—81no—
220C2×D4—81no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4C2×D412yes—
224C2×D4C2×D412yes—
225C2×D4C2×D412yes—
226C2×D4C2×D412yes—
227C2×D4#4112yes—
228C22×C4—24no—
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—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2411yescharacteristic
259SD16—44no—
260D8—28no—
261D8—44no—
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—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8C2×D412yes—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4C2×D412yes—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—22no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C42C2×D412yes—
297C42C2×D412yes—
298C23—21no—
299C23—22no—
300C23—21no—
301C23—22no—
302C23—21no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—21no—
309C23—22no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—48no—
317C23—48no—
318C23—41no—
319C23—41no—
320C23—41no—
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—84no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—84no—
339C23#13512yes—
340C23C22×D411yescharacteristic
341Q8—22no—
342Q8—22no—
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—22no—
350D4—22no—
351D4—22no—
352D4—44no—
353D4—48no—
354D4—48no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—84no—
360D4—84no—
361D4—84no—
362D4#12114yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—48no—
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—84no—
398C2×C4—84no—
399C2×C4#12012yes—
400C2×C4#12012yes—
401C2×C4C22×D411yescharacteristic
402C2×C4C22×D411yescharacteristic
403C2×C4C22×D411yescharacteristic
404C2×C4C2×#4111yescharacteristic
405C2×C4H(5,𝔽2)11yescharacteristic
406C2×C4H(5,𝔽2)11yescharacteristic
407C8—22no—
408C8—41no—
409C8—41no—
410C8—41no—
411C22—24no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—41no—
416C22—42no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—42no—
425C22—41no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—41no—
433C22—42no—
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—
447C22C2×#12011yescharacteristic
448C22D4211yescharacteristic
449C22#54611yescharacteristic
450C22C2×#13511yescharacteristic
451C22#57511yescharacteristic
452C22#57611yescharacteristic
453C22#57611yescharacteristic
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4C2×#12111yescharacteristic
464C4#53511yescharacteristic
465C4D4211yescharacteristic
466C4#54611yescharacteristic
467C2—42no—
468C2—44no—
469C2—81no—
470C2—82no—
471C2—82no—
472C2#240811yescharacteristic
473C2#300211yescharacteristic
474C2#300811yescharacteristic
475C2#327311yescharacteristic
476C2#328011yescharacteristic
477C2#329711yescharacteristic
478C2#332211yescharacteristic
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×#463C2semidirect
3. C2×#449C2semidirect
4. C2×#447C2semidirect
5. C2×C4×D8C2semidirect
6. #2535C2semidirect
7. #2036C2semidirect
8. #2033C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1958C2semidirect
12. #1954C2semidirect
13. #1804C2semidirect
14. #1788C2semidirect
15. #1786C2semidirect
18. C22×SD16C22semidirect
29. C2×#121C22semidirect
33. C2×C4×D4C22semidirect
39. C2×#107C22semidirect
40. C2×C4⋊C8C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
50. C2×#102C22semidirect
53. C2×C22⋊C8C22semidirect
54. C2×C22⋊C8C22semidirect
55. C2×C4×C8C22semidirect
60. #382C22semidirect
62. #380C22semidirect
63. #340C22semidirect
64. #340C22semidirect
101. C2×D8D4semidirect
107. #124D4semidirect
124. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4632
C2×#4492
C2×#4472
C2×C4×D82
#25352
#20362
#20332
#20002
#19992
#19582
#19542
#18042
#17882
#17862