Finite Group Databasebeta

← #22423

Subgroups — #22423

Order 256.

965 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×D42C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2352C211yesmaximal, characteristic, complemented
6#1982C211yesmaximal, characteristic, complemented
7#1961C211yesmaximal, characteristic, complemented
8#1895C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1804C211yesmaximal, characteristic, complemented
13#1786C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C23×D4—21no—
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×M16C2211yescharacteristic, complemented
20C23×C8C2211yescharacteristic, complemented
21D42—24no—
22D42C418yescomplemented
23C2×#127C2211yescharacteristic
24C2×#121—21no—
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#120—21no—
28C2×#120—21no—
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic, complemented
32C22×C4⋊C4C2211yescharacteristic, complemented
33#435—24no—
34C2×C4⋊C8C2211yescharacteristic, complemented
35#418—24no—
36C2×#102—24no—
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×#102C2211yescharacteristic, complemented
48C2×#102C2211yescharacteristic, complemented
49C2×C22⋊C8C2211yescharacteristic, complemented
50C2×C22⋊C8C2211yescharacteristic, complemented
51C2×#97C2211yescharacteristic, complemented
52#388—21no—
53#387—21no—
54#386—21no—
55#382C2211yescharacteristic, complemented
56#380C2211yescharacteristic, complemented
57#380C2211yescharacteristic, complemented
58C4×C4⋊C4C2211yescharacteristic, complemented
59C4×C22⋊C4—21no—
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64C25—21no—
65C25—21no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—28no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—21no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—44no—
84C22×D4—44no—
85C22×D4C2×C414yes—
86C22×D4C2×C414yescomplemented
87C22×D4D411yescharacteristic
88C22×D4C2311yescharacteristic
89C22×D4C2311yescharacteristic
90C22×D4C2311yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—41no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105#127—22no—
106#127C2×C414yescomplemented
107#121—22no—
108#121—22no—
109#121—28no—
110#121C2×C414yes—
111#121C2×C414yescomplemented
112#120—28no—
113#120—28no—
114C4×D4—22no—
115C4×D4C2×C414yes—
116C4×D4C2×C414yescomplemented
117C4×D4D414yes—
118#117—22no—
119#117—41no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4D411yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D411yescharacteristic
136C2×C22⋊C4D411yescharacteristic
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2311yescharacteristic
140C4⋊C8—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—22no—
148#102—41no—
149#102—41no—
150#102—41no—
151#102—41no—
152#102D414yescomplemented
153C22⋊C8—22no—
154C22⋊C8—22no—
155#97—22no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—24no—
165C24—24no—
166C24—42no—
167C24—42no—
168C24—42no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—42no—
174C24—81no—
175C24—81no—
176C24—81no—
177C24—81no—
178C24C2×D411yescharacteristic
179C24C2×D411yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—28no—
183C2×D4—28no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—21no—
191C2×D4—21no—
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—44no—
204C2×D4—44no—
205C2×D4—44no—
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×D4—42no—
215C2×D4—432no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—81no—
221C2×D4—81no—
222C2×D4C22⋊C414yes—
223C2×D4C22×C412yes—
224C2×D4C22×C412yes—
225C2×D4C22×C412yes—
226C2×D4C2×D412yes—
227C2×D4#4112yes—
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—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4C22×C412yes—
248C22×C4C22×C412yes—
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D412yes—
254C22×C4C2×D412yes—
255C22×C4C2411yescharacteristic
256M16—44no—
257C2×C8—21no—
258C2×C8—21no—
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—44no—
269C2×C8C2×D412yescomplemented
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—42no—
279C4⋊C4C22⋊C414yes—
280C4⋊C4C22×C412yes—
281C4⋊C4C2×D412yescomplemented
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—81no—
288C22⋊C4C22⋊C414yes—
289C22⋊C4C22⋊C414yes—
290C42—42no—
291C42—81no—
292C42C22×C412yes—
293C42C2×D412yes—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—21no—
300C23—22no—
301C23—21no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—48no—
310C23—48no—
311C23—48no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—48no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23—84no—
334C23—84no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—84no—
339C23C2×C22⋊C412yes—
340C23C2×C22⋊C412yes—
341C23AGL(1,ℤ/8)12yes—
342C23AGL(1,ℤ/8)12yes—
343C23C22×D411yescharacteristic
344D4—28no—
345D4—22no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—22no—
350D4—22no—
351D4—22no—
352D4—48no—
353D4—48no—
354D4—48no—
355D4—48no—
356D4—48no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—84no—
362D4C4×D414yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—48no—
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—84no—
396C2×C4C2×C22⋊C412yes—
397C2×C4C2×C22⋊C412yes—
398C2×C4C2×C22⋊C412yes—
399C2×C4C2×C22⋊C412yes—
400C2×C4C23×C411yescharacteristic
401C2×C4C22×D411yescharacteristic
402C2×C4C22×D411yescharacteristic
403C2×C4C2×#4111yescharacteristic
404C2×C4H(5,𝔽2)11yescharacteristic
405C2×C4H(5,𝔽2)11yescharacteristic
406C8—22no—
407C8—41no—
408C8—41no—
409C8—41no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—24no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—41no—
423C22—42no—
424C22—41no—
425C22—42no—
426C22—41no—
427C22—42no—
428C22—41no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—816no—
437C22—816no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#41814yescomplemented
447C22C22×C22⋊C411yescharacteristic
448C22D4211yescharacteristic
449C22#54611yescharacteristic
450C22C2×AGL(1,ℤ/8)11yescharacteristic
451C22C2×AGL(1,ℤ/8)11yescharacteristic
452C22#57611yescharacteristic
453C22#57711yescharacteristic
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4C2×C4×D411yescharacteristic
464C4#51811yescharacteristic
465C4D4211yescharacteristic
466C4#54611yescharacteristic
467C2—24no—
468C2—42no—
469C2—44no—
470C2—82no—
471C2—82no—
472C2D4×C22⋊C411yescharacteristic
473C2C2×#41811yescharacteristic
474C2#289711yescharacteristic
475C2#326411yescharacteristic
476C2#326511yescharacteristic
477C2#331011yescharacteristic
478C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×D42C2
LCSG ▷ C2×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
2. C2×#435C2semidirect
3. C2×#418C2semidirect
4. C22×#102C2semidirect
5. #2352C2semidirect
6. #1982C2semidirect
7. #1961C2semidirect
8. #1895C2semidirect
9. #1893C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1804C2semidirect
13. #1786C2semidirect
14. #1786C2semidirect
15. #1762C2semidirect
19. C22×M16C22semidirect
20. C23×C8C22semidirect
22. D42C4semidirect
31. C2×#117C22semidirect
32. C22×C4⋊C4C22semidirect
34. C2×C4⋊C8C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#102C22semidirect
49. C2×C22⋊C8C22semidirect
50. C2×C22⋊C8C22semidirect
51. C2×#97C22semidirect
55. #382C22semidirect
56. #380C22semidirect
57. #380C22semidirect
58. C4×C4⋊C4C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #340C22semidirect
86. C22×D4C2×C4semidirect
106. #127C2×C4semidirect
111. #121C2×C4semidirect
116. C4×D4C2×C4semidirect
152. #102D4semidirect
269. C2×C8C2×D4semidirect
281. C4⋊C4C2×D4semidirect
446. C22#418semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4352
C2×#4182
C22×#1022
#23522
#19822
#19612
#18952
#18932
#18752
#18742
#18042
#17862
#17622