Finite Group Databasebeta

← #25124

Subgroups — #25124

Order 256.

681 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2416C211yesmaximal, characteristic, complemented
6#2408C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1936C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic, complemented
20C23×C8C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic, complemented
26C2×#120—21no—
27C2×C4×D4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29#469—24no—
30#447—24no—
31#435—24no—
32C2×#106C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#102—24no—
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic, complemented
46C2×#97C2211yescharacteristic, complemented
47#396—21no—
48#394—21no—
49#394—21no—
50#392—21no—
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#390C2211yescharacteristic, complemented
54#389C2211yescharacteristic, complemented
55#386—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic, complemented
58#379—21no—
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25—21no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—22no—
71C22×D4—21no—
72C22×D4—22no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4D411yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4—21no—
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8—41no—
87C2×D8—41no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—41no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98#121—22no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—22no—
103#121—41no—
104#121—41no—
105#121—41no—
106#121—44no—
107#120—44no—
108C4×D4—22no—
109C4×D4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4D411yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42C2311yescharacteristic
135#106—22no—
136#106—41no—
137C4⋊C8—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—41no—
145#102D414yescomplemented
146C22⋊C8—22no—
147C22⋊C8D414yescomplemented
148#97—22no—
149#95—21no—
150#95—21no—
151#95—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—42no—
160C24—81no—
161C24—81no—
162C24C2×D411yescharacteristic
163C24C2×D411yescharacteristic
164C2×D4—28no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4C2×D412yes—
203C2×D4#4112yes—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D412yes—
227C22×C4#4112yes—
228C22×C4C2411yescharacteristic
229D8—44no—
230D8—44no—
231M16—44no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—44no—
242C2×C8C2×D412yes—
243C2×C8C2×D412yes—
244C2×C8C2×D412yes—
245C4⋊C4—22no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4C2×D412yes—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—21no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—48no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—84no—
308C23—84no—
309C23—84no—
310C23—84no—
311C23—84no—
312C23—84no—
313C23#12112yes—
314C23AGL(1,ℤ/8)12yes—
315C23AGL(1,ℤ/8)12yes—
316C23C22×D411yescharacteristic
317D4—22no—
318D4—22no—
319D4—22no—
320D4—22no—
321D4—48no—
322D4—44no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—84no—
332D4—84no—
333D4—84no—
334D4—84no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—84no—
370C2×C4#12112yes—
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C2×#4111yescharacteristic
374C2×C4C2×#4111yescharacteristic
375C2×C4H(5,𝔽2)11yescharacteristic
376C2×C4H(5,𝔽2)11yescharacteristic
377C8—22no—
378C8—22no—
379C8—22no—
380C8—41no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—42no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—816no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#46914yescomplemented
417C22C2×#12111yescharacteristic
418C22D4211yescharacteristic
419C22#54611yescharacteristic
420C22C2×AGL(1,ℤ/8)11yescharacteristic
421C22C2×AGL(1,ℤ/8)11yescharacteristic
422C22#57611yescharacteristic
423C22#57711yescharacteristic
424C4—22no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4#52511yescharacteristic
434C4#53811yescharacteristic
435C4D4211yescharacteristic
436C4#54611yescharacteristic
437C2—24no—
438C2—81no—
439C2—82no—
440C2—82no—
441C2—161no—
442C2#242411yescharacteristic
443C2C2×#46911yescharacteristic
444C2#306211yescharacteristic
445C2#326411yescharacteristic
446C2#326511yescharacteristic
447C2#330611yescharacteristic
448C2#331811yescharacteristic
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×#469C2semidirect
2. C2×#447C2semidirect
3. C2×#435C2semidirect
4. C22×#102C2semidirect
5. #2416C2semidirect
6. #2408C2semidirect
7. #2065C2semidirect
8. #2054C2semidirect
9. #2034C2semidirect
10. #1999C2semidirect
11. #1972C2semidirect
12. #1936C2semidirect
13. #1895C2semidirect
14. #1893C2semidirect
15. #1853C2semidirect
19. C22×M16C22semidirect
25. C2×#121C22semidirect
32. C2×#106C22semidirect
33. C2×#106C22semidirect
34. C2×C4⋊C8C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
45. C2×C22⋊C8C22semidirect
46. C2×#97C22semidirect
53. #390C22semidirect
54. #389C22semidirect
57. #380C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect
145. #102D4semidirect
147. C22⋊C8D4semidirect
416. C22#469semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4472
C2×#4352
C22×#1022
#24162
#24082
#20652
#20542
#20342
#19992
#19722
#19362
#18952
#18932
#18532