Finite Group Databasebeta

← #30501

Subgroups — #30501

Order 256.

663 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5#2540C211yesmaximal, characteristic, complemented
6#2408C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic, complemented
14#1925C211yesmaximal, characteristic, complemented
15#1869C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C23×C8C2211yescharacteristic, complemented
22C2×#124C2211yescharacteristic, complemented
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic, complemented
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic, complemented
29C2×#120—21no—
30C2×C4×D4C2211yescharacteristic
31#466—24no—
32#465—24no—
33#450—24no—
34#447—24no—
35C2×#107C2211yescharacteristic, complemented
36C2×#106C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic, complemented
48#399—21no—
49#394—21no—
50#392—21no—
51#392—21no—
52#390—21no—
53#390—21no—
54#390C2211yescharacteristic
55#390C2211yescharacteristic
56#380C2211yescharacteristic
57#379—21no—
58C4×C4⋊C4C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4D411yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C22×D4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—41no—
86C2×SD16—41no—
87C2×SD16—41no—
88C2×D8—22no—
89C2×D8—22no—
90C2×D8—22no—
91C2×D8—41no—
92C2×D8—41no—
93C2×D8—41no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—41no—
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101#124—41no—
102#124—41no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—41no—
106C22⋊Q8—41no—
107#121—22no—
108#121—22no—
109#121—22no—
110#121—22no—
111#121—41no—
112#121—41no—
113#121—41no—
114#121—41no—
115#120—44no—
116C4×D4—41no—
117C4×D4—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4D411yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4D411yescharacteristic
139C2×C22⋊C4D411yescharacteristic
140C2×C42—21no—
141C2×C42C2311yescharacteristic
142#107—22no—
143#106—22no—
144#103—22no—
145#103—41no—
146#102—22no—
147#102—22no—
148#102—22no—
149#102—41no—
150#102D414yescomplemented
151#102D414yescomplemented
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—41no—
155#95—21no—
156#95—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—42no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—42no—
165C24—81no—
166C24—81no—
167C24C2×D411yescharacteristic
168C24C2×D411yescharacteristic
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4—81no—
212C2×D4—81no—
213C2×D4C2×D412yes—
214C2×D4C2×D412yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
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—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2411yescharacteristic
237SD16—44no—
238SD16—44no—
239D8—44no—
240D8—44no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—44no—
252C2×C8C2×D412yes—
253C2×C8C2×D412yes—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4C2×D412yes—
264C4⋊C4C2×D412yes—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—81no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—22no—
296C23—22no—
297C23—48no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23—84no—
321C23—84no—
322C23—84no—
323C23#13512yes—
324C23AGL(1,ℤ/8)12yes—
325C23C22×D411yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330D4—22no—
331D4—22no—
332D4—22no—
333D4—22no—
334D4—44no—
335D4—48no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—84no—
345D4—84no—
346D4—84no—
347D4—84no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
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×C4C22×D411yescharacteristic
384C2×C4C22×D411yescharacteristic
385C2×C4H(5,𝔽2)11yescharacteristic
386C2×C4H(5,𝔽2)11yescharacteristic
387C2×C4H(5,𝔽2)11yescharacteristic
388C2×C4H(5,𝔽2)11yescharacteristic
389C8—22no—
390C8—22no—
391C8—41no—
392C8—41no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—816no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#53411yescharacteristic
430C22D4211yescharacteristic
431C22D4211yescharacteristic
432C22C2×#13511yescharacteristic
433C22C2×AGL(1,ℤ/8)11yescharacteristic
434C22#57611yescharacteristic
435C22#57711yescharacteristic
436C4—41no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4#53511yescharacteristic
446C4#53511yescharacteristic
447C4D4211yescharacteristic
448C4#56011yescharacteristic
449C2—42no—
450C2—42no—
451C2—82no—
452C2—82no—
453C2—161no—
454C2#266911yescharacteristic
455C2#318811yescharacteristic
456C2#319811yescharacteristic
457C2#326411yescharacteristic
458C2#326511yescharacteristic
459C2#328011yescharacteristic
460C2#328211yescharacteristic
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×#466C2semidirect
2. C2×#465C2semidirect
3. C2×#450C2semidirect
4. C2×#447C2semidirect
5. #2540C2semidirect
6. #2408C2semidirect
7. #2036C2semidirect
8. #2034C2semidirect
9. #2012C2semidirect
10. #2012C2semidirect
11. #2000C2semidirect
12. #1999C2semidirect
13. #1926C2semidirect
14. #1925C2semidirect
15. #1869C2semidirect
21. C23×C8C22semidirect
22. C2×#124C22semidirect
24. C2×C22⋊Q8C22semidirect
28. C2×#121C22semidirect
35. C2×#107C22semidirect
36. C2×#106C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
47. C2×C22⋊C8C22semidirect
58. C4×C4⋊C4C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
150. #102D4semidirect
151. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4652
C2×#4502
C2×#4472
#25402
#24082
#20362
#20342
#20122
#20002
#19992
#19262
#19252
#18692