Finite Group Databasebeta

← #31057

Subgroups — #31057

Order 256.

691 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5#2408C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#2034C211yesmaximal, characteristic, complemented
11#2034C211yesmaximal, characteristic, complemented
12#2033C211yesmaximal, characteristic, complemented
13#1970C211yesmaximal, characteristic, complemented
14#1968C211yesmaximal, characteristic, complemented
15#1954C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic, complemented
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C22×D8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#120—21no—
27C2×C4×D4C2211yescharacteristic
28#450—24no—
29#449—24no—
30#449—24no—
31#447—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic, complemented
46C2×C22⋊C8C2211yescharacteristic, complemented
47C2×C4×C8C2211yescharacteristic, complemented
48#394—21no—
49#394—21no—
50#392—21no—
51#392—21no—
52#391C2211yescharacteristic, complemented
53#390C2211yescharacteristic
54#390C2211yescharacteristic
55#390C2211yescharacteristic, complemented
56#386—21no—
57#385—21no—
58#380C2211yescharacteristic
59#380C2211yescharacteristic, complemented
60#379—21no—
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25—21no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41—41no—
70C22×Q8C2311yescharacteristic
71C22×D4—22no—
72C22×D4—22no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4D411yescharacteristic
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×SD16—22no—
90C2×SD16—22no—
91C2×SD16—41no—
92C2×SD16—41no—
93C2×SD16—41no—
94C2×SD16D414yescomplemented
95C2×D8—22no—
96C2×D8—22no—
97C2×D8—41no—
98C2×D8—41no—
99C2×D8—41no—
100C2×D8D414yescomplemented
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22⋊Q8—22no—
106C22⋊Q8—41no—
107#121—22no—
108#121—22no—
109#121—22no—
110#121—41no—
111#121—41no—
112#121—41no—
113#120—44no—
114C4×D4—41no—
115C4×D4—41no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4D411yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4D411yescharacteristic
135C2×C22⋊C4D411yescharacteristic
136C2×C42C2311yescharacteristic
137#107—41no—
138#106—41no—
139C4⋊C8—41no—
140#103—22no—
141#103—22no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C4×C8—41no—
153#95—21no—
154#95—21no—
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
169#41—44no—
170#41—44no—
171#41—44no—
172#41—44no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8C2×D412yes—
179C2×D4—22no—
180C2×D4—22no—
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—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—81no—
214C2×D4—81no—
215C2×D4—81no—
216C2×D4C2×D412yes—
217C2×D4C2×D412yes—
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—21no—
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—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243C22×C4C2411yescharacteristic
244SD16—44no—
245SD16—44no—
246D8—44no—
247D8—44no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8C2×D412yes—
259C2×C8C2×D412yes—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C42—21no—
285C42—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—48no—
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—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—84no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23#13512yes—
328C23C22×D411yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—41no—
332Q8—41no—
333D4—22no—
334D4—22no—
335D4—22no—
336D4—22no—
337D4—22no—
338D4—22no—
339D4—44no—
340D4—48no—
341D4—48no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—84no—
349D4—84no—
350D4—84no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—48no—
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×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4AGL(1,ℤ/8)12yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4H(5,𝔽2)11yescharacteristic
390C2×C4H(5,𝔽2)11yescharacteristic
391C2×C4H(5,𝔽2)11yescharacteristic
392C2×C4H(5,𝔽2)11yescharacteristic
393C8—22no—
394C8—22no—
395C8—41no—
396C8—41no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—41no—
417C22—42no—
418C22—41no—
419C22—42no—
420C22—816no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22#53411yescharacteristic
434C22D4211yescharacteristic
435C22D4211yescharacteristic
436C22C2×#13511yescharacteristic
437C22C2×AGL(1,ℤ/8)11yescharacteristic
438C22#57611yescharacteristic
439C22#57711yescharacteristic
440C4—22no—
441C4—41no—
442C4—41no—
443C4—42no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4#53411yescharacteristic
450C4#53411yescharacteristic
451C4D4211yescharacteristic
452C4H(3,𝔽4)11yescharacteristic
453C2—42no—
454C2—81no—
455C2—82no—
456C2—82no—
457C2—82no—
458C2#267911yescharacteristic
459C2#318811yescharacteristic
460C2#319811yescharacteristic
461C2#327211yescharacteristic
462C2#327411yescharacteristic
463C2#328011yescharacteristic
464C2#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×#449C2semidirect
2. C2×#449C2semidirect
3. C2×#450C2semidirect
4. C2×#447C2semidirect
5. #2408C2semidirect
6. #2407C2semidirect
7. #2055C2semidirect
8. #2054C2semidirect
9. #2036C2semidirect
10. #2034C2semidirect
11. #2034C2semidirect
12. #2033C2semidirect
13. #1970C2semidirect
14. #1968C2semidirect
15. #1954C2semidirect
16. C22×#41C22semidirect
32. C2×#107C22semidirect
33. C2×#106C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
45. C2×C22⋊C8C22semidirect
46. C2×C22⋊C8C22semidirect
47. C2×C4×C8C22semidirect
52. #391C22semidirect
55. #390C22semidirect
59. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
94. C2×SD16D4semidirect
100. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C2×#4502
C2×#4472
#24082
#24072
#20552
#20542
#20362
#20342
#20332
#19702
#19682
#19542