Finite Group Databasebeta

← #25890

Subgroups — #25890

Order 256.

716 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×#488C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2562C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1983C211yesmaximal, characteristic, complemented
10#1954C211yesmaximal, characteristic, complemented
11#1788C211yesmaximal, characteristic, complemented
12#1788C211yesmaximal, characteristic, complemented
13#1786C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18#546—24no—
19#546—24no—
20C2×#125C2211yescharacteristic, complemented
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121—21no—
25C2×#121C2211yescharacteristic
26C2×#120—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C22×C22⋊C4—21no—
30#488—24no—
31#478—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic, complemented
50C2×C22⋊C8C2211yescharacteristic, complemented
51C2×C4×C8C2211yescharacteristic, complemented
52#400—21no—
53#394—21no—
54#388—21no—
55#388—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic, complemented
59#380C2211yescharacteristic, complemented
60C4×C4⋊C4C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25—21no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—21no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4D411yescharacteristic
81C22×D4C2311yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91#125—22no—
92#125—22no—
93#124—22no—
94#124D414yes—
95#123—28no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98#121—28no—
99#121—22no—
100#121—22no—
101#120—28no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4D414yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—28no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42C2311yescharacteristic
129#107—22no—
130#106—22no—
131C4⋊C8—22no—
132#103—22no—
133#103—22no—
134#103—41no—
135#103—41no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143#102—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C4×C8—22no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—24no—
156C24—21no—
157C24—42no—
158C24—42no—
159C24—42no—
160C24—42no—
161C24—81no—
162C24—81no—
163C24C2×D411yescharacteristic
164C24C2×D411yescharacteristic
165#41—44no—
166#41—44no—
167#41—44no—
168#41—44no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8C2×D412yes—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—22no—
178C2×D4—24no—
179C2×D4—22no—
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—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—44no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4C2×D412yes—
200C2×D4C2×D412yes—
201C2×D4#4112yes—
202C2×D4#4112yes—
203C22×C4—24no—
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—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2411yescharacteristic
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—21no—
242C2×C8—21no—
243C2×C8—21no—
244C4⋊C4—24no—
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—24no—
259C22⋊C4—24no—
260C22⋊C4—22no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—22no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C42—81no—
270C42C2×D412yes—
271C42C2×D412yes—
272C23—21no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—48no—
290C23—48no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23#13512yes—
307C23C22×D411yescharacteristic
308Q8—22no—
309Q8—22no—
310Q8—41no—
311Q8—41no—
312D4—22no—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—44no—
318D4—48no—
319D4—48no—
320D4—41no—
321D4—41no—
322D4—84no—
323D4#12414yes—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—48no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
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—84no—
362C2×C4#12012yes—
363C2×C4#12012yes—
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D411yescharacteristic
366C2×C4C2×#4111yescharacteristic
367C2×C4C2×#4111yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C8—41no—
371C8—41no—
372C8—41no—
373C8—41no—
374C22—24no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—42no—
386C22—41no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—816no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22C2×#12011yescharacteristic
404C22#54611yescharacteristic
405C22#54611yescharacteristic
406C22C2×#13511yescharacteristic
407C22#57511yescharacteristic
408C22#57611yescharacteristic
409C22#57711yescharacteristic
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4C2×#12411yescharacteristic
421C4#53511yescharacteristic
422C4#54611yescharacteristic
423C4#54611yescharacteristic
424C2—42no—
425C2—44no—
426C2—81no—
427C2—82no—
428C2#243511yescharacteristic
429C2#300211yescharacteristic
430C2#301211yescharacteristic
431C2#329611yescharacteristic
432C2#329711yescharacteristic
433C2#332211yescharacteristic
434C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
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
2. C2×#488C2semidirect
3. C2×#478C2semidirect
4. #2562C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1983C2semidirect
10. #1954C2semidirect
11. #1788C2semidirect
12. #1788C2semidirect
13. #1786C2semidirect
14. #1786C2semidirect
15. C4×#102C2semidirect
20. C2×#125C22semidirect
32. C2×#107C22semidirect
33. C2×#106C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
49. C2×C22⋊C8C22semidirect
50. C2×C22⋊C8C22semidirect
51. C2×C4×C8C22semidirect
58. #380C22semidirect
59. #380C22semidirect
60. C4×C4⋊C4C22semidirect
61. #340C22semidirect
62. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4882
C2×#4782
#25622
#20362
#20352
#20342
#20332
#19832
#19542
#17882
#17862
C4×#1022