Finite Group Databasebeta

← #24050

Subgroups — #24050

Order 256.

725 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×M16C2211yescharacteristic
19#546—24no—
20#546—24no—
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121—21no—
25C2×#121C2211yescharacteristic
26C2×#120—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C22⋊C4—21no—
31#485—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34#418—24no—
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, complemented
49C2×C22⋊C8C2211yescharacteristic, complemented
50#397—21no—
51#394—21no—
52#389C2211yescharacteristic, complemented
53#389C2211yescharacteristic, complemented
54#388—21no—
55#387—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic, complemented
58#380C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, 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—22no—
72C22×D4—22no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—22no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4D414yes—
80C22×D4D411yescharacteristic
81C22×D4C2311yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C2×M16—22no—
88C2×M16—41no—
89C2×M16—41no—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94#124—22no—
95#124—22no—
96#123—28no—
97C22⋊Q8—22no—
98C22⋊Q8D414yes—
99#121—28no—
100#121—22no—
101#121—22no—
102#120—28no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107#117—22no—
108#117—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—28no—
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⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4D411yescharacteristic
129C2×C22⋊C4D411yescharacteristic
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#106—22no—
134#103—22no—
135#103—22no—
136#103—41no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—41no—
145#102—41no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24—24no—
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—28no—
176C2×D4—24no—
177C2×D4—22no—
178C2×D4—24no—
179C2×D4—24no—
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—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4C2×D412yes—
200C2×D4C2×D412yes—
201C2×D4#4112yes—
202C2×D4#4112yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
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—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D412yes—
230C22×C4C2×D412yes—
231C22×C4C2411yescharacteristic
232M16—44no—
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—
244C2×C8—21no—
245C4⋊C4—24no—
246C4⋊C4—22no—
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—21no—
256C4⋊C4—42no—
257C4⋊C4—81no—
258C4⋊C4C2×D412yes—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—42no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—48no—
291C23—48no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23#12012yes—
308C23C22×D411yescharacteristic
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—22no—
318D4—48no—
319D4—44no—
320D4—48no—
321D4—41no—
322D4—41no—
323D4—84no—
324D4#12314yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—48no—
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—41no—
362C2×C4—41no—
363C2×C4—84no—
364C2×C4#12012yes—
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4C2×#4111yescharacteristic
368C2×C4C2×#4111yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C2×C4H(5,𝔽2)11yescharacteristic
371C8—41no—
372C8—41no—
373C8—41no—
374C8—41no—
375C22—22no—
376C22—24no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—42no—
389C22—41no—
390C22—42no—
391C22—41no—
392C22—42no—
393C22—41no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—816no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22C2×#12011yescharacteristic
405C22#54611yescharacteristic
406C22#54611yescharacteristic
407C22#57511yescharacteristic
408C22#57511yescharacteristic
409C22#57611yescharacteristic
410C22#57711yescharacteristic
411C4—22no—
412C4—41no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4C2×#12311yescharacteristic
422C4#53411yescharacteristic
423C4#54611yescharacteristic
424C4#54611yescharacteristic
425C2—42no—
426C2—44no—
427C2—81no—
428C2—82no—
429C2#240511yescharacteristic
430C2#300511yescharacteristic
431C2#301211yescharacteristic
432C2#331111yescharacteristic
433C2#331911yescharacteristic
434C2#332211yescharacteristic
435C2#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×#485C2semidirect
3. C2×#418C2semidirect
4. #2485C2semidirect
5. #2042C2semidirect
6. #2042C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1894C2semidirect
10. #1870C2semidirect
11. #1864C2semidirect
12. #1862C2semidirect
13. #1788C2semidirect
14. #1788C2semidirect
15. #1786C2semidirect
32. C2×#107C22semidirect
33. C2×#106C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
48. C2×C22⋊C8C22semidirect
49. C2×C22⋊C8C22semidirect
52. #389C22semidirect
53. #389C22semidirect
57. #380C22semidirect
58. #380C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4852
C2×#4182
#24852
#20422
#20342
#20332
#18942
#18702
#18642
#18622
#17882
#17862