Finite Group Databasebeta

← #30527

Subgroups — #30527

Order 256.

608 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3#2492C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic
5#2055C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2014C211yesmaximal, characteristic, complemented
9#2014C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic, complemented
14#1923C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic, complemented
20C2×C4⋊Q8C2211yescharacteristic, complemented
21C2×#127C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120—21no—
26C2×#120—21no—
27#486—24no—
28#450—24no—
29C2×#106C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×#102C2211yescharacteristic, complemented
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic, complemented
44#400—21no—
45#393—21no—
46#392—21no—
47#392—21no—
48#391—21no—
49#391C2211yescharacteristic, complemented
50#391C2211yescharacteristic, complemented
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#388—21no—
54#380C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic, complemented
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
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×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—41no—
84C2×SD16—41no—
85C2×SD16—41no—
86C2×SD16—41no—
87C2×SD16—41no—
88C2×SD16—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C4⋊Q8—22no—
94C4⋊Q8—41no—
95#127—22no—
96#127—22no—
97#127—41no—
98#127—41no—
99C22⋊Q8—22no—
100C22⋊Q8—41no—
101#121—41no—
102#121—41no—
103#120—44no—
104#120—44no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123#106—41no—
124C4⋊C8—41no—
125#103—41no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—41no—
131#102—41no—
132#102—41no—
133#102—41no—
134#102D414yescomplemented
135C22⋊C8—22no—
136C22⋊C8—41no—
137C4×C8—22no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95D411yescharacteristic
145#95D411yescharacteristic
146C24—21no—
147C24—21no—
148C24—21no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—81no—
155C24—81no—
156C24C2×D411yescharacteristic
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—81no—
166C2×D4—21no—
167C2×D4—21no—
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—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4C2×D412yes—
205C2×D4#4112yes—
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×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2411yescharacteristic
223SD16—44no—
224SD16—44no—
225SD16—44no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8C2×D412yes—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4C2×D412yes—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—81no—
256C22⋊C4SD1614yescomplemented
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—81no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—48no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—84no—
291C23—84no—
292C23—84no—
293C23—84no—
294C23—84no—
295C23—84no—
296C23C2×SD1612yes—
297C23C22×D411yescharacteristic
298Q8—41no—
299Q8—41no—
300Q8—41no—
301Q8—41no—
302Q8—84no—
303D4—22no—
304D4—22no—
305D4—22no—
306D4—22no—
307D4—48no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—84no—
317D4—84no—
318D4—84no—
319D4—84no—
320D4—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
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×C4C2×SD1612yes—
353C2×C4C2×SD1612yes—
354C2×C4AGL(1,ℤ/8)12yes—
355C2×C4C22×D411yescharacteristic
356C2×C4C2×#4111yescharacteristic
357C2×C4H(5,𝔽2)11yescharacteristic
358C2×C4H(5,𝔽2)11yescharacteristic
359C2×C4H(5,𝔽2)11yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C8—22no—
362C8—41no—
363C8—41no—
364C8—41no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—816no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#53511yescharacteristic
395C22D4211yescharacteristic
396C22#54611yescharacteristic
397C22C22×SD1611yescharacteristic
398C22C2×AGL(1,ℤ/8)11yescharacteristic
399C22#57611yescharacteristic
400C22#57711yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—41no—
405C4—42no—
406C4—41no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4#53511yescharacteristic
412C4#53811yescharacteristic
413C4#54611yescharacteristic
414C4H(3,𝔽4)11yescharacteristic
415C2—42no—
416C2—82no—
417C2—82no—
418C2—161no—
419C2#267011yescharacteristic
420C2#320811yescharacteristic
421C2#322411yescharacteristic
422C2#326411yescharacteristic
423C2D4×SD1611yescharacteristic
424C2#329511yescharacteristic
425C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#486C2semidirect
2. C2×#450C2semidirect
3. #2492C2semidirect
5. #2055C2semidirect
6. #2034C2semidirect
7. #2023C2semidirect
8. #2014C2semidirect
9. #2014C2semidirect
10. #2000C2semidirect
11. #1982C2semidirect
12. #1968C2semidirect
13. #1926C2semidirect
14. #1923C2semidirect
15. #1786C2semidirect
19. C22×SD16C22semidirect
20. C2×C4⋊Q8C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×#103C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
40. C2×#102C22semidirect
43. C2×C4×C8C22semidirect
49. #391C22semidirect
50. #391C22semidirect
55. C4×C4⋊C4C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
134. #102D4semidirect
256. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
C2×#4502
#24922
#24332
#20552
#20342
#20232
#20142
#20002
#19822
#19682
#19262
#19232
#17862