Finite Group Databasebeta

← #29744

Subgroups — #29744

Order 256.

584 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic, complemented
3#2848C211yesmaximal, characteristic, complemented
4#2847C211yesmaximal, characteristic
5#2076C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1894C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×SD16C2211yescharacteristic, complemented
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#120—21no—
27#468—24no—
28C2×#107C2211yescharacteristic, complemented
29C2×#107C2211yescharacteristic, complemented
30C2×#106C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2211yescharacteristic, complemented
33C2×#103C2211yescharacteristic, complemented
34C2×#102C2211yescharacteristic, complemented
35C2×#102C2211yescharacteristic, complemented
36C2×#102C2211yescharacteristic, complemented
37#407—24no—
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42#386—21no—
43#386—21no—
44#386—21no—
45#385—21no—
46#385—21no—
47#385—21no—
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#379—21no—
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25D411yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4D411yescharacteristic
74C23×C4D411yescharacteristic
75C23×C4D411yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×SD16—22no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×M16—22no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16D414yescomplemented
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98#121—22no—
99#121—22no—
100#121—41no—
101#121—41no—
102#121—41no—
103#121—41no—
104#120—44no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
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⋊C4—21no—
126#107—22no—
127#107—41no—
128#106—41no—
129#103—22no—
130#103—41no—
131#103—41no—
132#102—22no—
133#102—41no—
134#102—41no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—21no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24C2×D411yescharacteristic
154C24C2×D411yescharacteristic
155C24C2×D411yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D412yes—
217C22×C4#4112yes—
218C22×C4C2411yescharacteristic
219SD16—44no—
220M16—44no—
221M16—44no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8C2×D412yes—
233C2×C8C2×D412yes—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#12112yes—
294C23AGL(1,ℤ/8)12yes—
295C23#13712yes—
296C23C22×D411yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—41no—
300Q8—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—84no—
306D4—84no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
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×C4#12112yes—
353C2×C4C22×D411yescharacteristic
354C2×C4C2×#4111yescharacteristic
355C2×C4H(5,𝔽2)11yescharacteristic
356C2×C4H(5,𝔽2)11yescharacteristic
357C2×C4H(5,𝔽2)11yescharacteristic
358C2×C4H(5,𝔽2)11yescharacteristic
359C8—22no—
360C8—22no—
361C8—41no—
362C8—41no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—416no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C2×#12111yescharacteristic
393C22#53411yescharacteristic
394C22#53411yescharacteristic
395C22C2×AGL(1,ℤ/8)11yescharacteristic
396C22C2×#13711yescharacteristic
397C22#57511yescharacteristic
398C22#57511yescharacteristic
399C4—22no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4#53411yescharacteristic
410C4#53511yescharacteristic
411C4#53811yescharacteristic
412C4#53811yescharacteristic
413C2—42no—
414C2—42no—
415C2—42no—
416C2—161no—
417C2#261711yescharacteristic
418C2#305511yescharacteristic
419C2#305611yescharacteristic
420C2#319311yescharacteristic
421C2#319411yescharacteristic
422C2#319611yescharacteristic
423C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#468C2semidirect
2. C2×#407C2semidirect
3. #2848C2semidirect
5. #2076C2semidirect
6. #2075C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2034C2semidirect
10. #2033C2semidirect
11. #2000C2semidirect
12. #2000C2semidirect
13. #1895C2semidirect
14. #1894C2semidirect
15. #1809C2semidirect
17. C22×SD16C22semidirect
28. C2×#107C22semidirect
29. C2×#107C22semidirect
30. C2×#106C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
87. C2×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4072
#28482
#28472
#20762
#20752
#20362
#20352
#20342
#20332
#20002
#18952
#18942
#18092