Finite Group Databasebeta

← #25093

Subgroups — #25093

Order 256.

661 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#460C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#418C211yesmaximal, characteristic, complemented
5#2433C211yesmaximal, characteristic
6#2362C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2041C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic, complemented
15#1851C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic, complemented
20C22×M16C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120—21no—
26C2×#120—21no—
27C2×C4×Q8C2211yescharacteristic, complemented
28C2×#117C2211yescharacteristic
29#468—24no—
30#460—24no—
31#450—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×C4⋊C8C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35#418—24no—
36C2×#103C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×C22⋊C8C2211yescharacteristic
47#392—21no—
48#392—21no—
49#391C2211yescharacteristic, complemented
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic, complemented
53#388—21no—
54#388—21no—
55#387—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2211yescharacteristic, complemented
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4—22no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C22×D4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—41no—
88C2×SD16—41no—
89C2×SD16—41no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—41no—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97#127—22no—
98#127—22no—
99#127—41no—
100#127—41no—
101C22⋊Q8—22no—
102C22⋊Q8D414yescomplemented
103#121—22no—
104#121—22no—
105#120—44no—
106#120—44no—
107C4×Q8—22no—
108C4×Q8—41no—
109#117—22no—
110#117—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131C2×C42C2311yescharacteristic
132#107—22no—
133C4⋊C8—22no—
134C4⋊C8—41no—
135#103—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—41no—
144#102D414yescomplemented
145C22⋊C8—22no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95D411yescharacteristic
151#95D411yescharacteristic
152C24—21no—
153C24—21no—
154C24—21no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—42no—
160C24—81no—
161C24—81no—
162C24C2×D411yescharacteristic
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C2×D412yes—
169C2×D4—28no—
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—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4C2×D412yes—
208C2×D4#4112yes—
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×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D412yes—
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4#4112yes—
225C22×C4C2411yescharacteristic
226SD16—44no—
227SD16—44no—
228SD16—44no—
229M16—44no—
230C2×C8—21no—
231C2×C8—21no—
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×C8C2×D412yes—
242C4⋊C4—24no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4C2×D412yes—
252C4⋊C4C2×D412yes—
253C22⋊C4—24no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—42no—
267C42—42no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—22no—
277C23—48no—
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—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—84no—
298C23—84no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23#12112yes—
304C23C22×D411yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—41no—
308Q8—41no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—48no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—84no—
323D4—84no—
324D4—84no—
325D4—84no—
326D4—84no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
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#12112yes—
359C2×C4AGL(1,ℤ/8)12yes—
360C2×C4AGL(1,ℤ/8)12yes—
361C2×C4C22×D411yescharacteristic
362C2×C4C22×D411yescharacteristic
363C2×C4C2×#4111yescharacteristic
364C2×C4C2×#4111yescharacteristic
365C2×C4H(5,𝔽2)11yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C8—22no—
368C8—41no—
369C8—41no—
370C8—41no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—816no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×#12111yescharacteristic
401C22D4211yescharacteristic
402C22#54611yescharacteristic
403C22C2×AGL(1,ℤ/8)11yescharacteristic
404C22C2×AGL(1,ℤ/8)11yescharacteristic
405C22#57711yescharacteristic
406C22#57711yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—81no—
416C4—81no—
417C4#52511yescharacteristic
418C4#53811yescharacteristic
419C4D4211yescharacteristic
420C4#54611yescharacteristic
421C2—42no—
422C2—82no—
423C2—82no—
424C2—161no—
425C2#242411yescharacteristic
426C2#303811yescharacteristic
427C2#304311yescharacteristic
428C2#326511yescharacteristic
429C2#327211yescharacteristic
430C2#331011yescharacteristic
431C2#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×#468C2semidirect
2. C2×#460C2semidirect
3. C2×#450C2semidirect
4. C2×#418C2semidirect
6. #2362C2semidirect
7. #2067C2semidirect
8. #2055C2semidirect
9. #2041C2semidirect
10. #2014C2semidirect
11. #1982C2semidirect
12. #1925C2semidirect
13. #1880C2semidirect
14. #1875C2semidirect
15. #1851C2semidirect
19. C22×SD16C22semidirect
27. C2×C4×Q8C22semidirect
32. C2×#107C22semidirect
33. C2×C4⋊C8C22semidirect
34. C2×C4⋊C8C22semidirect
36. C2×#103C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
49. #391C22semidirect
52. #389C22semidirect
57. C4×C4⋊C4C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
102. C22⋊Q8D4semidirect
144. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4602
C2×#4502
C2×#4182
#24332
#23622
#20672
#20552
#20412
#20142
#19822
#19252
#18802
#18752
#18512