Finite Group Databasebeta

← #25097

Subgroups — #25097

Order 256.

689 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic, complemented
2C2×#459C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic, complemented
4C2×#418C211yesmaximal, characteristic, complemented
5#2433C211yesmaximal, characteristic, complemented
6#2363C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2041C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1926C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic, complemented
15#1851C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic, complemented
20C22×M16C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120—21no—
26C2×#120—21no—
27C2×C4×D4C2211yescharacteristic, complemented
28C2×#117C2211yescharacteristic
29#469—24no—
30#459—24no—
31#447—24no—
32C2×#106C2211yescharacteristic, complemented
33C2×C4⋊C8C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35#418—24no—
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×C22⋊C8C2211yescharacteristic
47#392—21no—
48#392—21no—
49#390—21no—
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic, complemented
53#389C2211yescharacteristic, complemented
54#388—21no—
55#386—21no—
56C4×C4⋊C4C2211yescharacteristic, complemented
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
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
80C22×D4C2311yescharacteristic
81C23×C4—21no—
82C23×C4C2311yescharacteristic
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8—22no—
87C2×D8—22no—
88C2×D8—22no—
89C2×D8—41no—
90C2×D8—41no—
91C2×D8—41no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99#127—22no—
100#127—22no—
101#127—41no—
102#127—41no—
103#121—22no—
104#121—22no—
105#121—22no—
106#121D414yescomplemented
107#120—44no—
108#120—44no—
109C4×D4—22no—
110C4×D4—41no—
111#117—22no—
112#117—41no—
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⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4D411yescharacteristic
129C2×C22⋊C4D411yescharacteristic
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#106—22no—
135C4⋊C8—22no—
136C4⋊C8—41no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102—41no—
146#102D414yescomplemented
147C22⋊C8—22no—
148#95—21no—
149#95—21no—
150#95D411yescharacteristic
151#95D411yescharacteristic
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—42no—
160C24—42no—
161C24—81no—
162C24—81no—
163C24C2×D411yescharacteristic
164C2×D4—28no—
165C2×D4—24no—
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—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—44no—
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—42no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C2×D4—81no—
210C2×D4C2×D412yes—
211C2×D4C2×D412yes—
212C2×D4#4112yes—
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×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D412yes—
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4#4112yes—
231C22×C4C2411yescharacteristic
232D8—44no—
233D8—44no—
234D8—44no—
235M16—44no—
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—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8C2×D412yes—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—81no—
255C4⋊C4C2×D412yes—
256C4⋊C4C2×D412yes—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—42no—
274C42—81no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
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—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—84no—
311C23—84no—
312C23—84no—
313C23—84no—
314C23—84no—
315C23—84no—
316C23#12112yes—
317C23C22×D411yescharacteristic
318D4—22no—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—48no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—84no—
336D4—84no—
337D4—84no—
338D4—84no—
339D4—84no—
340D4—84no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—84no—
370C2×C4#12112yes—
371C2×C4AGL(1,ℤ/8)12yes—
372C2×C4AGL(1,ℤ/8)12yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D411yescharacteristic
375C2×C4C2×#4111yescharacteristic
376C2×C4C2×#4111yescharacteristic
377C2×C4H(5,𝔽2)11yescharacteristic
378C2×C4H(5,𝔽2)11yescharacteristic
379C8—22no—
380C8—41no—
381C8—41no—
382C8—41no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—816no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22C2×#12111yescharacteristic
420C22D4211yescharacteristic
421C22#54611yescharacteristic
422C22C2×AGL(1,ℤ/8)11yescharacteristic
423C22C2×AGL(1,ℤ/8)11yescharacteristic
424C22#57611yescharacteristic
425C22#57611yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—42no—
430C4—41no—
431C4—42no—
432C4—41no—
433C4—81no—
434C4—81no—
435C4#52511yescharacteristic
436C4#53811yescharacteristic
437C4D4211yescharacteristic
438C4#54611yescharacteristic
439C2—42no—
440C2—82no—
441C2—82no—
442C2—82no—
443C2—161no—
444C2#242411yescharacteristic
445C2#303811yescharacteristic
446C2#304111yescharacteristic
447C2#326411yescharacteristic
448C2#327411yescharacteristic
449C2#331811yescharacteristic
450C2#332511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#469C2semidirect
2. C2×#459C2semidirect
3. C2×#447C2semidirect
4. C2×#418C2semidirect
5. #2433C2semidirect
6. #2363C2semidirect
7. #2065C2semidirect
8. #2054C2semidirect
9. #2041C2semidirect
10. #2011C2semidirect
11. #1982C2semidirect
12. #1926C2semidirect
13. #1879C2semidirect
14. #1875C2semidirect
15. #1851C2semidirect
19. C22×D8C22semidirect
27. C2×C4×D4C22semidirect
32. C2×#106C22semidirect
33. C2×C4⋊C8C22semidirect
34. C2×C4⋊C8C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
52. #390C22semidirect
53. #389C22semidirect
56. C4×C4⋊C4C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
106. #121D4semidirect
146. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4592
C2×#4472
C2×#4182
#24332
#23632
#20652
#20542
#20412
#20112
#19822
#19262
#18792
#18752
#18512