Finite Group Databasebeta

← #30456

Subgroups — #30456

Order 256.

666 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#492C211yesmaximal, characteristic, complemented
2C2×#460C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5#2433C211yesmaximal, characteristic
6#2429C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#2014C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#1934C211yesmaximal, characteristic, complemented
14#1923C211yesmaximal, characteristic, complemented
15#1880C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22C22×SD16C2211yescharacteristic, complemented
23C2×C4⋊Q8C2211yescharacteristic, complemented
24C2×#127C2211yescharacteristic
25C2×#127C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#120—21no—
30C2×#120—21no—
31C2×C4×Q8C2211yescharacteristic, complemented
32#492—24no—
33#460—24no—
34#450—24no—
35#450—24no—
36C2×C4⋊C8C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C4×C8C2211yescharacteristic, complemented
48#398—21no—
49#396—21no—
50#392—21no—
51#392—21no—
52#391C2211yescharacteristic, complemented
53#390C2211yescharacteristic
54#390C2211yescharacteristic
55#389—21no—
56#389C2211yescharacteristic, complemented
57#388—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4C2311yescharacteristic
79C22×D4C2311yescharacteristic
80C22×D4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×SD16—41no—
91C2×SD16—41no—
92C2×SD16—41no—
93C2×SD16—41no—
94C2×SD16—41no—
95C2×SD16—41no—
96C2×SD16—41no—
97C2×SD16—41no—
98C2×SD16—41no—
99C2×SD16D414yescomplemented
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C4⋊Q8—22no—
105C4⋊Q8—41no—
106#127—22no—
107#127—22no—
108#127—41no—
109#127—41no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—41no—
113C22⋊Q8—41no—
114#121—41no—
115#121—41no—
116#120—44no—
117#120—44no—
118C4×Q8—22no—
119C4×Q8—41no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4D411yescharacteristic
137C2×C22⋊C4D411yescharacteristic
138C2×C42—21no—
139C2×C42C2311yescharacteristic
140C2×C42C2311yescharacteristic
141C4⋊C8—22no—
142#103—41no—
143#103—41no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—22no—
148#102—41no—
149#102D414yescomplemented
150C22⋊C8—22no—
151C22⋊C8—22no—
152C4×C8—22no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95D411yescharacteristic
157#95D411yescharacteristic
158C24—21no—
159C24—21no—
160C24—21no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—42no—
165C24—42no—
166C24—81no—
167C24—81no—
168C24C2×D411yescharacteristic
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8C2×D412yes—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—81no—
216C2×D4—81no—
217C2×D4—81no—
218C2×D4—81no—
219C2×D4C2×D412yes—
220C2×D4C2×D412yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2411yescharacteristic
238SD16—44no—
239SD16—44no—
240SD16—44no—
241SD16—44no—
242SD16—44no—
243SD16—44no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8C2×D412yes—
255C2×C8C2×D412yes—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4C2×D412yes—
268C22⋊C4—22no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4SD1614yescomplemented
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—42no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—48no—
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—
316C23C2×SD1612yes—
317C23C22×D411yescharacteristic
318Q8—22no—
319Q8—22no—
320Q8—41no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4—22no—
330D4—48no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—84no—
340D4—84no—
341D4—84no—
342D4—84no—
343D4—84no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
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—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4C2×SD1612yes—
376C2×C4C2×SD1612yes—
377C2×C4AGL(1,ℤ/8)12yes—
378C2×C4C22×D411yescharacteristic
379C2×C4C22×D411yescharacteristic
380C2×C4H(5,𝔽2)11yescharacteristic
381C2×C4H(5,𝔽2)11yescharacteristic
382C2×C4H(5,𝔽2)11yescharacteristic
383C2×C4H(5,𝔽2)11yescharacteristic
384C8—22no—
385C8—22no—
386C8—41no—
387C8—41no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—816no—
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#53511yescharacteristic
418C22D4211yescharacteristic
419C22D4211yescharacteristic
420C22C22×SD1611yescharacteristic
421C22C2×AGL(1,ℤ/8)11yescharacteristic
422C22#57711yescharacteristic
423C22#57711yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—42no—
429C4—41no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4#53411yescharacteristic
435C4#53511yescharacteristic
436C4D4211yescharacteristic
437C4#56011yescharacteristic
438C2—42no—
439C2—82no—
440C2—82no—
441C2—161no—
442C2#266911yescharacteristic
443C2#320811yescharacteristic
444C2#322211yescharacteristic
445C2#326511yescharacteristic
446C2#327211yescharacteristic
447C2D4×SD1611yescharacteristic
448C2D4×SD1611yescharacteristic
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×#492C2semidirect
2. C2×#460C2semidirect
3. C2×#450C2semidirect
4. C2×#450C2semidirect
6. #2429C2semidirect
7. #2055C2semidirect
8. #2041C2semidirect
9. #2014C2semidirect
10. #2013C2semidirect
11. #2000C2semidirect
12. #2000C2semidirect
13. #1934C2semidirect
14. #1923C2semidirect
15. #1880C2semidirect
21. C22×SD16C22semidirect
22. C22×SD16C22semidirect
23. C2×C4⋊Q8C22semidirect
31. C2×C4×Q8C22semidirect
36. C2×C4⋊C8C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
47. C2×C4×C8C22semidirect
52. #391C22semidirect
56. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
99. C2×SD16D4semidirect
149. #102D4semidirect
276. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4922
C2×#4602
C2×#4502
#24332
#24292
#20552
#20412
#20142
#20132
#20002
#19342
#19232
#18802