Finite Group Databasebeta

← #29601

Subgroups — #29601

Order 256.

731 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4#2848C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1779C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic, complemented
20#525—24no—
21#525—24no—
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#120—21no—
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33C2×#117C2211yescharacteristic
34#485—24no—
35#452—24no—
36C2×#107C2211yescharacteristic, complemented
37C2×#106C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#103C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52#385—21no—
53#385—21no—
54#385—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#379—21no—
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25D411yescharacteristic
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41D414yes—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4D411yescharacteristic
84C23×C4D411yescharacteristic
85C23×C4D411yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×Q16—22no—
90C2×Q16—41no—
91C2×Q16—41no—
92C2×SD16—22no—
93C2×SD16—41no—
94C2×SD16—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99#123—28no—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110#121—22no—
111#121—22no—
112#121—22no—
113#121—22no—
114#120—28no—
115C4×D4—28no—
116C4×D4—28no—
117#117—22no—
118#117D414yes—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C42—21no—
141#107—22no—
142#106—22no—
143#103—22no—
144#103—22no—
145#103—22no—
146#103—41no—
147#103—41no—
148#102—22no—
149#102—41no—
150#102—41no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157#95—21no—
158#95—21no—
159#95—21no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—21no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24C2×D411yescharacteristic
169C24C2×D411yescharacteristic
170C24C2×D411yescharacteristic
171#41—44no—
172#41—44no—
173#41—44no—
174#41—44no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8C2×D412yes—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4C2×D412yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
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—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D412yes—
245C22×C4C2×D412yes—
246C22×C4C2411yescharacteristic
247Q16—44no—
248SD16—44no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4C2×D412yes—
275C4⋊C4C2×D412yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—24no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23#12012yes—
321C23#13512yes—
322C23#13712yes—
323C23C22×D411yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332D4—22no—
333D4—22no—
334D4—48no—
335D4—48no—
336D4—48no—
337D4—41no—
338D4—41no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—44no—
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×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4#12012yes—
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D411yescharacteristic
387C2×C4H(5,𝔽2)11yescharacteristic
388C2×C4H(5,𝔽2)11yescharacteristic
389C2×C4H(5,𝔽2)11yescharacteristic
390C2×C4H(5,𝔽2)11yescharacteristic
391C8—41no—
392C8—41no—
393C8—41no—
394C8—41no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—416no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C2×#12011yescharacteristic
425C22#53411yescharacteristic
426C22#53411yescharacteristic
427C22C2×#13511yescharacteristic
428C22C2×#13711yescharacteristic
429C22#57511yescharacteristic
430C22#57511yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4#53411yescharacteristic
442C4#53411yescharacteristic
443C4#53511yescharacteristic
444C4#53511yescharacteristic
445C2—42no—
446C2—42no—
447C2—42no—
448C2—82no—
449C2#261311yescharacteristic
450C2#300211yescharacteristic
451C2#300711yescharacteristic
452C2#319011yescharacteristic
453C2#319111yescharacteristic
454C2#319611yescharacteristic
455C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#525C2
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
2. C2×#485C2semidirect
3. C2×#452C2semidirect
4. #2848C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #2001C2semidirect
10. #2001C2semidirect
11. #2000C2semidirect
12. #2000C2semidirect
13. #1825C2semidirect
14. #1788C2semidirect
15. #1779C2semidirect
18. C22×Q16C22semidirect
19. C22×SD16C22semidirect
36. C2×#107C22semidirect
37. C2×#106C22semidirect
38. C2×#103C22semidirect
39. C2×#103C22semidirect
40. C2×#103C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4852
C2×#4522
#28482
#20362
#20352
#20342
#20332
#20012
#20002
#18252
#17882
#17792