Finite Group Databasebeta

← #24442

Subgroups — #24442

Order 256.

785 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5C2×#418C211yesmaximal, characteristic, complemented
6#2415C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic, complemented
13#1870C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22#546—24no—
23#546—24no—
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic, complemented
29C2×#121—21no—
30C2×#121C2211yescharacteristic
31C2×#120—21no—
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×#117C2211yescharacteristic
35C22×C22⋊C4—21no—
36#470—24no—
37#452—24no—
38#450—24no—
39C2×#106C2211yescharacteristic, complemented
40#418—24no—
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#103C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic, complemented
53C2×C22⋊C8C2211yescharacteristic, complemented
54#397—21no—
55#394—21no—
56#391C2211yescharacteristic, complemented
57#389—21no—
58#389C2211yescharacteristic, complemented
59#388—21no—
60#380C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4D414yes—
83C22×D4D411yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—41no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C2×SD16—41no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—41no—
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106#124—22no—
107#124—22no—
108#123—28no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—41no—
114C22⋊Q8D414yes—
115#121—28no—
116#121—22no—
117#121—22no—
118#120—28no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123#117—22no—
124#117—41no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4D411yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C22⋊C4—28no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4D411yescharacteristic
145C2×C22⋊C4D411yescharacteristic
146C2×C42—21no—
147C2×C42C2311yescharacteristic
148#106—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—41no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102D414yescomplemented
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24—21no—
168C24—24no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—81no—
174C24—81no—
175C24C2×D411yescharacteristic
176C24C2×D411yescharacteristic
177#41—44no—
178#41—44no—
179#41—44no—
180#41—44no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8C2×D412yes—
193C2×D4—28no—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—81no—
216C2×D4—81no—
217C2×D4C2×D412yes—
218C2×D4C2×D412yes—
219C2×D4C2×D412yes—
220C2×D4#4112yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D412yes—
248C22×C4C2×D412yes—
249C22×C4C2411yescharacteristic
250Q16—44no—
251SD16—44no—
252SD16—44no—
253M16—44no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8C2×D412yes—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4C2×D412yes—
278C4⋊C4C2×D412yes—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C42—21no—
291C42—21no—
292C42—42no—
293C23—22no—
294C23—21no—
295C23—22no—
296C23—21no—
297C23—22no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—48no—
311C23—48no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23#12012yes—
328C23C22×D411yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337D4—22no—
338D4—22no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—48no—
343D4—44no—
344D4—48no—
345D4—41no—
346D4—41no—
347D4—84no—
348D4#12114yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—48no—
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—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—84no—
388C2×C4#12012yes—
389C2×C4C22×D411yescharacteristic
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C2×#4111yescharacteristic
393C2×C4H(5,𝔽2)11yescharacteristic
394C2×C4H(5,𝔽2)11yescharacteristic
395C8—22no—
396C8—41no—
397C8—41no—
398C8—41no—
399C22—22no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—42no—
417C22—41no—
418C22—42no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—816no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C2×#12011yescharacteristic
429C22D4211yescharacteristic
430C22#54611yescharacteristic
431C22#57511yescharacteristic
432C22#57511yescharacteristic
433C22#57711yescharacteristic
434C22#57711yescharacteristic
435C4—22no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4C2×#12111yescharacteristic
446C4#53411yescharacteristic
447C4D4211yescharacteristic
448C4#54611yescharacteristic
449C2—42no—
450C2—44no—
451C2—81no—
452C2—82no—
453C2#241011yescharacteristic
454C2#300511yescharacteristic
455C2#300911yescharacteristic
456C2#326711yescharacteristic
457C2#327511yescharacteristic
458C2#331111yescharacteristic
459C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#470C2semidirect
3. C2×#452C2semidirect
4. C2×#450C2semidirect
5. C2×#418C2semidirect
6. #2415C2semidirect
7. #2042C2semidirect
8. #2034C2semidirect
9. #2015C2semidirect
10. #2000C2semidirect
11. #1877C2semidirect
12. #1873C2semidirect
13. #1870C2semidirect
14. #1864C2semidirect
15. #1788C2semidirect
18. C22×Q16C22semidirect
28. C2×C22⋊Q8C22semidirect
39. C2×#106C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
44. C2×#103C22semidirect
52. C2×C22⋊C8C22semidirect
53. C2×C22⋊C8C22semidirect
56. #391C22semidirect
58. #389C22semidirect
60. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
158. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4702
C2×#4522
C2×#4502
C2×#4182
#24152
#20422
#20342
#20152
#20002
#18772
#18732
#18702
#18642
#17882