Finite Group Databasebeta

← #24389

Subgroups — #24389

Order 256.

784 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#407C211yesmaximal, characteristic, complemented
6#2848C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic, complemented
12#1894C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic, complemented
20C22×M16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22#525—24no—
23#525—24no—
24C2×#123—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×#120—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×#117C2211yescharacteristic
36#470—24no—
37#468—24no—
38#452—24no—
39C2×#107C2211yescharacteristic, complemented
40C2×#106C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#103C2211yescharacteristic, complemented
45C2×#103C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×#102C2211yescharacteristic, complemented
48C2×#102C2211yescharacteristic, complemented
49#407—24no—
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54#385—21no—
55#385—21no—
56#385—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#380C2211yescharacteristic
60#379—21no—
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64#340C2211yescharacteristic, complemented
65C25D411yescharacteristic
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4C2311yescharacteristic
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4D411yescharacteristic
86C23×C4D411yescharacteristic
87C23×C4D411yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—41no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—41no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—41no—
102C2×M16—41no—
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107#123—28no—
108C22⋊Q8—22no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—41no—
116C22⋊Q8—41no—
117C22⋊Q8D414yes—
118#121—22no—
119#121—22no—
120#121—22no—
121#121D414yes—
122#120—28no—
123C4×D4—28no—
124C4×D4—28no—
125#117—22no—
126#117—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149#107—22no—
150#106—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—41no—
156#102—22no—
157#102—22no—
158#102—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8D414yescomplemented
163#95—21no—
164#95—21no—
165#95—21no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—21no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24C2×D411yescharacteristic
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—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4C2×D412yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D412yes—
248C22×C4C2×D412yes—
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D412yes—
251C22×C4#4112yes—
252C22×C4C2411yescharacteristic
253Q16—44no—
254SD16—44no—
255M16—44no—
256M16—44no—
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×C8—21no—
266C2×C8—21no—
267C2×C8C2×D412yes—
268C2×C8C2×D412yes—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4C2×D412yes—
283C4⋊C4C2×D412yes—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C42—24no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#12012yes—
329C23#12112yes—
330C23#13712yes—
331C23C22×D411yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338Q8—41no—
339Q8—41no—
340D4—22no—
341D4—22no—
342D4—48no—
343D4—48no—
344D4—48no—
345D4—41no—
346D4—41no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
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—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4#12012yes—
392C2×C4#12112yes—
393C2×C4C22×D411yescharacteristic
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D411yescharacteristic
396C2×C4C2×#4111yescharacteristic
397C2×C4H(5,𝔽2)11yescharacteristic
398C2×C4H(5,𝔽2)11yescharacteristic
399C8—22no—
400C8—22no—
401C8—41no—
402C8—41no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—416no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22C2×#12011yescharacteristic
433C22C2×#12111yescharacteristic
434C22#53411yescharacteristic
435C22C2×#13711yescharacteristic
436C22#57511yescharacteristic
437C22#57511yescharacteristic
438C22#57511yescharacteristic
439C4—22no—
440C4—22no—
441C4—41no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4D4211yescharacteristic
450C4D4211yescharacteristic
451C4#54611yescharacteristic
452C4#54611yescharacteristic
453C2—42no—
454C2—42no—
455C2—42no—
456C2—82no—
457C2#241011yescharacteristic
458C2#300511yescharacteristic
459C2#300711yescharacteristic
460C2#305411yescharacteristic
461C2#305611yescharacteristic
462C2#319611yescharacteristic
463C2#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×#470C2semidirect
3. C2×#468C2semidirect
4. C2×#452C2semidirect
5. C2×#407C2semidirect
6. #2848C2semidirect
7. #2036C2semidirect
8. #2034C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #1895C2semidirect
12. #1894C2semidirect
13. #1873C2semidirect
14. #1854C2semidirect
15. #1788C2semidirect
18. C22×Q16C22semidirect
19. C22×SD16C22semidirect
39. C2×#107C22semidirect
40. C2×#106C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
44. C2×#103C22semidirect
45. C2×#103C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#102C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #340C22semidirect
64. #340C22semidirect
162. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4702
C2×#4682
C2×#4522
C2×#4072
#28482
#20362
#20342
#20012
#20002
#18952
#18942
#18732
#18542
#17882