Finite Group Databasebeta

← #62128

Subgroups — #62128

Order 256.

1009 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3499C211yesmaximal, characteristic, complemented
2#3481C211yesmaximal, characteristic, complemented
3#3384C212yesmaximal, complemented
4#3373C212yesmaximal, complemented
5#3345C214yesmaximal, complemented
6#3335C212yesmaximal, complemented
7#3306C214yesmaximal, complemented
8#3294C212yesmaximal, complemented
9#3253C211yesmaximal, characteristic, complemented
10#3247C212yesmaximal, complemented
11#3238C212yesmaximal, complemented
12#3227C211yesmaximal, characteristic, complemented
13#3109C212yesmaximal, complemented
14#3106C211yesmaximal, characteristic, complemented
15#3079C211yesmaximal, characteristic, complemented
16C2×#476C212yesmaximal, complemented
17#2990C211yesmaximal, characteristic, complemented
18C2×2−1+4—21no—
19#559C2211yescharacteristic
20#554C2211yescharacteristic
21#554C2211yescharacteristic
22#553—21no—
23#553C2212yes—
24#551—22no—
25D4×Q8—21no—
26D4×Q8C2212yescomplemented
27#547—21no—
28#547—22no—
29#547C2212yes—
30#547C2214yescomplemented
31#546—21no—
32#546C2212yes—
33D42C2211yescharacteristic
34#536—22no—
35#536—21no—
36#532C2211yescharacteristic
37#532C2211yescharacteristic, complemented
38#532C2212yescomplemented
39C2×C4⋊Q8C2211yescharacteristic, complemented
40C2×C4⋊Q8C2212yescomplemented
41#529—21no—
42#529C2212yes—
43#525—22no—
44C4×#41C2211yescharacteristic
45C4×#41C2211yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2212yes—
48#501C2214yescomplemented
49#498C2212yescomplemented
50#489C2214yescomplemented
51#486C2212yes—
52#483C2214yescomplemented
53#482C2212yescomplemented
54#481C2214yescomplemented
55#481C2214yescomplemented
56#480C2214yes—
57#478C2214yescomplemented
58#477C2212yescomplemented
59#477C2211yescharacteristic, complemented
60#476C2212yes—
61#476C2211yescharacteristic
62#476C2214yes—
63#476C2212yescomplemented
64#476C2214yescomplemented
65#476C2212yescomplemented
66#476C2212yescomplemented
67#476C2212yescomplemented
68#475C2212yescomplemented
69#474C2212yes—
70#469C2214yescomplemented
71#466C2214yes—
72#465C2214yescomplemented
73#463C2214yescomplemented
74#460C2211yescharacteristic, complemented
75#459C2211yescharacteristic
76#459C2212yes—
77#459C2212yescomplemented
78#449C2212yes—
79#447C2214yes—
80#446C2212yescomplemented
81#442C2214yes—
82#442C2214yescomplemented
83C4×SD16C2212yescomplemented
84C4×D8C2212yes—
85#435C2214yescomplemented
86C8×D4C2212yes—
87#432C2212yescomplemented
88#431C2211yescharacteristic, complemented
89#428C2212yescomplemented
90C2×#106C2214yescomplemented
91#423C2211yescharacteristic, complemented
92C2×C4⋊C8C2212yescomplemented
93#417C2212yes—
94C2×#102C2214yes—
952−1+4—48no—
96C2×#41—22no—
97C2×#41—24no—
98C2×#41—22no—
99C2×#41—21no—
100C2×#41—21no—
101C2×#41D411yescharacteristic
102C2×#41C2311yescharacteristic
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—21no—
106C22×D4—21no—
107C22×D4C2312yes—
108C23×C4—22no—
109C2×SD16C2312yes—
110C2×D8C2312yes—
111C2×D8C2314yes—
112C2×M16C2312yes—
113C22×C8C2314yes—
114C4⋊Q8—21no—
115C4⋊Q8—22no—
116C4⋊Q8—21no—
117C4⋊Q8—22no—
118C4⋊Q8C2312yes—
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2312yes—
121C4⋊Q8C2314yes—
122C4⋊Q8C2311yescharacteristic
123#127C2311yescharacteristic
124#127C2311yescharacteristic
125#126—22no—
126#126—22no—
127#125—21no—
128#125C2312yes—
129#124—21no—
130#124—21no—
131#124—21no—
132#124—22no—
133#124C2312yes—
134#123—22no—
135#123—22no—
136#123—22no—
137#123—22no—
138#123—24no—
139#123—24no—
140#123—24no—
141C22⋊Q8—21no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—24no—
145C22⋊Q8—22no—
146C22⋊Q8—24no—
147C22⋊Q8—22no—
148C22⋊Q8C2312yes—
149#121—22no—
150#121—21no—
151#121—22no—
152#121—21no—
153#121—21no—
154#121—22no—
155#121C2312yes—
156#121C2312yes—
157#121C2314yes—
158#121C2312yes—
159#120—22no—
160#120—22no—
161C4×Q8D411yescharacteristic
162C4×Q8C2311yescharacteristic
163C4×Q8C2311yescharacteristic
164C4×Q8C2311yescharacteristic
165C4×D4—21no—
166C4×D4—22no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4—21no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—21no—
175C4×D4—22no—
176C4×D4D411yescharacteristic
177C4×D4D412yes—
178C4×D4C2311yescharacteristic
179C4×D4C2312yes—
180C4×D4C2312yes—
181C4×D4C2312yes—
182C4×D4C2311yescharacteristic
183C4×D4C2312yes—
184C4×D4C2314yes—
185C4×D4C2311yescharacteristic
186C4×D4C2312yes—
187#117—21no—
188#117—22no—
189#117D412yes—
190#117D411yescharacteristic
191#117C2311yescharacteristic
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2312yes—
196C2×C4⋊C4C2314yes—
197C2×C4⋊C4C2312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C42—21no—
201C2×C42C2312yes—
202C2×C42C2311yescharacteristic
203#107C2314yes—
204#107C2312yes—
205#106C2312yes—
206#106C2314yes—
207#106C2314yes—
208#106C2314yes—
209#106C2314yes—
210C4⋊C8C2311yescharacteristic
211C4⋊C8C2312yes—
212C4⋊C8C2314yes—
213C4⋊C8C2312yes—
214C4⋊C8C2311yescharacteristic
215#103C2314yes—
216#103C2312yes—
217#102C2312yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221#102C2312yes—
222#102C2312yes—
223#102C2314yes—
224#102C2314yes—
225C22⋊C8C2312yes—
226C22⋊C8C2314yes—
227#97C2314yes—
228C4×C8C2312yes—
229C24—22no—
230#41—24no—
231#41—22no—
232#41—41no—
233#41—48no—
234#41—44no—
235#41—416no—
236#41—48no—
237#41—82no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—48no—
244C2×Q8—44no—
245C2×Q8—48no—
246C2×Q8C2×D411yescharacteristic
247C2×Q8C2411yescharacteristic
248C2×D4—24no—
249C2×D4—21no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4C2×D412yes—
262C2×D4C2×D411yescharacteristic
263C2×D4C2412yes—
264C2×D4C2412yes—
265C2×D4C2411yescharacteristic
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—84no—
278C22×C4C2×D412yes—
279C22×C4C2×D412yes—
280C22×C4C2×D411yescharacteristic
281C22×C4C2×D411yescharacteristic
282C22×C4C2412yes—
283C22×C4C2411yescharacteristic
284SD16—42no—
285D8—24no—
286D8—44no—
287M16—42no—
288C2×C8—44no—
289C2×C8C2412yes—
290C2×C8C2412yes—
291C2×C8C2414yes—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4C2×D411yescharacteristic
306C4⋊C4C2×D412yes—
307C4⋊C4C2×D411yescharacteristic
308C4⋊C4C2×D412yes—
309C4⋊C4C2412yes—
310C4⋊C4C2414yes—
311C4⋊C4C2411yescharacteristic
312C4⋊C4C2412yes—
313C4⋊C4C2411yescharacteristic
314C22⋊C4—22no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—21no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4C2×D412yes—
330C22⋊C4C2×D412yes—
331C22⋊C4C2×D412yes—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D411yescharacteristic
336C42C2×D411yescharacteristic
337C42C2×D412yes—
338C42C2×D412yes—
339C42C2411yescharacteristic
340C42C2412yes—
341C42C2411yescharacteristic
342C23—21no—
343C23—22no—
344C23—22no—
345C23—44no—
346C23—44no—
347C23—82no—
348C23C22×D411yescharacteristic
349C23C22×D412yes—
350Q8—22no—
351Q8—21no—
352Q8—21no—
353Q8—44no—
354Q8—42no—
355Q8—48no—
356Q8—44no—
357Q8—81no—
358D4—24no—
359D4—22no—
360D4—22no—
361D4—21no—
362D4—22no—
363D4—22no—
364D4—44no—
365D4—42no—
366D4—42no—
367D4—44no—
368D4—44no—
369D4—48no—
370D4—42no—
371D4—42no—
372D4—81no—
373D4—82no—
374D4AGL(1,ℤ/8)14yes—
375D42−1+412yescomplemented
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4C22×D411yescharacteristic
402C2×C4C22×D411yescharacteristic
403C2×C4C22×D412yes—
404C2×C4C22×D412yes—
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D412yes—
407C2×C4C22×D411yescharacteristic
408C2×C4C22×D411yescharacteristic
409C2×C4C2511yescharacteristic
410C8—22no—
411C8—22no—
412C8—24no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—88no—
428C22C2×AGL(1,ℤ/8)14yes—
429C22C23×D411yescharacteristic
430C4—21no—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—22no—
435C4—21no—
436C4—41no—
437C4—44no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—42no—
442C4C2×AGL(1,ℤ/8)12yes—
443C4#57712yes—
444C4C2×2−1+411yescharacteristic
445C4#58511yescharacteristic
446C2—24no—
447C2—41no—
448C2—42no—
449C2—82no—
450C2—82no—
451C2#348911yescharacteristic
452C2C22×AGL(1,ℤ/8)11yescharacteristic
453C2C2×#57711yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3499C2semidirect
2. #3481C2semidirect
3. #3384C2semidirect
4. #3373C2semidirect
5. #3345C2semidirect
6. #3335C2semidirect
7. #3306C2semidirect
8. #3294C2semidirect
9. #3253C2semidirect
10. #3247C2semidirect
11. #3238C2semidirect
12. #3227C2semidirect
13. #3109C2semidirect
14. #3106C2semidirect
15. #3079C2semidirect
16. C2×#476C2semidirect
17. #2990C2semidirect
26. D4×Q8C22semidirect
30. #547C22semidirect
37. #532C22semidirect
38. #532C22semidirect
39. C2×C4⋊Q8C22semidirect
40. C2×C4⋊Q8C22semidirect
48. #501C22semidirect
49. #498C22semidirect
50. #489C22semidirect
52. #483C22semidirect
53. #482C22semidirect
54. #481C22semidirect
55. #481C22semidirect
57. #478C22semidirect
58. #477C22semidirect
59. #477C22semidirect
63. #476C22semidirect
64. #476C22semidirect
65. #476C22semidirect
66. #476C22semidirect
67. #476C22semidirect
68. #475C22semidirect
70. #469C22semidirect
72. #465C22semidirect
73. #463C22semidirect
74. #460C22semidirect
77. #459C22semidirect
80. #446C22semidirect
82. #442C22semidirect
83. C4×SD16C22semidirect
85. #435C22semidirect
87. #432C22semidirect
88. #431C22semidirect
89. #428C22semidirect
90. C2×#106C22semidirect
91. #423C22semidirect
92. C2×C4⋊C8C22semidirect
375. D42−1+4semidirect

Maximal subgroups

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

subgroupindex
#34992
#34812
#33842
#33732
#33452
#33352
#33062
#32942
#32532
#32472
#32382
#32272
#31092
#31062
#30792
C2×#4762
#29902