Finite Group Databasebeta

← #61996

Subgroups — #61996

Order 256.

957 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
1#3511C211yesmaximal, characteristic, complemented
2#3481C211yesmaximal, characteristic, complemented
3#3394C212yesmaximal, complemented
4#3360C212yesmaximal, complemented
5#3331C212yesmaximal, complemented
6#3326C214yesmaximal, complemented
7#3275C214yesmaximal, complemented
8#3272C212yesmaximal, complemented
9#3253C211yesmaximal, characteristic, complemented
10#3252C212yesmaximal, complemented
11#3212C212yesmaximal, complemented
12#3206C211yesmaximal, characteristic, complemented
13#3106C211yesmaximal, characteristic, complemented
14#3101C212yesmaximal, complemented
15#3039C211yesmaximal, characteristic, complemented
16#3036C212yesmaximal, complemented
17#2991C211yesmaximal, characteristic, complemented
18C2×#137C2212yes—
19C2×AGL(1,ℤ/8)C2214yes—
20#559C2211yescharacteristic
21#556C2212yescomplemented
22#555—21no—
23#554—21no—
24#554C2211yescharacteristic
25#554C2211yescharacteristic
26#553—22no—
27#553—21no—
28#553C2212yes—
29#551—22no—
30#548C2214yescomplemented
31#547C2212yes—
32#546—21no—
33#546C2212yes—
34D42C2211yescharacteristic
35#542—22no—
36#537—21no—
37#533C2212yescomplemented
38#532C2211yescharacteristic
39#532C2212yescomplemented
40#532C2211yescharacteristic, complemented
41C2×C4⋊Q8C2211yescharacteristic, complemented
42#529—21no—
43#529C2212yes—
44#525—22no—
45#520—21no—
46C4×#41C2211yescharacteristic
47C4×#41C2211yescharacteristic
48C2×C4×D4—21no—
49C2×C4×D4C2212yes—
50#495C2214yescomplemented
51#492C2212yes—
52#488C2214yescomplemented
53#487C2212yescomplemented
54#483C2214yescomplemented
55#482C2212yescomplemented
56#479C2214yescomplemented
57#477C2211yescharacteristic, complemented
58#476C2211yescharacteristic
59#476C2212yescomplemented
60#476C2212yescomplemented
61#470C2214yescomplemented
62#469C2214yescomplemented
63#468C2214yes—
64#464C2214yescomplemented
65#463C2214yes—
66#463C2214yescomplemented
67#462C2212yescomplemented
68#462C2212yescomplemented
69#461C2212yes—
70#461C2212yes—
71#461C2212yescomplemented
72#461C2212yescomplemented
73#460C2211yescharacteristic, complemented
74#459C2211yescharacteristic
75#459C2212yes—
76#459C2212yescomplemented
77#452C2214yescomplemented
78#450C2214yes—
79#449C2212yes—
80#449C2214yes—
81#448C2214yescomplemented
82C8×Q8C2212yescomplemented
83C4×SD16C2214yes—
84C4×SD16C2212yescomplemented
85C4×SD16C2214yescomplemented
86C4×D8C2212yes—
87#436C2212yes—
88#436C2214yescomplemented
89#431C2212yescomplemented
90#431C2211yescharacteristic, complemented
91#423C2212yescomplemented
92#423C2211yescharacteristic, complemented
93#419C2212yescomplemented
94#417C2214yescomplemented
95C2×#41—21no—
96C2×#41—21no—
97C2×#41D411yescharacteristic
98C2×#41C2311yescharacteristic
99C2×#41C2314yes—
100C22×Q8C2312yes—
101C22×D4—21no—
102C22×D4C2312yes—
103C23×C4—22no—
104#137—44no—
105AGL(1,ℤ/8)—48no—
106C2×Q16C2314yes—
107C2×SD16C2314yes—
108C2×SD16C2314yes—
109C2×SD16C2314yes—
110C2×SD16C2312yes—
111C2×D8C2312yes—
112C2×D8C2314yes—
113C2×M16C2314yes—
114C2×M16C2312yes—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C2312yes—
118C4⋊Q8C2312yes—
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2311yescharacteristic
121#127C2311yescharacteristic
122#127C2311yescharacteristic
123#126—22no—
124#126—24no—
125#126—22no—
126#125—21no—
127#125—22no—
128#125C2312yes—
129#124—21no—
130#124—24no—
131#124—21no—
132#124—21no—
133#124C2312yes—
134#124C2314yes—
135#123—22no—
136#123—22no—
137#123—22no—
138#123—22no—
139#123—24no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—24no—
145C22⋊Q8—22no—
146C22⋊Q8C2312yes—
147C22⋊Q8C2314yes—
148#121—22no—
149#121—22no—
150#121—24no—
151#121—21no—
152#121—22no—
153#121—21no—
154#121—22no—
155#121—21no—
156#121—22no—
157#121C2312yes—
158#121C2312yes—
159#121C2312yes—
160#120—22no—
161#120—22no—
162C4×Q8—22no—
163C4×Q8D411yescharacteristic
164C4×Q8C2311yescharacteristic
165C4×Q8C2311yescharacteristic
166C4×Q8C2314yes—
167C4×Q8C2311yescharacteristic
168C4×D4—21no—
169C4×D4—22no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—24no—
175C4×D4—21no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—21no—
181C4×D4D411yescharacteristic
182C4×D4D412yes—
183C4×D4C2311yescharacteristic
184C4×D4C2312yes—
185C4×D4C2312yes—
186C4×D4C2312yes—
187C4×D4C2311yescharacteristic
188C4×D4C2312yes—
189C4×D4C2311yescharacteristic
190C4×D4C2312yes—
191#117—22no—
192#117—22no—
193#117—21no—
194#117—22no—
195#117D411yescharacteristic
196#117D412yes—
197#117C2311yescharacteristic
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C2312yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42C2312yes—
207C2×C42C2311yescharacteristic
208#107C2312yes—
209#106C2314yes—
210#106C2312yes—
211C4⋊C8C2311yescharacteristic
212C4⋊C8C2312yes—
213C4⋊C8C2312yes—
214C4⋊C8C2314yes—
215C4⋊C8C2311yescharacteristic
216#103C2314yes—
217#103C2314yes—
218#103C2314yes—
219#103C2314yes—
220#103C2312yes—
221#102C2312yes—
222#102C2314yes—
223#102C2312yes—
224#102C2312yes—
225#102C2314yes—
226C22⋊C8C2314yes—
227C22⋊C8C2312yes—
228C4×C8C2312yes—
229C4×C8C2314yes—
230C24—22no—
231#41—22no—
232#41—42no—
233#41—41no—
234#41—44no—
235#41—44no—
236#41—44no—
237#41—82no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C2×D411yescharacteristic
244C2×Q8C2414yes—
245C2×Q8C2411yescharacteristic
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—24no—
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4C2×D411yescharacteristic
260C2×D4C2×D412yes—
261C2×D4C2412yes—
262C2×D4C2412yes—
263C2×D4C2411yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—84no—
275C22×C4C2×D411yescharacteristic
276C22×C4C2×D411yescharacteristic
277C22×C4C2×D412yes—
278C22×C4C2×D412yes—
279C22×C4C2412yes—
280C22×C4C2411yescharacteristic
281Q16—44no—
282SD16—28no—
283SD16—44no—
284SD16—42no—
285SD16—44no—
286D8—24no—
287D8—44no—
288M16—44no—
289M16—42no—
290C2×C8C2412yes—
291C2×C8C2414yes—
292C2×C8C2412yes—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4C2×D411yescharacteristic
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2×D412yes—
310C4⋊C4C2×D412yes—
311C4⋊C4C2412yes—
312C4⋊C4C2411yescharacteristic
313C4⋊C4C2412yes—
314C4⋊C4C2411yescharacteristic
315C22⋊C4—22no—
316C22⋊C4—24no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—24no—
325C22⋊C4—21no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—24no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4C2×D412yes—
332C22⋊C4C2×D412yes—
333C22⋊C4C2×D412yes—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—24no—
338C42C2×D411yescharacteristic
339C42C2×D411yescharacteristic
340C42C2×D412yes—
341C42C2×D412yes—
342C42C2411yescharacteristic
343C42C2412yes—
344C42C2411yescharacteristic
345C23—21no—
346C23—22no—
347C23—22no—
348C23—44no—
349C23—44no—
350C23—82no—
351C23C22×D412yes—
352C23C22×D411yescharacteristic
353Q8—22no—
354Q8—24no—
355Q8—21no—
356Q8—21no—
357Q8—24no—
358Q8—42no—
359Q8—82no—
360Q8—81no—
361D4—22no—
362D4—22no—
363D4—22no—
364D4—21no—
365D4—22no—
366D4—22no—
367D4—42no—
368D4—44no—
369D4—42no—
370D4—42no—
371D4—44no—
372D4—44no—
373D4—42no—
374D4—42no—
375D4—81no—
376D4—84no—
377D4—82no—
378D4H(5,𝔽2)12yes—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4C22×D412yes—
404C2×C4C22×D411yescharacteristic
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D412yes—
407C2×C4C22×D412yes—
408C2×C4C22×D411yescharacteristic
409C2×C4C22×D411yescharacteristic
410C2×C4C22×D411yescharacteristic
411C2×C4C2511yescharacteristic
412C8—22no—
413C8—24no—
414C8—22no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—21no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—41no—
423C22—41no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—88no—
431C22C23×D411yescharacteristic
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—22no—
437C4—21no—
438C4—41no—
439C4—44no—
440C4—41no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4#57512yes—
445C4#57712yes—
446C4C2×H(5,𝔽2)11yescharacteristic
447C4#58511yescharacteristic
448C2—41no—
449C2—42no—
450C2—42no—
451C2—82no—
452C2—82no—
453C2#348811yescharacteristic
454C2C2×#57511yescharacteristic
455C2C2×#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. #3511C2semidirect
2. #3481C2semidirect
3. #3394C2semidirect
4. #3360C2semidirect
5. #3331C2semidirect
6. #3326C2semidirect
7. #3275C2semidirect
8. #3272C2semidirect
9. #3253C2semidirect
10. #3252C2semidirect
11. #3212C2semidirect
12. #3206C2semidirect
13. #3106C2semidirect
14. #3101C2semidirect
15. #3039C2semidirect
16. #3036C2semidirect
17. #2991C2semidirect
21. #556C22semidirect
30. #548C22semidirect
37. #533C22semidirect
39. #532C22semidirect
40. #532C22semidirect
41. C2×C4⋊Q8C22semidirect
50. #495C22semidirect
52. #488C22semidirect
53. #487C22semidirect
54. #483C22semidirect
55. #482C22semidirect
56. #479C22semidirect
57. #477C22semidirect
59. #476C22semidirect
60. #476C22semidirect
61. #470C22semidirect
62. #469C22semidirect
64. #464C22semidirect
66. #463C22semidirect
67. #462C22semidirect
68. #462C22semidirect
71. #461C22semidirect
72. #461C22semidirect
73. #460C22semidirect
76. #459C22semidirect
77. #452C22semidirect
81. #448C22semidirect
82. C8×Q8C22semidirect
84. C4×SD16C22semidirect
85. C4×SD16C22semidirect
88. #436C22semidirect
89. #431C22semidirect
90. #431C22semidirect
91. #423C22semidirect
92. #423C22semidirect
93. #419C22semidirect
94. #417C22semidirect

Maximal subgroups

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

subgroupindex
#35112
#34812
#33942
#33602
#33312
#33262
#32752
#32722
#32532
#32522
#32122
#32062
#31062
#31012
#30392
#30362
#29912