Finite Group Databasebeta

← #60929

Subgroups — #60929

Order 256.

934 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic, complemented
2#3460C211yesmaximal, characteristic, complemented
3#3263C214yesmaximal, complemented
4#3257C214yesmaximal, complemented
5#3253C212yesmaximal, complemented
6#3252C212yesmaximal, complemented
7#3247C212yesmaximal, complemented
8#3246C212yesmaximal, complemented
9#3084C212yesmaximal, complemented
10#3079C211yesmaximal, characteristic, complemented
11C2×#476C211yesmaximal, characteristic, complemented
12#3062C212yesmaximal, complemented
13#3060C212yesmaximal, complemented
14#3045C212yesmaximal, complemented
15#3036C211yesmaximal, characteristic, complemented
16#3035C211yesmaximal, characteristic, complemented
17#2906C211yesmaximal, characteristic, complemented
18C2×AGL(1,ℤ/8)C2212yes—
19C2×#135C2212yes—
20C2×#131C2212yes—
21#556—21no—
22#555—21no—
23#554—21no—
24#553—22no—
25#548—22no—
26#547—21no—
27#544C2211yescharacteristic
28#542—21no—
29#542C2212yes—
30#542C2214yescomplemented
31#540—21no—
32#540C2212yes—
33#538—22no—
34#537C2211yescharacteristic
35#537C2212yes—
36#535C2211yescharacteristic
37#535C2211yescharacteristic
38#533C2212yescomplemented
39#532—21no—
40#532C2211yescharacteristic
41#532C2212yes—
42#532C2212yescomplemented
43C2×C4⋊Q8C2211yescharacteristic
44#529—21no—
45#529C2212yes—
46#525—22no—
47#520C2212yes—
48C4×#41C2211yescharacteristic
49C4×#41C2211yescharacteristic
50C2×C4×D4C2211yescharacteristic
51#479C2214yescomplemented
52#478C2214yescomplemented
53#477C2212yescomplemented
54#476C2212yes—
55#476C2212yes—
56#476C2214yes—
57#476C2212yescomplemented
58#476C2214yescomplemented
59#475C2212yescomplemented
60#475C2214yescomplemented
61#474C2212yescomplemented
62#470C2214yescomplemented
63#469C2214yes—
64#469C2214yescomplemented
65#468C2214yes—
66#466C2214yes—
67#465C2214yes—
68#465C2214yescomplemented
69#465C2214yescomplemented
70#464C2214yes—
71#464C2212yescomplemented
72#463C2214yes—
73#463C2214yes—
74#463C2212yescomplemented
75#462C2211yescharacteristic, complemented
76#461C2212yes—
77#461C2211yescharacteristic, complemented
78#460C2214yes—
79#460C2211yescharacteristic, complemented
80#459C2212yes—
81#459C2214yes—
82#459C2211yescharacteristic, complemented
83#428C2212yes—
84C2×#106C2212yes—
85#424C2212yes—
86#424C2214yescomplemented
87#423C2211yescharacteristic
88#423C2212yescomplemented
89C2×C4⋊C8C2211yescharacteristic
90C2×C4⋊C8C2212yescomplemented
91#417C2212yes—
92#417C2212yes—
93#416C2212yes—
94C2×#102C2212yes—
95C2×#41—22no—
96C2×#41—22no—
97C2×#41C2311yescharacteristic
98C2×#41C2312yes—
99C2×#41C2311yescharacteristic
100C22×Q8—21no—
101C22×D4C2311yescharacteristic
102C23×C4—21no—
103AGL(1,ℤ/8)—44no—
104#135—44no—
105C2×Q16C2312yes—
106C2×SD16C2312yes—
107C2×SD16C2312yes—
108C2×SD16C2314yes—
109C2×D8C2312yes—
110C2×D8C2314yes—
111#131—44no—
112C2×M16C2312yes—
113C2×M16C2314yes—
114C22×C8C2312yes—
115C22×C8C2314yes—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2312yes—
121C4⋊Q8C2312yes—
122C4⋊Q8C2312yes—
123#127—21no—
124#127C2311yescharacteristic
125#127C2312yes—
126#126—22no—
127#126—24no—
128#126—22no—
129#125—21no—
130#125—22no—
131#125C2312yes—
132#124—21no—
133#124—22no—
134#124—24no—
135#124—22no—
136#124—22no—
137#124C2312yes—
138#124C2312yes—
139#123—22no—
140#123—22no—
141#123—22no—
142#123—22no—
143#123—24no—
144C22⋊Q8—21no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—24no—
149C22⋊Q8—22no—
150C22⋊Q8C2312yes—
151C22⋊Q8C2314yes—
152#121—22no—
153#121—21no—
154#121—22no—
155#121—24no—
156#121—22no—
157#121—21no—
158#121—21no—
159#121—22no—
160#121—21no—
161#121—21no—
162#121—22no—
163#121—22no—
164#121C2312yes—
165#121C2312yes—
166#121C2312yes—
167#121C2314yes—
168#120—22no—
169#120—22no—
170C4×Q8—22no—
171C4×Q8D411yescharacteristic
172C4×Q8D411yescharacteristic
173C4×Q8C2311yescharacteristic
174C4×Q8C2311yescharacteristic
175C4×Q8C2314yes—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—22no—
179C4×D4—24no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—22no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—22no—
186C4×D4D412yes—
187C4×D4D411yescharacteristic
188C4×D4D412yes—
189C4×D4D411yescharacteristic
190C4×D4C2312yes—
191C4×D4C2312yes—
192C4×D4C2311yescharacteristic
193C4×D4C2311yescharacteristic
194C4×D4C2314yes—
195#117—22no—
196#117—22no—
197#117—21no—
198#117—22no—
199#117C2311yescharacteristic
200#117C2311yescharacteristic
201#117C2312yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C2311yescharacteristic
205C2×C4⋊C4C2312yes—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42C2311yescharacteristic
209C2×C42C2312yes—
210#107C2314yes—
211#106C2314yes—
212#106C2314yes—
213#106C2314yes—
214C4⋊C8C2311yescharacteristic
215C4⋊C8C2311yescharacteristic
216C4⋊C8C2312yes—
217C4⋊C8C2312yes—
218C4⋊C8C2314yes—
219C4⋊C8C2312yes—
220C4⋊C8C2314yes—
221#103C2312yes—
222#103C2312yes—
223#103C2314yes—
224#103C2314yes—
225#102C2314yes—
226#102C2314yes—
227#102C2312yes—
228#102C2312yes—
229#102C2314yes—
230#102C2314yes—
231C24—21no—
232#41—22no—
233#41—22no—
234#41—22no—
235#41—24no—
236#41—22no—
237#41—24no—
238#41—84no—
239#41—84no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—82no—
244C2×Q8C2×D411yescharacteristic
245C2×Q8C2411yescharacteristic
246C2×Q8C2412yes—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4C2×D412yes—
261C2×D4C2×D411yescharacteristic
262C2×D4C2412yes—
263C2×D4C2412yes—
264C2×D4C2411yescharacteristic
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—82no—
275C22×C4C2×D411yescharacteristic
276C22×C4C2×D412yes—
277C22×C4C2×D411yescharacteristic
278C22×C4C2×D412yes—
279C22×C4C2412yes—
280C22×C4C2411yescharacteristic
281Q16—42no—
282SD16—42no—
283SD16—42no—
284SD16—44no—
285D8—42no—
286D8—44no—
287M16—24no—
288M16—44no—
289C2×C8—24no—
290C2×C8—44no—
291C2×C8C2412yes—
292C2×C8C2412yes—
293C2×C8C2414yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2×D411yescharacteristic
310C4⋊C4C2×D412yes—
311C4⋊C4C2×D412yes—
312C4⋊C4C2414yes—
313C4⋊C4C2412yes—
314C4⋊C4C2411yescharacteristic
315C4⋊C4C2411yescharacteristic
316C22⋊C4—24no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—24no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4C2×D412yes—
333C22⋊C4C2×D412yes—
334C22⋊C4C2×D412yes—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—24no—
339C42C2×D411yescharacteristic
340C42C2×D412yes—
341C42C2×D411yescharacteristic
342C42C2×D412yes—
343C42C2411yescharacteristic
344C42C2411yescharacteristic
345C42C2412yes—
346C23—22no—
347C23—22no—
348C23—21no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23C22×D412yes—
353C23C22×D411yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—21no—
357Q8—21no—
358Q8—22no—
359Q8—44no—
360Q8—82no—
361Q8—82no—
362D4—24no—
363D4—22no—
364D4—22no—
365D4—22no—
366D4—21no—
367D4—22no—
368D4—22no—
369D4—21no—
370D4—42no—
371D4—42no—
372D4—44no—
373D4—42no—
374D4—42no—
375D4—82no—
376D4—82no—
377D4—82no—
378D4—84no—
379D4—82no—
380D4—82no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4C22×D412yes—
403C2×C4C22×D411yescharacteristic
404C2×C4C22×D411yescharacteristic
405C2×C4C22×D412yes—
406C2×C4C22×D411yescharacteristic
407C2×C4C22×D412yes—
408C2×C4C22×D411yescharacteristic
409C2×C4C22×D411yescharacteristic
410C2×C4H(5,𝔽2)12yes—
411C2×C42−1+412yes—
412C2×C4C2511yescharacteristic
413C8—22no—
414C8—22no—
415C8—24no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—41no—
423C22—42no—
424C22—42no—
425C22—41no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—42no—
430C22—84no—
431C22#53712yes—
432C22C23×D411yescharacteristic
433C4—21no—
434C4—22no—
435C4—21no—
436C4—22no—
437C4—21no—
438C4—22no—
439C4—44no—
440C4—41no—
441C4—41no—
442C4—42no—
443C4—42no—
444C4—41no—
445C4#53712yes—
446C4#57712yes—
447C4C2×H(5,𝔽2)11yescharacteristic
448C4C2×2−1+411yescharacteristic
449C2—22no—
450C2—42no—
451C2—82no—
452C2—81no—
453C2—82no—
454C2C2×#53711yescharacteristic
455C2C2×#57711yescharacteristic
456C2#358511yescharacteristic
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. #3460C2semidirect
3. #3263C2semidirect
4. #3257C2semidirect
5. #3253C2semidirect
6. #3252C2semidirect
7. #3247C2semidirect
8. #3246C2semidirect
9. #3084C2semidirect
10. #3079C2semidirect
11. C2×#476C2semidirect
12. #3062C2semidirect
13. #3060C2semidirect
14. #3045C2semidirect
15. #3036C2semidirect
16. #3035C2semidirect
17. #2906C2semidirect
30. #542C22semidirect
38. #533C22semidirect
42. #532C22semidirect
51. #479C22semidirect
52. #478C22semidirect
53. #477C22semidirect
57. #476C22semidirect
58. #476C22semidirect
59. #475C22semidirect
60. #475C22semidirect
61. #474C22semidirect
62. #470C22semidirect
64. #469C22semidirect
68. #465C22semidirect
69. #465C22semidirect
71. #464C22semidirect
74. #463C22semidirect
75. #462C22semidirect
77. #461C22semidirect
79. #460C22semidirect
82. #459C22semidirect
86. #424C22semidirect
88. #423C22semidirect
90. C2×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35112
#34602
#32632
#32572
#32532
#32522
#32472
#32462
#30842
#30792
C2×#4762
#30622
#30602
#30452
#30362
#30352
#29062