Finite Group Databasebeta

← #61712

Subgroups — #61712

Order 256.

881 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic
2C2×#556C211yesmaximal, characteristic, complemented
3#3350C214yesmaximal, complemented
4#3349C214yesmaximal, complemented
5#3313C214yesmaximal, complemented
6#3311C214yesmaximal, complemented
7#3238C212yesmaximal, complemented
8#3195C211yesmaximal, characteristic, complemented
9#3193C211yesmaximal, characteristic, complemented
10#3190C212yesmaximal, complemented
11#3117C212yesmaximal, complemented
12#3095C211yesmaximal, characteristic, complemented
13#3094C211yesmaximal, characteristic, complemented
14#3091C212yesmaximal, complemented
15#2973C211yesmaximal, characteristic, complemented
16#556—22no—
17#556—22no—
18#556C2214yes—
19#556C2214yescomplemented
20#553—22no—
21#553C2212yes—
22#552C2212yes—
23#551—22no—
24#548C2212yes—
25#546C2212yes—
26#539—21no—
27#538—21no—
28#534—21no—
29C2×C4⋊Q8C2211yescharacteristic, complemented
30#529C2212yes—
31C2×#126—21no—
32C2×#126—21no—
33C2×#124—21no—
34#525C2212yes—
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2211yescharacteristic, complemented
38C2×#121C2211yescharacteristic
39#518—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8C2211yescharacteristic, complemented
42C2×C4×D4C2211yescharacteristic
43C2×#117C2211yescharacteristic
44C2×#117C2211yescharacteristic, complemented
45#501C2214yescomplemented
46#489C2214yescomplemented
47#485C2214yescomplemented
48#485C2214yescomplemented
49#485C2214yescomplemented
50#484C2212yescomplemented
51#484C2212yescomplemented
52#483C2212yescomplemented
53#483C2212yescomplemented
54#483C2212yescomplemented
55#482C2214yescomplemented
56#482C2214yescomplemented
57#481C2212yes—
58#480C2212yes—
59#480C2212yes—
60#479C2214yescomplemented
61#477C2214yescomplemented
62#476C2214yescomplemented
63#470C2214yescomplemented
64#468C2214yescomplemented
65#467C2214yescomplemented
66#465C2214yescomplemented
67#464C2214yescomplemented
68#452C2214yescomplemented
69#451C2212yescomplemented
70#451C2212yescomplemented
71#450C2212yescomplemented
72#448C2212yes—
73#441C2214yescomplemented
74#440C2214yescomplemented
75#435C2214yes—
76#435C2214yescomplemented
77#432C2212yescomplemented
78#428C2212yescomplemented
79#428C2212yescomplemented
80#427C2214yescomplemented
81#419C2212yescomplemented
82#418C2212yescomplemented
83#416C2214yescomplemented
84#407C2211yescharacteristic
85#407C2211yescharacteristic
86C2×C22⋊C8C2212yescomplemented
87C2×#41C2312yes—
88C22×Q8—21no—
89C22×Q8C2311yescharacteristic
90C22×Q8C2311yescharacteristic
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C2×Q16C2314yes—
97C2×SD16C2314yes—
98C2×M16C2312yes—
99C2×M16C2312yes—
100C22×C8C2314yes—
101C4⋊Q8—22no—
102C4⋊Q8C2314yes—
103#126—22no—
104#126—24no—
105#126—24no—
106#126—24no—
107#126—24no—
108#126—22no—
109#126—22no—
110#125—22no—
111#125C2312yes—
112#124—24no—
113#124—24no—
114#124—22no—
115#124C2312yes—
116#123—22no—
117#123—22no—
118#123—22no—
119#123—22no—
120#123—22no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8C2312yes—
128C22⋊Q8C2312yes—
129C22⋊Q8C2312yes—
130C22⋊Q8C2312yes—
131C22⋊Q8C2314yes—
132C22⋊Q8C2312yes—
133#121—22no—
134#121—22no—
135#121—21no—
136#121—21no—
137#121—22no—
138#121C2312yes—
139#121C2312yes—
140#121C2312yes—
141#120—22no—
142#120—22no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—22no—
146C4×Q8C2314yes—
147C4×Q8C2312yes—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4D412yes—
154C4×D4D412yes—
155C4×D4C2312yes—
156C4×D4C2312yes—
157C4×D4C2312yes—
158#117—21no—
159#117—21no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117C2312yes—
164#117C2312yes—
165#117C2314yes—
166#117C2312yes—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4D411yescharacteristic
173C2×C4⋊C4C2311yescharacteristic
174C2×C4⋊C4C2311yescharacteristic
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C4⋊C4C2311yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4D411yescharacteristic
186C2×C22⋊C4D411yescharacteristic
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C2311yescharacteristic
191#107C2314yes—
192#107C2314yes—
193#107C2314yes—
194#106C2314yes—
195#106C2314yes—
196#106C2314yes—
197C4⋊C8C2314yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#103C2314yes—
202#103C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208C22⋊C8C2314yes—
209C22⋊C8C2312yes—
210C22⋊C8C2312yes—
211#97C2314yes—
212C24—21no—
213C24C2×D411yescharacteristic
214C24C2×D411yescharacteristic
215#41—44no—
216C2×Q8—21no—
217C2×Q8—24no—
218C2×Q8—21no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8C2412yes—
223C2×Q8C2412yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—44no—
232C2×D4C2×D412yes—
233C2×D4C2×D412yes—
234C2×D4C2412yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2412yes—
257Q16—44no—
258SD16—44no—
259M16—42no—
260M16—42no—
261C2×C8—44no—
262C2×C8C2414yes—
263C2×C8C2414yes—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4C2×D412yes—
281C4⋊C4C2×D412yes—
282C4⋊C4C2412yes—
283C4⋊C4C2412yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C4⋊C4C2412yes—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—24no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—24no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4C2×D412yes—
304C22⋊C4C2×D412yes—
305C22⋊C4C2×D412yes—
306C22⋊C4C2×D412yes—
307C42—24no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42C2×D412yes—
314C42C2412yes—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23C22×D411yescharacteristic
326C23C22×D412yes—
327C23C22×D411yescharacteristic
328C23C22×D411yescharacteristic
329Q8—24no—
330Q8—24no—
331Q8—42no—
332Q8—48no—
333Q8—81no—
334D4—24no—
335D4—42no—
336D4—42no—
337D4—44no—
338D4—44no—
339D4—81no—
340D4—82no—
341D4—82no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4C22×D412yes—
370C2×C4C22×D412yes—
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C22×D412yes—
374C2×C4C22×D411yescharacteristic
375C2×C4C2511yescharacteristic
376C8—24no—
377C8—24no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—82no—
393C22—82no—
394C22—162no—
395C22#57512yes—
396C22C23×D411yescharacteristic
397C4—22no—
398C4—21no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4#58512yes—
411C2—22no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—82no—
416C2#348511yescharacteristic
417C2C2×#57511yescharacteristic
418C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3505C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#556C2semidirect
3. #3350C2semidirect
4. #3349C2semidirect
5. #3313C2semidirect
6. #3311C2semidirect
7. #3238C2semidirect
8. #3195C2semidirect
9. #3193C2semidirect
10. #3190C2semidirect
11. #3117C2semidirect
12. #3095C2semidirect
13. #3094C2semidirect
14. #3091C2semidirect
15. #2973C2semidirect
19. #556C22semidirect
29. C2×C4⋊Q8C22semidirect
37. C2×C22⋊Q8C22semidirect
41. C2×C4×Q8C22semidirect
44. C2×#117C22semidirect
45. #501C22semidirect
46. #489C22semidirect
47. #485C22semidirect
48. #485C22semidirect
49. #485C22semidirect
50. #484C22semidirect
51. #484C22semidirect
52. #483C22semidirect
53. #483C22semidirect
54. #483C22semidirect
55. #482C22semidirect
56. #482C22semidirect
60. #479C22semidirect
61. #477C22semidirect
62. #476C22semidirect
63. #470C22semidirect
64. #468C22semidirect
65. #467C22semidirect
66. #465C22semidirect
67. #464C22semidirect
68. #452C22semidirect
69. #451C22semidirect
70. #451C22semidirect
71. #450C22semidirect
73. #441C22semidirect
74. #440C22semidirect
76. #435C22semidirect
77. #432C22semidirect
78. #428C22semidirect
79. #428C22semidirect
80. #427C22semidirect
81. #419C22semidirect
82. #418C22semidirect
83. #416C22semidirect
86. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5562
#33502
#33492
#33132
#33112
#32382
#31952
#31932
#31902
#31172
#30952
#30942
#30912
#29732