Finite Group Databasebeta

← #61716

Subgroups — #61716

Order 256.

883 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×#552C211yesmaximal, characteristic, complemented
3#3352C214yesmaximal, complemented
4#3351C214yesmaximal, complemented
5#3327C214yesmaximal, complemented
6#3323C214yesmaximal, complemented
7#3232C212yesmaximal, complemented
8#3196C212yesmaximal, complemented
9#3195C211yesmaximal, characteristic, complemented
10#3193C211yesmaximal, characteristic, complemented
11#3108C212yesmaximal, complemented
12#3095C211yesmaximal, characteristic, complemented
13#3094C211yesmaximal, characteristic, complemented
14#3092C212yesmaximal, complemented
15#2975C211yesmaximal, characteristic, complemented
16#553—22no—
17#553C2212yes—
18#552—22no—
19#552—22no—
20#552C2212yes—
21#552C2214yes—
22#552C2214yescomplemented
23#551—22no—
24#548C2212yes—
25#546C2212yes—
26#539—21no—
27#538—21no—
28#534—21no—
29#529C2212yes—
30C2×#126—21no—
31C2×#125—21no—
32C2×#125C2211yescharacteristic, complemented
33#525C2212yes—
34C2×#123—21no—
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2211yescharacteristic, complemented
38C2×#121C2211yescharacteristic
39#518—21no—
40C2×C4×Q8C2211yescharacteristic, complemented
41C2×C4×D4C2211yescharacteristic
42C2×#117—21no—
43C2×#117C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic, complemented
45#499C2214yescomplemented
46#488C2214yescomplemented
47#485C2214yescomplemented
48#485C2214yescomplemented
49#484C2214yescomplemented
50#484C2212yescomplemented
51#484C2212yescomplemented
52#483C2212yescomplemented
53#483C2212yescomplemented
54#483C2212yescomplemented
55#482C2214yescomplemented
56#481C2212yes—
57#481C2214yescomplemented
58#480C2212yes—
59#480C2212yes—
60#479C2214yescomplemented
61#479C2214yescomplemented
62#478C2214yescomplemented
63#470C2214yescomplemented
64#470C2214yescomplemented
65#468C2214yescomplemented
66#468C2214yescomplemented
67#464C2214yescomplemented
68#452C2214yescomplemented
69#451C2212yescomplemented
70#451C2212yescomplemented
71#450C2212yescomplemented
72#448C2212yes—
73C4×Q16C2214yescomplemented
74C4×SD16C2214yescomplemented
75#436C2214yes—
76#436C2214yescomplemented
77#431C2212yescomplemented
78#428C2214yescomplemented
79#428C2212yescomplemented
80#428C2212yescomplemented
81#419C2212yescomplemented
82#418C2212yescomplemented
83#417C2214yescomplemented
84#407C2211yescharacteristic
85#407C2211yescharacteristic
86#407C2212yescomplemented
87C2×#41C2312yes—
88C22×Q8C2311yescharacteristic
89C22×Q8C2311yescharacteristic
90C22×D4—21no—
91C22×D4D411yescharacteristic
92C22×D4C2311yescharacteristic
93C23×C4—21no—
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C2×Q16C2314yes—
97C2×SD16C2314yes—
98C2×M16C2312yes—
99C2×M16C2312yes—
100C2×M16C2314yes—
101#126—22no—
102#126—24no—
103#126—24no—
104#126—22no—
105#126—22no—
106#125—24no—
107#125—24no—
108#125—22no—
109#125—22no—
110#125C2314yes—
111#125C2312yes—
112#124—22no—
113#124C2312yes—
114#123—24no—
115#123—24no—
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—22no—
144C4×Q8C2314yes—
145C4×Q8C2312yes—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4D412yes—
152C4×D4D412yes—
153C4×D4C2312yes—
154C4×D4C2312yes—
155C4×D4C2312yes—
156#117—21no—
157#117—21no—
158#117—24no—
159#117—24no—
160#117—22no—
161#117—22no—
162#117C2312yes—
163#117C2312yes—
164#117C2312yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4D411yescharacteristic
174C2×C4⋊C4C2311yescharacteristic
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2314yes—
177C2×C4⋊C4C2311yescharacteristic
178C2×C4⋊C4C2311yescharacteristic
179C2×C4⋊C4C2311yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4D411yescharacteristic
187C2×C22⋊C4D411yescharacteristic
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⋊C8C2312yes—
209C22⋊C8C2312yes—
210C22⋊C8C2314yes—
211C4×C8C2314yes—
212C24—21no—
213C24C2×D411yescharacteristic
214C24C2×D411yescharacteristic
215#41—44no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8C2412yes—
221C2×Q8C2412yes—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—21no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4C2×D412yes—
231C2×D4C2×D412yes—
232C2×D4C2412yes—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257Q16—44no—
258SD16—44no—
259M16—42no—
260M16—44no—
261M16—42no—
262C2×C8C2414yes—
263C2×C8C2414yes—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4C2×D412yes—
283C4⋊C4C2×D412yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C4⋊C4C2412yes—
287C4⋊C4C2412yes—
288C4⋊C4C2412yes—
289C22⋊C4—22no—
290C22⋊C4—24no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
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⋊C4—22no—
304C22⋊C4C2×D412yes—
305C22⋊C4C2×D412yes—
306C22⋊C4C2×D412yes—
307C22⋊C4C2×D412yes—
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×D411yescharacteristic
327C23C22×D412yes—
328C23C22×D411yescharacteristic
329Q8—24no—
330Q8—24no—
331Q8—42no—
332Q8—81no—
333D4—24no—
334D4—42no—
335D4—42no—
336D4—44no—
337D4—44no—
338D4—81no—
339D4—82no—
340D4—82no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
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—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—48no—
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×D411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D412yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D412yes—
375C2×C4C2511yescharacteristic
376C8—24no—
377C8—24no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—21no—
386C22—22no—
387C22—21no—
388C22—21no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
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—42no—
413C2—41no—
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×#552C2semidirect
3. #3352C2semidirect
4. #3351C2semidirect
5. #3327C2semidirect
6. #3323C2semidirect
7. #3232C2semidirect
8. #3196C2semidirect
9. #3195C2semidirect
10. #3193C2semidirect
11. #3108C2semidirect
12. #3095C2semidirect
13. #3094C2semidirect
14. #3092C2semidirect
15. #2975C2semidirect
22. #552C22semidirect
32. C2×#125C22semidirect
37. C2×C22⋊Q8C22semidirect
40. C2×C4×Q8C22semidirect
44. C22×C4⋊C4C22semidirect
45. #499C22semidirect
46. #488C22semidirect
47. #485C22semidirect
48. #485C22semidirect
49. #484C22semidirect
50. #484C22semidirect
51. #484C22semidirect
52. #483C22semidirect
53. #483C22semidirect
54. #483C22semidirect
55. #482C22semidirect
57. #481C22semidirect
60. #479C22semidirect
61. #479C22semidirect
62. #478C22semidirect
63. #470C22semidirect
64. #470C22semidirect
65. #468C22semidirect
66. #468C22semidirect
67. #464C22semidirect
68. #452C22semidirect
69. #451C22semidirect
70. #451C22semidirect
71. #450C22semidirect
73. C4×Q16C22semidirect
74. C4×SD16C22semidirect
76. #436C22semidirect
77. #431C22semidirect
78. #428C22semidirect
79. #428C22semidirect
80. #428C22semidirect
81. #419C22semidirect
82. #418C22semidirect
83. #417C22semidirect
86. #407C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5522
#33522
#33512
#33272
#33232
#32322
#31962
#31952
#31932
#31082
#30952
#30942
#30922
#29752