Finite Group Databasebeta

← #61709

Subgroups — #61709

Order 256.

903 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic
2C2×#554C211yesmaximal, characteristic, complemented
3#3346C214yesmaximal, complemented
4#3344C214yesmaximal, complemented
5#3316C214yesmaximal, complemented
6#3314C214yesmaximal, complemented
7#3239C212yesmaximal, complemented
8#3196C212yesmaximal, complemented
9#3189C211yesmaximal, characteristic, complemented
10#3187C211yesmaximal, characteristic, complemented
11#3117C212yesmaximal, complemented
12#3092C212yesmaximal, complemented
13C2×#484C211yesmaximal, characteristic, complemented
14C2×#481C211yesmaximal, characteristic, complemented
15#2973C211yesmaximal, characteristic, complemented
16#554—22no—
17#554—22no—
18#554C2214yes—
19#554C2214yescomplemented
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×#125—21no—
32#525C2212yes—
33C2×C22⋊Q8—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2211yescharacteristic, complemented
38C2×#121C2211yescharacteristic
39#518—21no—
40C2×C4×Q8C2211yescharacteristic, complemented
41C2×C4×D4—21no—
42C2×C4×D4C2211yescharacteristic
43C22×C4⋊C4C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic, complemented
45#501C2214yescomplemented
46#489C2214yescomplemented
47#485C2214yescomplemented
48#484C2214yescomplemented
49#484C2212yescomplemented
50#484C2212yescomplemented
51#484C2212yescomplemented
52#484C2212yescomplemented
53#483C2212yescomplemented
54#483C2214yescomplemented
55#481C2212yes—
56#481C2212yes—
57#481C2212yes—
58#481C2214yescomplemented
59#480C2214yescomplemented
60#479C2214yescomplemented
61#475C2214yescomplemented
62#474C2214yescomplemented
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#428C2214yescomplemented
79C2×#107C2212yescomplemented
80C2×#106C2212yescomplemented
81#417C2214yescomplemented
82C2×#103C2212yescomplemented
83C2×#102C2212yescomplemented
84#407C2212yescomplemented
85C2×C22⋊C8C2211yescharacteristic
86C2×C22⋊C8C2211yescharacteristic
87C2×#41C2312yes—
88C22×Q8—21no—
89C22×Q8C2311yescharacteristic
90C22×Q8C2311yescharacteristic
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C2×Q16C2314yes—
99C2×SD16C2314yes—
100C2×M16C2314yes—
101C22×C8C2312yes—
102C22×C8C2312yes—
103C4⋊Q8—22no—
104C4⋊Q8C2314yes—
105#126—22no—
106#126—22no—
107#126—22no—
108#125—24no—
109#125—24no—
110#125—22no—
111#125C2312yes—
112#124—22no—
113#124C2312yes—
114#123—22no—
115#123—22no—
116#123—22no—
117#123—22no—
118#123—22no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C22⋊Q8—24no—
128C22⋊Q8—24no—
129C22⋊Q8C2312yes—
130C22⋊Q8C2312yes—
131C22⋊Q8C2312yes—
132C22⋊Q8C2312yes—
133C22⋊Q8C2314yes—
134C22⋊Q8C2312yes—
135#121—22no—
136#121—22no—
137#121—21no—
138#121—21no—
139#121—22no—
140#121C2312yes—
141#121C2312yes—
142#121C2312yes—
143#120—22no—
144#120—22no—
145C4×Q8—22no—
146C4×Q8C2314yes—
147C4×Q8C2312yes—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4D412yes—
156C4×D4D412yes—
157C4×D4C2312yes—
158C4×D4C2312yes—
159C4×D4C2312yes—
160#117—22no—
161#117—22no—
162#117C2312yes—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
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⋊C4C2312yes—
177C2×C4⋊C4C2312yes—
178C2×C4⋊C4C2311yescharacteristic
179C2×C4⋊C4C2314yes—
180C2×C4⋊C4C2311yescharacteristic
181C2×C4⋊C4C2311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4D411yescharacteristic
188C2×C22⋊C4D411yescharacteristic
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—
211#97C2314yes—
212C24—21no—
213C24C2×D411yescharacteristic
214C24C2×D411yescharacteristic
215#41—44no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—82no—
222C2×Q8C2412yes—
223C2×Q8C2412yes—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
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—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D412yes—
258C22×C4C2411yescharacteristic
259C22×C4C2411yescharacteristic
260C22×C4C2411yescharacteristic
261C22×C4C2412yes—
262Q16—44no—
263SD16—44no—
264M16—44no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8C2414yes—
268C2×C8C2414yes—
269C4⋊C4—21no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4C2×D412yes—
289C4⋊C4C2×D412yes—
290C4⋊C4C2412yes—
291C4⋊C4C2412yes—
292C4⋊C4C2412yes—
293C4⋊C4C2412yes—
294C4⋊C4C2412yes—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4C2×D412yes—
309C22⋊C4C2×D412yes—
310C22⋊C4C2×D412yes—
311C22⋊C4C2×D412yes—
312C42—24no—
313C42—22no—
314C42—22no—
315C42C2×D412yes—
316C42C2412yes—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—81no—
325C23—81no—
326C23#13714yes—
327C23C22×D412yes—
328C23C22×D411yescharacteristic
329C23C22×D411yescharacteristic
330C23C22×D411yescharacteristic
331Q8—24no—
332Q8—24no—
333Q8—42no—
334Q8—81no—
335Q8—84no—
336D4—24no—
337D4—24no—
338D4—24no—
339D4—44no—
340D4—44no—
341D4—81no—
342D4—82no—
343D4—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—81no—
374C2×C4C22×D412yes—
375C2×C4C22×D412yes—
376C2×C4C22×D412yes—
377C2×C4C22×D411yescharacteristic
378C2×C4C22×D411yescharacteristic
379C2×C4C22×D411yescharacteristic
380C2×C4C2511yescharacteristic
381C8—24no—
382C8—24no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—44no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—162no—
398C22C2×#13712yes—
399C22C2×#13712yes—
400C22C2×#13712yes—
401C22C23×D411yescharacteristic
402C4—22no—
403C4—21no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4#58512yes—
416C2—22no—
417C2—22no—
418C2—22no—
419C2—42no—
420C2—82no—
421C2#348511yescharacteristic
422C2C22×#13711yescharacteristic
423C2#358511yescharacteristic
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×#554C2semidirect
3. #3346C2semidirect
4. #3344C2semidirect
5. #3316C2semidirect
6. #3314C2semidirect
7. #3239C2semidirect
8. #3196C2semidirect
9. #3189C2semidirect
10. #3187C2semidirect
11. #3117C2semidirect
12. #3092C2semidirect
13. C2×#484C2semidirect
14. C2×#481C2semidirect
15. #2973C2semidirect
19. #554C22semidirect
29. C2×C4⋊Q8C22semidirect
37. C2×C22⋊Q8C22semidirect
40. C2×C4×Q8C22semidirect
44. C22×C4⋊C4C22semidirect
45. #501C22semidirect
46. #489C22semidirect
47. #485C22semidirect
48. #484C22semidirect
49. #484C22semidirect
50. #484C22semidirect
51. #484C22semidirect
52. #484C22semidirect
53. #483C22semidirect
54. #483C22semidirect
58. #481C22semidirect
59. #480C22semidirect
60. #479C22semidirect
61. #475C22semidirect
62. #474C22semidirect
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. C2×#107C22semidirect
80. C2×#106C22semidirect
81. #417C22semidirect
82. C2×#103C22semidirect
83. C2×#102C22semidirect
84. #407C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5542
#33462
#33442
#33162
#33142
#32392
#31962
#31892
#31872
#31172
#30922
C2×#4842
C2×#4812
#29732