Finite Group Databasebeta

← #61713

Subgroups — #61713

Order 256.

913 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic, complemented
2C2×#553C211yesmaximal, characteristic, complemented
3#3350C214yesmaximal, complemented
4#3347C214yesmaximal, complemented
5#3309C214yesmaximal, complemented
6#3307C214yesmaximal, complemented
7#3236C211yesmaximal, characteristic, complemented
8#3234C211yesmaximal, characteristic, complemented
9#3194C212yesmaximal, complemented
10#3188C212yesmaximal, complemented
11#3110C211yesmaximal, characteristic, complemented
12#3109C211yesmaximal, characteristic, complemented
13#3092C212yesmaximal, complemented
14#3091C212yesmaximal, complemented
15#2973C211yesmaximal, characteristic, complemented
16#559—22no—
17#556C2212yescomplemented
18#555C2212yescomplemented
19#553—22no—
20#553—22no—
21#553—22no—
22#553C2214yes—
23#553C2214yes—
24#550C2212yescomplemented
25#547C2212yescomplemented
26#538—21no—
27#538—21no—
28#535—21no—
29#533C2211yescharacteristic
30#532C2211yescharacteristic
31C2×#126—21no—
32C2×#125C2211yescharacteristic
33#525C2212yes—
34#525C2212yes—
35C2×#123—21no—
36C2×#121—21no—
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic
39#520—21no—
40C2×C4×D4—21no—
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic
43C2×#117C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2212yescomplemented
46#501C2212yescomplemented
47#491C2212yescomplemented
48#490C2212yescomplemented
49#485C2214yescomplemented
50#484C2212yes—
51#484C2212yes—
52#483C2212yes—
53#483C2214yescomplemented
54#482C2214yescomplemented
55#482C2214yescomplemented
56#482C2214yescomplemented
57#481C2212yes—
58#481C2212yes—
59#480C2212yes—
60#480C2214yescomplemented
61#479C2214yescomplemented
62#478C2214yescomplemented
63#476C2212yescomplemented
64#474C2212yescomplemented
65#469C2214yescomplemented
66#468C2214yescomplemented
67#466C2214yescomplemented
68#465C2214yescomplemented
69#461C2212yescomplemented
70#459C2212yescomplemented
71#450C2212yes—
72#449C2214yescomplemented
73#449C2214yescomplemented
74#447C2212yes—
75#442C2214yescomplemented
76#440C2214yescomplemented
77#435C2214yescomplemented
78#435C2214yescomplemented
79#432C2211yescharacteristic
80#432C2211yescharacteristic
81#428C2214yescomplemented
82#427C2214yescomplemented
83#417C2214yescomplemented
84#416C2214yescomplemented
85#407C2212yes—
86C2×C22⋊C8C2212yes—
87C2×#41—21no—
88C2×#41C2312yes—
89C2×#41C2312yes—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C2×SD16C2314yes—
99C2×D8C2314yes—
100C2×M16C2314yes—
101C22×C8C2314yes—
102C4⋊Q8C2311yescharacteristic
103C4⋊Q8C2311yescharacteristic
104C4⋊Q8C2311yescharacteristic
105#127—22no—
106#127C2311yescharacteristic
107#126—24no—
108#126—24no—
109#126—22no—
110#126—22no—
111#125—21no—
112#125—21no—
113#125—21no—
114#125C2312yes—
115#125C2312yes—
116#124—22no—
117#124—22no—
118#124—22no—
119#124—21no—
120#123—22no—
121#123—22no—
122#123—24no—
123#123—24no—
124#123—22no—
125#123—22no—
126C22⋊Q8—21no—
127C22⋊Q8C2312yes—
128C22⋊Q8C2312yes—
129C22⋊Q8C2312yes—
130#121—24no—
131#121—24no—
132#121—22no—
133#121—22no—
134#121—22no—
135#121—22no—
136#121—21no—
137#121—22no—
138#121—22no—
139#121C2314yes—
140#121C2314yes—
141#121C2312yes—
142#121C2312yes—
143#121C2312yes—
144#120—22no—
145#120—22no—
146C4×Q8—22no—
147C4×Q8—21no—
148C4×Q8C2312yes—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4D412yes—
160C4×D4D412yes—
161C4×D4C2312yes—
162C4×D4C2312yes—
163C4×D4C2312yes—
164C4×D4C2312yes—
165C4×D4C2312yes—
166#117—22no—
167#117—21no—
168#117—22no—
169#117C2314yes—
170#117C2312yes—
171#117C2312yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4D411yescharacteristic
178C2×C4⋊C4C2311yescharacteristic
179C2×C4⋊C4C2311yescharacteristic
180C2×C4⋊C4C2314yes—
181C2×C4⋊C4C2311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4D411yescharacteristic
187C2×C22⋊C4D411yescharacteristic
188C2×C42—21no—
189C2×C42C2311yescharacteristic
190#107C2314yes—
191#107C2314yes—
192#107C2314yes—
193#106C2314yes—
194#106C2314yes—
195#106C2314yes—
196C4⋊C8C2312yes—
197C4⋊C8C2312yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#102C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208C22⋊C8C2314yes—
209C22⋊C8C2314yes—
210#97C2312yes—
211#97C2312yes—
212C24—21no—
213C24C2×D411yescharacteristic
214C24C2×D411yescharacteristic
215#41—44no—
216#41—44no—
217#41—44no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8C2412yes—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—24no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—82no—
235C2×D4C2×D412yes—
236C2×D4C2×D412yes—
237C2×D4C2412yes—
238C2×D4C2412yes—
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—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D412yes—
257C22×C4C2×D412yes—
258C22×C4C2411yescharacteristic
259C22×C4C2412yes—
260C22×C4C2412yes—
261SD16—44no—
262D8—44no—
263M16—44no—
264C2×C8—44no—
265C2×C8C2414yes—
266C2×C8C2414yes—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—24no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4C2×D412yes—
284C4⋊C4C2×D412yes—
285C4⋊C4C2412yes—
286C4⋊C4C2412yes—
287C4⋊C4C2412yes—
288C4⋊C4C2412yes—
289C4⋊C4C2412yes—
290C22⋊C4—24no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—24no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—24no—
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—24no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42C2×D411yescharacteristic
313C42C2×D411yescharacteristic
314C42C2411yescharacteristic
315C42C2411yescharacteristic
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23C22×D412yes—
328C23C22×D412yes—
329C23C22×D411yescharacteristic
330Q8—24no—
331Q8—42no—
332Q8—81no—
333D4—22no—
334D4—24no—
335D4—22no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—42no—
340D4—48no—
341D4—44no—
342D4—81no—
343D4—82no—
344D4—82no—
345D4—84no—
346D4—82no—
347D4—82no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4C22×D412yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D412yes—
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D411yescharacteristic
377C2×C4C22×D411yescharacteristic
378C2×C4C22×D411yescharacteristic
379C2×C4C2511yescharacteristic
380C8—24no—
381C8—24no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—44no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—84no—
399C22#57512yes—
400C22C23×D411yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—22no—
405C4—22no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4#58512yes—
415C2—22no—
416C2—42no—
417C2—42no—
418C2—82no—
419C2—82no—
420C2#348511yescharacteristic
421C2C2×#57511yescharacteristic
422C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
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
1. #3506C2semidirect
2. C2×#553C2semidirect
3. #3350C2semidirect
4. #3347C2semidirect
5. #3309C2semidirect
6. #3307C2semidirect
7. #3236C2semidirect
8. #3234C2semidirect
9. #3194C2semidirect
10. #3188C2semidirect
11. #3110C2semidirect
12. #3109C2semidirect
13. #3092C2semidirect
14. #3091C2semidirect
15. #2973C2semidirect
17. #556C22semidirect
18. #555C22semidirect
24. #550C22semidirect
25. #547C22semidirect
45. #501C22semidirect
46. #501C22semidirect
47. #491C22semidirect
48. #490C22semidirect
49. #485C22semidirect
53. #483C22semidirect
54. #482C22semidirect
55. #482C22semidirect
56. #482C22semidirect
60. #480C22semidirect
61. #479C22semidirect
62. #478C22semidirect
63. #476C22semidirect
64. #474C22semidirect
65. #469C22semidirect
66. #468C22semidirect
67. #466C22semidirect
68. #465C22semidirect
69. #461C22semidirect
70. #459C22semidirect
72. #449C22semidirect
73. #449C22semidirect
75. #442C22semidirect
76. #440C22semidirect
77. #435C22semidirect
78. #435C22semidirect
81. #428C22semidirect
82. #427C22semidirect
83. #417C22semidirect
84. #416C22semidirect

Maximal subgroups

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

subgroupindex
#35062
C2×#5532
#33502
#33472
#33092
#33072
#32362
#32342
#31942
#31882
#31102
#31092
#30922
#30912
#29732