Finite Group Databasebeta

← #61895

Subgroups — #61895

Order 256.

897 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507C211yesmaximal, characteristic, complemented
2C2×#559C211yesmaximal, characteristic, complemented
3#3403C214yesmaximal, complemented
4#3401C214yesmaximal, complemented
5#3371C214yesmaximal, complemented
6#3370C214yesmaximal, complemented
7#3217C212yesmaximal, complemented
8#3215C211yesmaximal, characteristic, complemented
9#3213C211yesmaximal, characteristic, complemented
10#3211C212yesmaximal, complemented
11#3117C212yesmaximal, complemented
12#3110C211yesmaximal, characteristic, complemented
13#3109C211yesmaximal, characteristic, complemented
14#3105C212yesmaximal, complemented
15#2983C211yesmaximal, characteristic, complemented
16#559—22no—
17#559—22no—
18#559C2214yes—
19#559C2214yes—
20#559C2212yescomplemented
21#557C2212yescomplemented
22#556—22no—
23#556C2212yescomplemented
24#554C2212yescomplemented
25#552—22no—
26#539—21no—
27#539—21no—
28#536—21no—
29#533C2211yescharacteristic
30#532C2211yescharacteristic
31C2×#127C2211yescharacteristic
32#529C2212yes—
33#529C2212yes—
34C2×#124—21no—
35C2×#124C2211yescharacteristic
36C2×#124C2211yescharacteristic
37C2×#123—21no—
38C2×#123—21no—
39#520—21no—
40C2×C4×Q8C2211yescharacteristic
41C2×C4×Q8C2211yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4C2211yescharacteristic
44C2×C4×D4C2211yescharacteristic
45#497C2212yescomplemented
46#496C2214yes—
47#496C2212yescomplemented
48#495C2214yescomplemented
49#490C2214yes—
50#489C2212yescomplemented
51#489C2212yescomplemented
52#488C2214yescomplemented
53#484C2212yes—
54#481C2212yes—
55#479C2214yescomplemented
56#478C2214yescomplemented
57#476C2212yescomplemented
58#474C2212yescomplemented
59#464C2214yescomplemented
60#464C2214yescomplemented
61#464C2214yescomplemented
62#463C2214yescomplemented
63#463C2214yescomplemented
64#463C2214yescomplemented
65#461C2212yescomplemented
66#460C2214yes—
67#459C2214yes—
68#459C2212yescomplemented
69#451C2212yes—
70#450C2212yes—
71#450C2212yes—
72#448C2212yes—
73#447C2212yes—
74#447C2212yes—
75#446C2214yescomplemented
76#446C2214yescomplemented
77#442C2214yescomplemented
78#441C2214yescomplemented
79#440C2214yescomplemented
80#440C2214yescomplemented
81C4×SD16C2214yescomplemented
82C4×D8C2214yescomplemented
83#432C2211yescharacteristic
84#432C2211yescharacteristic
85#432C2212yes—
86#431C2212yes—
87C2×#41—21no—
88C22×Q8D411yescharacteristic
89C22×Q8C2311yescharacteristic
90C22×D4—21no—
91C22×D4—21no—
92C22×D4C2311yescharacteristic
93C22×D4C2311yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C2×Q16C2314yes—
97C2×SD16C2314yes—
98C2×SD16C2314yes—
99C2×SD16C2314yes—
100C2×D8C2314yes—
101C2×D8C2314yes—
102C4⋊Q8—22no—
103C4⋊Q8C2311yescharacteristic
104C4⋊Q8C2311yescharacteristic
105C4⋊Q8C2311yescharacteristic
106#127—22no—
107#127C2314yes—
108#127C2311yescharacteristic
109#126—22no—
110#126—22no—
111#126—22no—
112#126—22no—
113#125—22no—
114#125—22no—
115#125—21no—
116#125C2312yes—
117#125C2312yes—
118#124—24no—
119#124—24no—
120#124—22no—
121#124—22no—
122#124—21no—
123#124—21no—
124#124—21no—
125#124C2312yes—
126#124C2314yes—
127#124C2312yes—
128#124C2312yes—
129#124C2312yes—
130#123—22no—
131#123—22no—
132#123—24no—
133#123—24no—
134#123—24no—
135#123—24no—
136#123—22no—
137#123—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8C2312yes—
144#121—21no—
145#121C2312yes—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—21no—
149C4×Q8D412yes—
150C4×Q8D412yes—
151C4×Q8C2314yes—
152C4×Q8C2312yes—
153C4×Q8C2312yes—
154C4×Q8C2312yes—
155C4×Q8C2312yes—
156C4×Q8C2312yes—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4C2314yes—
167C4×D4C2312yes—
168C4×D4C2312yes—
169C4×D4C2312yes—
170C4×D4C2312yes—
171C4×D4C2312yes—
172#117—21no—
173#117—22no—
174#117—22no—
175#117—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4D411yescharacteristic
179C2×C4⋊C4D411yescharacteristic
180C2×C4⋊C4D411yescharacteristic
181C2×C4⋊C4C2311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42—21no—
189C2×C42C2311yescharacteristic
190C2×C42C2311yescharacteristic
191C2×C42C2311yescharacteristic
192#107C2314yes—
193#106C2314yes—
194C4⋊C8C2314yes—
195C4⋊C8C2314yes—
196C4⋊C8C2312yes—
197C4⋊C8C2312yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#102C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208#97C2312yes—
209#97C2314yes—
210#97C2312yes—
211C4×C8C2314yes—
212C24—21no—
213C24—21no—
214#41—44no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8C2×D412yes—
221C2×Q8C2×D412yes—
222C2×Q8C2412yes—
223C2×Q8C2412yes—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—24no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—82no—
235C2×D4C2412yes—
236C2×D4C2412yes—
237C2×D4C2412yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2411yescharacteristic
255Q16—44no—
256SD16—44no—
257SD16—44no—
258SD16—44no—
259D8—44no—
260D8—44no—
261C2×C8C2414yes—
262C2×C8C2414yes—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4C2×D412yes—
278C4⋊C4C2×D412yes—
279C4⋊C4C2×D412yes—
280C4⋊C4C2×D412yes—
281C4⋊C4C2×D412yes—
282C4⋊C4C2×D412yes—
283C4⋊C4C2412yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—24no—
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—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42C2×D412yes—
309C42C2×D412yes—
310C42C2×D411yescharacteristic
311C42C2×D411yescharacteristic
312C42C2412yes—
313C42C2411yescharacteristic
314C42C2411yescharacteristic
315C42C2412yes—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23C22×D411yescharacteristic
326Q8—24no—
327Q8—24no—
328Q8—42no—
329Q8—42no—
330Q8—42no—
331Q8—42no—
332Q8—81no—
333D4—24no—
334D4—22no—
335D4—24no—
336D4—22no—
337D4—42no—
338D4—42no—
339D4—48no—
340D4—42no—
341D4—44no—
342D4—81no—
343D4—84no—
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—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D412yes—
366C2×C4C22×D412yes—
367C2×C4C22×D412yes—
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D412yes—
370C2×C4C22×D411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C2511yescharacteristic
374C8—24no—
375C8—24no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—84no—
389C22#57512yes—
390C22C23×D411yescharacteristic
391C4—22no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—21no—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4#58512yes—
406C2—22no—
407C2—82no—
408C2—82no—
409C2—82no—
410C2#348711yescharacteristic
411C2C2×#57511yescharacteristic
412C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3507C2semidirect
2. C2×#559C2semidirect
3. #3403C2semidirect
4. #3401C2semidirect
5. #3371C2semidirect
6. #3370C2semidirect
7. #3217C2semidirect
8. #3215C2semidirect
9. #3213C2semidirect
10. #3211C2semidirect
11. #3117C2semidirect
12. #3110C2semidirect
13. #3109C2semidirect
14. #3105C2semidirect
15. #2983C2semidirect
20. #559C22semidirect
21. #557C22semidirect
23. #556C22semidirect
24. #554C22semidirect
45. #497C22semidirect
47. #496C22semidirect
48. #495C22semidirect
50. #489C22semidirect
51. #489C22semidirect
52. #488C22semidirect
55. #479C22semidirect
56. #478C22semidirect
57. #476C22semidirect
58. #474C22semidirect
59. #464C22semidirect
60. #464C22semidirect
61. #464C22semidirect
62. #463C22semidirect
63. #463C22semidirect
64. #463C22semidirect
65. #461C22semidirect
68. #459C22semidirect
75. #446C22semidirect
76. #446C22semidirect
77. #442C22semidirect
78. #441C22semidirect
79. #440C22semidirect
80. #440C22semidirect
81. C4×SD16C22semidirect
82. C4×D8C22semidirect

Maximal subgroups

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

subgroupindex
#35072
C2×#5592
#34032
#34012
#33712
#33702
#32172
#32152
#32132
#32112
#31172
#31102
#31092
#31052
#29832