Finite Group Databasebeta

← #44839

Subgroups — #44839

Order 256.

834 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2807C211yesmaximal, characteristic, complemented
2#2768C212yesmaximal, complemented
3#2738C212yesmaximal, complemented
4#2735C211yesmaximal, characteristic
5#2689C212yesmaximal, complemented
6#2611C211yesmaximal, characteristic, complemented
7#2587C212yesmaximal, complemented
8#2577C212yesmaximal, complemented
9#2523C211yesmaximal, characteristic, complemented
10#2515C212yesmaximal, complemented
11#2495C212yesmaximal, complemented
12#2482C211yesmaximal, characteristic
13#2480C212yesmaximal
14#2475C212yesmaximal, complemented
15#2437C211yesmaximal, characteristic
16#2396C211yesmaximal, characteristic, complemented
17#2373C212yesmaximal, complemented
18#2342C212yesmaximal
19#2323C212yesmaximal, complemented
20C2×#126C2212yes—
21C2×#125C2211yescharacteristic, complemented
22C2×C22⋊Q8C2212yes—
23C2×C4×Q8C2211yescharacteristic, complemented
24C2×#117C2212yes—
25C2×#117C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2211yescharacteristic
28#400C2212yescomplemented
29#400C2212yescomplemented
30#400C2211yescharacteristic, complemented
31#400C2211yescharacteristic, complemented
32#400C2212yescomplemented
33#399C2211yescharacteristic
34#399C2211yescharacteristic
35#399C2212yes—
36#398C2212yescomplemented
37#398C2211yescharacteristic, complemented
38#398C2212yescomplemented
39#398C2212yescomplemented
40#398C2212yescomplemented
41#397C2212yes—
42#397C2212yes—
43#397C2211yescharacteristic
44#397C2212yes—
45#397C2212yes—
46#397C2212yes—
47#396C2212yes—
48#396C2212yes—
49#396C2212yes—
50#395C2211yescharacteristic, complemented
51#393C2211yescharacteristic
52#391C2212yescomplemented
53#389C2212yescomplemented
54#389C2212yescomplemented
55#389C2212yescomplemented
56#389C2212yescomplemented
57#389C2212yescomplemented
58#389C2212yescomplemented
59#388C2212yes—
60#388C2212yes—
61#388C2212yes—
62#388C2212yes—
63#388C2212yes—
64#388C2212yes—
65#388C2212yes—
66#388C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#387C2212yescomplemented
72#387C2212yescomplemented
73#387C2212yescomplemented
74#387C2212yescomplemented
75#387C2212yescomplemented
76#387C2212yescomplemented
77#387C2212yescomplemented
78#387C2212yescomplemented
79#385C2212yes—
80#385C2211yescharacteristic
81#385C2212yes—
82#385C2212yes—
83#385C2212yes—
84#385C2212yes—
85#385C2212yes—
86#385C2211yescharacteristic
87#383C2212yescomplemented
88#383C2212yescomplemented
89#382C2212yescomplemented
90#381C2212yes—
91#381C2211yescharacteristic
92#380C2212yes—
93#380C2212yes—
94#380C2212yes—
95#379C2212yes—
96#379C2211yescharacteristic
97#379C2211yescharacteristic
98C4×C4⋊C4C2211yescharacteristic, complemented
99C4×C4⋊C4C2212yescomplemented
100C4×C4⋊C4C2212yescomplemented
101C4×C4⋊C4C2212yescomplemented
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106#376C2211yescharacteristic, complemented
107C25C2311yescharacteristic
108C22×Q8C2311yescharacteristic
109C23×C4C2312yes—
110C23×C4C2311yescharacteristic
111C23×C4C2312yes—
112C23×C4C2311yescharacteristic
113#126—28no—
114#125—42no—
115C22⋊Q8—28no—
116C4×Q8—42no—
117#117—42no—
118#117—44no—
119C2×C4⋊C4C2312yes—
120C2×C4⋊C4C2312yes—
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2312yes—
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2311yescharacteristic
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2311yescharacteristic
156C2×C22⋊C4C2311yescharacteristic
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2311yescharacteristic
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2312yes—
163C2×C22⋊C4C2312yes—
164C2×C42C2312yes—
165C2×C42C2311yescharacteristic
166C2×C42C2311yescharacteristic
167C2×C42C2312yes—
168C2×C42C2312yes—
169C2×C42C2312yes—
170C2×C42C2312yes—
171C2×C42C2312yes—
172C2×C42C2312yes—
173#95C2312yes—
174#95C2311yescharacteristic
175#95C2311yescharacteristic
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2311yescharacteristic
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2311yescharacteristic
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202C24—24no—
203C24—22no—
204C24—41no—
205C24—42no—
206C24—41no—
207C24C2412yes—
208C24C2411yescharacteristic
209C2×Q8—24no—
210C2×Q8—41no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C2412yes—
223C22×C4C2412yes—
224C22×C4C2412yes—
225C22×C4C2411yescharacteristic
226C22×C4C2412yes—
227C22×C4C2412yes—
228C22×C4C2412yes—
229C22×C4C2412yes—
230C22×C4C2411yescharacteristic
231C22×C4C2412yes—
232C22×C4C2411yescharacteristic
233C22×C4C2412yes—
234C22×C4C2411yescharacteristic
235C22×C4C2411yescharacteristic
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2411yescharacteristic
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—22no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4#4118yes—
282C42—24no—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—21no—
301C23—44no—
302C23—48no—
303C23—84no—
304C23—82no—
305C23—82no—
306C23C2×#4114yes—
307C232−1+412yes—
308C23C2511yescharacteristic
309Q8—44no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
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—21no—
353C2×C4—22no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4C2×#4114yes—
359C2×C4C2×#4114yes—
360C2×C4H(5,𝔽2)12yes—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—22no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—42no—
373C22—88no—
374C22C22×#4112yes—
375C22C2×H(5,𝔽2)11yescharacteristic
376C22C2×2−1+411yescharacteristic
377C22#58512yes—
378C22#58511yescharacteristic
379C4—24no—
380C4—24no—
381C4—22no—
382C4—41no—
383C4—42no—
384C4—41no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—42no—
396C2—44no—
397C2—42no—
398C2#348111yescharacteristic
399C2#349512yes—
400C2#352112yes—
401C2#353511yescharacteristic
402C2#355111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2807C2semidirect
2. #2768C2semidirect
3. #2738C2semidirect
5. #2689C2semidirect
6. #2611C2semidirect
7. #2587C2semidirect
8. #2577C2semidirect
9. #2523C2semidirect
10. #2515C2semidirect
11. #2495C2semidirect
14. #2475C2semidirect
16. #2396C2semidirect
17. #2373C2semidirect
19. #2323C2semidirect
21. C2×#125C22semidirect
23. C2×C4×Q8C22semidirect
28. #400C22semidirect
29. #400C22semidirect
30. #400C22semidirect
31. #400C22semidirect
32. #400C22semidirect
36. #398C22semidirect
37. #398C22semidirect
38. #398C22semidirect
39. #398C22semidirect
40. #398C22semidirect
50. #395C22semidirect
52. #391C22semidirect
53. #389C22semidirect
54. #389C22semidirect
55. #389C22semidirect
56. #389C22semidirect
57. #389C22semidirect
58. #389C22semidirect
71. #387C22semidirect
72. #387C22semidirect
73. #387C22semidirect
74. #387C22semidirect
75. #387C22semidirect
76. #387C22semidirect
77. #387C22semidirect
78. #387C22semidirect
87. #383C22semidirect
88. #383C22semidirect
89. #382C22semidirect
98. C4×C4⋊C4C22semidirect
99. C4×C4⋊C4C22semidirect
100. C4×C4⋊C4C22semidirect
101. C4×C4⋊C4C22semidirect
106. #376C22semidirect

Maximal subgroups

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

subgroupindex
#28072
#27682
#27382
#27352
#26892
#26112
#25872
#25772
#25232
#25152
#24952
#24822
#24802
#24752
#24372
#23962
#23732
#23422
#23232