Finite Group Databasebeta

← #35488

Subgroups — #35488

Order 256.

833 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2608C211yesmaximal, characteristic, complemented
2#2589C212yesmaximal, complemented
3#2577C212yesmaximal, complemented
4#2573C212yesmaximal, complemented
5#2561C211yesmaximal, characteristic
6#2530C212yesmaximal, complemented
7#2523C212yesmaximal, complemented
8#2517C212yesmaximal, complemented
9#2515C212yesmaximal, complemented
10#2499C211yesmaximal, characteristic, complemented
11#2496C212yesmaximal, complemented
12#2482C211yesmaximal, characteristic
13#2480C212yesmaximal
14#2480C212yesmaximal
15#2334C212yesmaximal, complemented
16#2332C211yesmaximal, characteristic, complemented
17C4×#126C212yesmaximal, complemented
18#2298C211yesmaximal, characteristic, complemented
19#2273C211yesmaximal, characteristic
20C2×#126C2212yes—
21C2×#126C2212yes—
22C2×C22⋊Q8C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2212yes—
26#400C2212yescomplemented
27#400C2212yescomplemented
28#400C2212yescomplemented
29#399C2212yes—
30#398C2212yescomplemented
31#397C2212yes—
32#397C2212yes—
33#397C2212yes—
34#397C2212yes—
35#397C2212yes—
36#397C2212yes—
37#397C2212yes—
38#397C2212yes—
39#396C2212yes—
40#396C2212yes—
41#393C2212yes—
42#391C2212yescomplemented
43#391C2211yescharacteristic, complemented
44#391C2211yescharacteristic, complemented
45#389C2212yescomplemented
46#389C2212yescomplemented
47#389C2212yescomplemented
48#388C2212yes—
49#388C2212yes—
50#388C2212yes—
51#388C2212yes—
52#388C2212yes—
53#388C2212yes—
54#388C2212yes—
55#388C2212yes—
56#388C2212yes—
57#388C2212yes—
58#387C2212yescomplemented
59#387C2212yescomplemented
60#387C2212yescomplemented
61#387C2212yescomplemented
62#387C2212yescomplemented
63#387C2212yescomplemented
64#387C2212yescomplemented
65#387C2212yescomplemented
66#387C2212yescomplemented
67#387C2212yescomplemented
68#387C2212yescomplemented
69#387C2212yescomplemented
70#387C2212yescomplemented
71#387C2212yescomplemented
72#385C2211yescharacteristic
73#385C2212yes—
74#385C2212yes—
75#385C2212yes—
76#385C2212yes—
77#385C2211yescharacteristic
78#384C2211yescharacteristic, complemented
79#383C2212yescomplemented
80#383C2212yescomplemented
81#382C2212yescomplemented
82#382C2211yescharacteristic, complemented
83#381C2212yes—
84#381C2211yescharacteristic
85#380C2212yes—
86#380C2212yes—
87#379C2212yes—
88#379C2211yescharacteristic
89#379C2211yescharacteristic
90C4×C4⋊C4C2212yescomplemented
91C4×C4⋊C4C2212yescomplemented
92C4×C4⋊C4C2212yescomplemented
93C4×C4⋊C4C2211yescharacteristic, complemented
94C4×C4⋊C4C2211yescharacteristic, complemented
95C4×C22⋊C4C2212yes—
96C4×C22⋊C4C2212yes—
97C4×C22⋊C4C2212yes—
98C4×C22⋊C4C2211yescharacteristic
99C4×C22⋊C4C2212yes—
100C4×C22⋊C4C2212yes—
101C4×C22⋊C4C2211yescharacteristic
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2211yescharacteristic
104C4×C22⋊C4C2212yes—
105#376C2211yescharacteristic, complemented
106C43C2211yescharacteristic, complemented
107C25C2311yescharacteristic
108C22×Q8C2311yescharacteristic
109C23×C4C2312yes—
110C23×C4C2312yes—
111C23×C4C2311yescharacteristic
112C23×C4C2311yescharacteristic
113#126—28no—
114#126—28no—
115C22⋊Q8—24no—
116#117—24no—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2312yes—
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4C2312yes—
138C2×C22⋊C4C2312yes—
139C2×C22⋊C4C2312yes—
140C2×C22⋊C4C2312yes—
141C2×C22⋊C4C2312yes—
142C2×C22⋊C4C2311yescharacteristic
143C2×C22⋊C4C2311yescharacteristic
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⋊C4C2311yescharacteristic
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C42C2311yescharacteristic
158C2×C42C2311yescharacteristic
159C2×C42C2312yes—
160C2×C42C2312yes—
161C2×C42C2311yescharacteristic
162C2×C42C2312yes—
163C2×C42C2311yescharacteristic
164C2×C42C2311yescharacteristic
165C2×C42C2311yescharacteristic
166C2×C42C2312yes—
167C2×C42C2311yescharacteristic
168C2×C42C2312yes—
169C2×C42C2311yescharacteristic
170C2×C42C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2312yes—
174#95C2312yes—
175#95C2312yes—
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200C24—22no—
201C24—24no—
202C24—41no—
203C24—41no—
204C24—42no—
205C24C2411yescharacteristic
206C24C2412yes—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4#4114yes—
221C22×C4C2412yes—
222C22×C4C2412yes—
223C22×C4C2412yes—
224C22×C4C2412yes—
225C22×C4C2411yescharacteristic
226C22×C4C2411yescharacteristic
227C22×C4C2412yes—
228C22×C4C2412yes—
229C22×C4C2411yescharacteristic
230C22×C4C2412yes—
231C22×C4C2411yescharacteristic
232C22×C4C2412yes—
233C22×C4C2412yes—
234C22×C4C2411yescharacteristic
235C22×C4C2411yescharacteristic
236C22×C4C2412yes—
237C22×C4C2412yes—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—24no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4#4118yes—
276C42—22no—
277C42—22no—
278C42—24no—
279C42—24no—
280C42—41no—
281C42—42no—
282C42—41no—
283C42—42no—
284C42—41no—
285C42—42no—
286C42—41no—
287C42—41no—
288C42—41no—
289C42—42no—
290C23—28no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—82no—
302C23—82no—
303C23—84no—
304C23—84no—
305C23C2×#4114yes—
306C23C2×#4112yes—
307C23C2511yescharacteristic
308Q8—82no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
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—44no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4C2×#4112yes—
357C2×C4C2×#4114yes—
358C2×C4C2×#4112yes—
359C2×C4C2×#4114yes—
360C22—22no—
361C22—24no—
362C22—22no—
363C22—22no—
364C22—24no—
365C22—24no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—88no—
373C22C22×#4111yescharacteristic
374C22C22×#4112yes—
375C22#58511yescharacteristic
376C22#58511yescharacteristic
377C22#58512yes—
378C4—22no—
379C4—22no—
380C4—24no—
381C4—24no—
382C4—41no—
383C4—41no—
384C4—42no—
385C4—42no—
386C4—41no—
387C4—41no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—42no—
395C2—42no—
396C2—44no—
397C2#344411yescharacteristic
398C2#349111yescharacteristic
399C2#351011yescharacteristic
400C2#352112yes—
401C2#352112yes—
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. #2608C2semidirect
2. #2589C2semidirect
3. #2577C2semidirect
4. #2573C2semidirect
6. #2530C2semidirect
7. #2523C2semidirect
8. #2517C2semidirect
9. #2515C2semidirect
10. #2499C2semidirect
11. #2496C2semidirect
15. #2334C2semidirect
16. #2332C2semidirect
17. C4×#126C2semidirect
18. #2298C2semidirect
26. #400C22semidirect
27. #400C22semidirect
28. #400C22semidirect
30. #398C22semidirect
42. #391C22semidirect
43. #391C22semidirect
44. #391C22semidirect
45. #389C22semidirect
46. #389C22semidirect
47. #389C22semidirect
58. #387C22semidirect
59. #387C22semidirect
60. #387C22semidirect
61. #387C22semidirect
62. #387C22semidirect
63. #387C22semidirect
64. #387C22semidirect
65. #387C22semidirect
66. #387C22semidirect
67. #387C22semidirect
68. #387C22semidirect
69. #387C22semidirect
70. #387C22semidirect
71. #387C22semidirect
78. #384C22semidirect
79. #383C22semidirect
80. #383C22semidirect
81. #382C22semidirect
82. #382C22semidirect
90. C4×C4⋊C4C22semidirect
91. C4×C4⋊C4C22semidirect
92. C4×C4⋊C4C22semidirect
93. C4×C4⋊C4C22semidirect
94. C4×C4⋊C4C22semidirect
105. #376C22semidirect
106. C43C22semidirect

Maximal subgroups

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

subgroupindex
#26082
#25892
#25772
#25732
#25612
#25302
#25232
#25172
#25152
#24992
#24962
#24822
#24802
#23342
#23322
C4×#1262
#22982
#22732