Finite Group Databasebeta

← #37445

Subgroups — #37445

Order 256.

1277 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic, complemented
2C2×#525C211yesmaximal, characteristic, complemented
3#2812C214yesmaximal, complemented
4#2663C212yesmaximal, complemented
5#2661C212yesmaximal, complemented
6#2638C212yesmaximal, complemented
7#2635C212yesmaximal, complemented
8#2625C211yesmaximal, characteristic, complemented
9#2624C211yesmaximal, characteristic, complemented
10#2616C211yesmaximal, characteristic, complemented
11#2615C211yesmaximal, characteristic, complemented
12#2602C214yesmaximal, complemented
13#2485C212yesmaximal, complemented
14#2415C212yesmaximal, complemented
15#2414C212yesmaximal, complemented
16#2407C212yesmaximal, complemented
17#2280C211yesmaximal, characteristic, complemented
18C22×#41C2211yescharacteristic
19C22×#41C2211yescharacteristic
20C24×C4C2211yescharacteristic
21#525—28no—
22#525—28no—
23C2×#123C2212yes—
24C2×#123C2212yes—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2212yescomplemented
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×#121C2211yescharacteristic
33C2×#121C2211yescharacteristic
34C2×#121C2212yescomplemented
35C2×#120C2212yes—
36C2×#120C2212yes—
37C2×C4×D4C2212yes—
38C2×C4×D4C2212yes—
39C2×C4×D4C2212yes—
40C2×C4×D4C2212yes—
41C2×#117C2211yescharacteristic
42C2×#117C2211yescharacteristic
43C2×#117C2212yes—
44#400C2212yescomplemented
45#399C2212yescomplemented
46#398C2212yescomplemented
47#397C2214yes—
48#397C2214yes—
49#397C2214yes—
50#397C2212yescomplemented
51#395C2212yescomplemented
52#394C2214yes—
53#394C2214yes—
54#394C2214yes—
55#394C2214yes—
56#394C2214yes—
57#394C2214yes—
58#394C2212yescomplemented
59#393C2212yescomplemented
60#392C2214yes—
61#392C2214yes—
62#392C2214yes—
63#392C2212yescomplemented
64#391C2212yescomplemented
65#390C2214yescomplemented
66#390C2212yescomplemented
67#389C2214yescomplemented
68#389C2212yescomplemented
69#389C2212yescomplemented
70#388C2214yescomplemented
71#388C2214yescomplemented
72#388C2214yescomplemented
73#387C2214yescomplemented
74#386C2214yescomplemented
75#385C2214yescomplemented
76#381C2212yes—
77#380C2211yescharacteristic
78#380C2211yescharacteristic
79#380C2212yes—
80#380C2212yescomplemented
81#380C2212yescomplemented
82#379C2212yes—
83C4×C22⋊C4C2212yes—
84C4×C22⋊C4C2212yes—
85C25C2311yescharacteristic
86C2×#41—24no—
87C2×#41—24no—
88C2×#41—44no—
89C2×#41—44no—
90C22×Q8C2311yescharacteristic
91C22×Q8C2311yescharacteristic
92C22×D4C2311yescharacteristic
93C22×D4C2312yes—
94C22×D4C2312yes—
95C22×D4C2312yes—
96C22×D4C2312yes—
97C22×D4C2312yes—
98C22×D4C2312yes—
99C22×D4C2311yescharacteristic
100C22×D4C2314yes—
101C23×C4—44no—
102C23×C4D418yescomplemented
103C23×C4C2312yes—
104C23×C4C2311yescharacteristic
105C23×C4C2311yescharacteristic
106C23×C4C2311yescharacteristic
107C23×C4C2312yes—
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110C23×C4C2311yescharacteristic
111#123—28no—
112#123—28no—
113C22⋊Q8—24no—
114C22⋊Q8—24no—
115C22⋊Q8—24no—
116C22⋊Q8—24no—
117C22⋊Q8—44no—
118#121—24no—
119#121—24no—
120#121—24no—
121#121—24no—
122#121—44no—
123#120—28no—
124#120—28no—
125C4×D4—28no—
126C4×D4—28no—
127C4×D4—28no—
128C4×D4—28no—
129#117—24no—
130#117—24no—
131#117—44no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2314yes—
143C2×C4⋊C4C2314yes—
144C2×C4⋊C4C2314yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2312yes—
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2314yes—
156C2×C22⋊C4C2314yes—
157C2×C22⋊C4C2314yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2314yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2314yes—
162C2×C22⋊C4C2314yes—
163C2×C22⋊C4C2314yes—
164C2×C22⋊C4C2314yes—
165C2×C42C2312yes—
166C2×C42C2312yes—
167C2×C42C2314yes—
168#95C2314yes—
169#95C2312yes—
170#95C2312yes—
171#95C2314yes—
172#95C2314yes—
173#95C2314yes—
174#95C2314yes—
175#95C2314yes—
176#95C2314yes—
177C24—42no—
178C24—41no—
179C24—42no—
180C24—41no—
181C24C2×D414yes—
182C24C2412yes—
183C24C2412yes—
184C24C2411yescharacteristic
185C24C2411yescharacteristic
186C24C2411yescharacteristic
187#41—416no—
188#41—416no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—24no—
192C2×Q8—42no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—44no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—416no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4C2×D414yes—
229C22×C4C2×D414yes—
230C22×C4C2×D414yes—
231C22×C4C2×D414yes—
232C22×C4C2×D414yes—
233C22×C4C2×D414yes—
234C22×C4C2414yes—
235C22×C4C2414yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2411yescharacteristic
243C22×C4C2411yescharacteristic
244C22×C4C2411yescharacteristic
245C22×C4C2411yescharacteristic
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
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—24no—
269C22⋊C4—42no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C42—24no—
280C42—24no—
281C42—44no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—24no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—48no—
310C23—48no—
311C23C22×D412yes—
312C23C22×D412yes—
313C23C22×D412yes—
314C23C2511yescharacteristic
315Q8—44no—
316Q8—44no—
317D4—44no—
318D4—48no—
319D4—48no—
320D4—48no—
321D4—48no—
322D4—48no—
323D4—48no—
324D4—44no—
325D4—88no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4C22×D412yes—
360C2×C4C22×D412yes—
361C2×C4C22×D412yes—
362C2×C4C22×D412yes—
363C2×C4H(5,𝔽2)14yes—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—416no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—44no—
391C22—42no—
392C22C23×D411yescharacteristic
393C22C2×H(5,𝔽2)12yes—
394C22#58511yescharacteristic
395C22#58512yes—
396C22#58511yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—44no—
407C4—44no—
408C4#53414yes—
409C2—42no—
410C2—42no—
411C2—42no—
412C2—82no—
413C2—82no—
414C2C2×#53411yescharacteristic
415C2#344911yescharacteristic
416C2#344911yescharacteristic
417C2#353912yes—
418C2#354012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#525C2semidirect
2. C2×#525C2semidirect
3. #2812C2semidirect
4. #2663C2semidirect
5. #2661C2semidirect
6. #2638C2semidirect
7. #2635C2semidirect
8. #2625C2semidirect
9. #2624C2semidirect
10. #2616C2semidirect
11. #2615C2semidirect
12. #2602C2semidirect
13. #2485C2semidirect
14. #2415C2semidirect
15. #2414C2semidirect
16. #2407C2semidirect
17. #2280C2semidirect
29. C2×C22⋊Q8C22semidirect
34. C2×#121C22semidirect
44. #400C22semidirect
45. #399C22semidirect
46. #398C22semidirect
50. #397C22semidirect
51. #395C22semidirect
58. #394C22semidirect
59. #393C22semidirect
63. #392C22semidirect
64. #391C22semidirect
65. #390C22semidirect
66. #390C22semidirect
67. #389C22semidirect
68. #389C22semidirect
69. #389C22semidirect
70. #388C22semidirect
71. #388C22semidirect
72. #388C22semidirect
73. #387C22semidirect
74. #386C22semidirect
75. #385C22semidirect
80. #380C22semidirect
81. #380C22semidirect
102. C23×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
#28122
#26632
#26612
#26382
#26352
#26252
#26242
#26162
#26152
#26022
#24852
#24152
#24142
#24072
#22802