Finite Group Databasebeta

← #36004

Subgroups — #36004

Order 256.

1413 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
1C2×#555C211yesmaximal, characteristic, complemented
2C2×#518C211yesmaximal, characteristic, complemented
3#2631C212yesmaximal, complemented
4#2624C211yesmaximal, characteristic, complemented
5#2598C212yesmaximal, complemented
6#2540C212yesmaximal, complemented
7#2472C214yesmaximal, complemented
8#2452C212yesmaximal, complemented
9#2433C211yesmaximal, characteristic, complemented
10#2428C212yesmaximal, complemented
11#2424C214yesmaximal, complemented
12#2408C211yesmaximal, characteristic, complemented
13#2395C211yesmaximal, characteristic, complemented
14#2392C212yesmaximal, complemented
15#2366C212yesmaximal, complemented
16#2363C212yesmaximal, complemented
17#2347C211yesmaximal, characteristic, complemented
18C23×D4C2211yescharacteristic
19#555—28no—
20C2×C4⋊Q8C2211yescharacteristic, complemented
21C2×#127C2212yes—
22C2×#124C2212yes—
23C2×#124C2212yescomplemented
24C2×C22⋊Q8C2214yescomplemented
25C2×#121C2212yes—
26C2×#121C2211yescharacteristic
27C2×#121C2212yes—
28C2×#121C2211yescharacteristic, complemented
29C2×#121C2214yescomplemented
30C2×#120C2212yes—
31C2×#120C2214yes—
32#518—28no—
33C2×C4×D4C2212yes—
34C2×C4×D4C2212yes—
35C2×C4×D4C2212yes—
36C2×C4×D4C2211yescharacteristic
37C2×C4×D4C2212yes—
38C2×C4×D4C2211yescharacteristic, complemented
39C2×C4×D4C2214yescomplemented
40C2×#117C2211yescharacteristic
41C2×#117C2211yescharacteristic
42C2×#117C2212yescomplemented
43C2×#117C2211yescharacteristic, complemented
44C22×C22⋊C4C2214yes—
45#400C2211yescharacteristic, complemented
46#399C2212yescomplemented
47#399C2211yescharacteristic, complemented
48#398C2211yescharacteristic, complemented
49#397C2212yes—
50#397C2212yescomplemented
51#397C2211yescharacteristic, complemented
52#396C2214yescomplemented
53#394C2212yes—
54#394C2214yes—
55#394C2212yes—
56#394C2214yescomplemented
57#393C2214yescomplemented
58#392C2212yes—
59#392C2212yes—
60#392C2212yes—
61#392C2214yescomplemented
62#390C2212yes—
63#390C2212yes—
64#390C2214yescomplemented
65#390C2212yescomplemented
66#390C2214yescomplemented
67#389C2212yescomplemented
68#389C2212yescomplemented
69#388C2214yes—
70#388C2214yescomplemented
71#386C2214yescomplemented
72#386C2214yescomplemented
73#386C2214yescomplemented
74#386C2214yescomplemented
75#385C2214yescomplemented
76#383C2212yescomplemented
77#380C2211yescharacteristic
78#379C2212yes—
79C4×C4⋊C4C2212yescomplemented
80C4×C4⋊C4C2212yescomplemented
81C4×C4⋊C4C2212yescomplemented
82C4×C22⋊C4C2211yescharacteristic
83C4×C22⋊C4C2214yescomplemented
84C4×C22⋊C4C2212yescomplemented
85C25C2312yes—
86C22×Q8C2312yes—
87C22×D4—28no—
88C22×D4—24no—
89C22×D4C2312yes—
90C22×D4C2311yescharacteristic
91C22×D4C2311yescharacteristic
92C22×D4C2314yes—
93C22×D4C2312yes—
94C22×D4C2312yes—
95C22×D4C2314yes—
96C22×D4C2314yes—
97C22×D4C2312yes—
98C23×C4C2311yescharacteristic
99C23×C4C2312yes—
100C23×C4C2311yescharacteristic
101C23×C4C2314yes—
102C23×C4C2314yes—
103C4⋊Q8—24no—
104#127—28no—
105#124—28no—
106#124—28no—
107C22⋊Q8—48no—
108#121—28no—
109#121—24no—
110#121—24no—
111#121—28no—
112#121—48no—
113#120—44no—
114#120—48no—
115C4×D4—24no—
116C4×D4—28no—
117C4×D4—28no—
118C4×D4—24no—
119C4×D4—28no—
120C4×D4—28no—
121C4×D4—48no—
122#117—24no—
123#117—24no—
124#117—28no—
125#117D418yescomplemented
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2314yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2314yes—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C22⋊C4—216no—
139C2×C22⋊C4C2312yes—
140C2×C22⋊C4C2314yes—
141C2×C22⋊C4C2314yes—
142C2×C22⋊C4C2312yes—
143C2×C22⋊C4C2312yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2311yescharacteristic
146C2×C22⋊C4C2311yescharacteristic
147C2×C22⋊C4C2314yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2314yes—
151C2×C22⋊C4C2314yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2314yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2314yes—
156C2×C22⋊C4C2314yes—
157C2×C22⋊C4C2314yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C42C2312yes—
161C2×C42C2312yes—
162C2×C42C2311yescharacteristic
163C2×C42C2312yes—
164C2×C42C2314yes—
165C2×C42C2312yes—
166#95C2311yescharacteristic
167#95C2311yescharacteristic
168#95C2312yes—
169#95C2312yes—
170#95C2314yes—
171#95C2312yes—
172#95C2314yes—
173#95C2312yes—
174#95C2312yes—
175#95C2314yes—
176C24—24no—
177C24—28no—
178C24—44no—
179C24—42no—
180C24—42no—
181C24C2411yescharacteristic
182C24C2412yes—
183C24C2411yescharacteristic
184C24C2414yes—
185C2×Q8—24no—
186C2×Q8—44no—
187C2×D4—232no—
188C2×D4—28no—
189C2×D4—22no—
190C2×D4—28no—
191C2×D4—24no—
192C2×D4—28no—
193C2×D4—28no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—28no—
202C2×D4—44no—
203C2×D4—41no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4#4118yes—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4C2×D414yes—
224C22×C4C2412yes—
225C22×C4C2411yescharacteristic
226C22×C4C2414yes—
227C22×C4C2412yes—
228C22×C4C2411yescharacteristic
229C22×C4C2412yes—
230C22×C4C2412yes—
231C22×C4C2411yescharacteristic
232C22×C4C2412yes—
233C22×C4C2412yes—
234C22×C4C2412yes—
235C22×C4C2411yescharacteristic
236C22×C4C2411yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—44no—
247C4⋊C4C2×D414yes—
248C4⋊C4C2×D414yes—
249C22⋊C4—24no—
250C22⋊C4—22no—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—28no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—28no—
260C22⋊C4—24no—
261C22⋊C4—28no—
262C22⋊C4—24no—
263C22⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4C2×D418yes—
270C42—24no—
271C42—24no—
272C42—22no—
273C42—24no—
274C42—44no—
275C42C2×D418yes—
276C23—216no—
277C23—24no—
278C23—22no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—21no—
284C23—24no—
285C23—24no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—24no—
291C23—22no—
292C23—22no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—84no—
298C23—84no—
299C23—88no—
300C23—88no—
301C23C22×D412yes—
302C23C2×#4118yes—
303C23C2511yescharacteristic
304Q8—84no—
305D4—216no—
306D4—216no—
307D4—44no—
308D4—416no—
309D4—44no—
310D4—84no—
311D4—84no—
312D4—88no—
313D4—88no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—28no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—88no—
344C2×C4—88no—
345C2×C4C22×D414yes—
346C2×C4C22×D412yes—
347C2×C4C22×D414yes—
348C2×C4C22×D412yes—
349C2×C4C2×#4114yes—
350C2×C4H(5,𝔽2)14yes—
351C2×C4H(5,𝔽2)12yes—
352C22—22no—
353C22—28no—
354C22—22no—
355C22—22no—
356C22—28no—
357C22—28no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—44no—
363C22—42no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—44no—
371C22—42no—
372C22—44no—
373C22—42no—
374C22—816no—
375C22C23×D411yescharacteristic
376C22C22×#4112yes—
377C22C2×H(5,𝔽2)11yescharacteristic
378C22C2×H(5,𝔽2)12yes—
379C22#58511yescharacteristic
380C4—22no—
381C4—22no—
382C4—24no—
383C4—24no—
384C4—24no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—41no—
390C4—42no—
391C4—44no—
392C4#52514yes—
393C2—48no—
394C2—42no—
395C2—81no—
396C2—82no—
397C2C2×#52511yescharacteristic
398C2#348811yescharacteristic
399C2#349811yescharacteristic
400C2#351412yes—
401C2#351912yes—
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×#555C2semidirect
2. C2×#518C2semidirect
3. #2631C2semidirect
4. #2624C2semidirect
5. #2598C2semidirect
6. #2540C2semidirect
7. #2472C2semidirect
8. #2452C2semidirect
9. #2433C2semidirect
10. #2428C2semidirect
11. #2424C2semidirect
12. #2408C2semidirect
13. #2395C2semidirect
14. #2392C2semidirect
15. #2366C2semidirect
16. #2363C2semidirect
17. #2347C2semidirect
20. C2×C4⋊Q8C22semidirect
23. C2×#124C22semidirect
24. C2×C22⋊Q8C22semidirect
28. C2×#121C22semidirect
29. C2×#121C22semidirect
38. C2×C4×D4C22semidirect
39. C2×C4×D4C22semidirect
42. C2×#117C22semidirect
43. C2×#117C22semidirect
45. #400C22semidirect
46. #399C22semidirect
47. #399C22semidirect
48. #398C22semidirect
50. #397C22semidirect
51. #397C22semidirect
52. #396C22semidirect
56. #394C22semidirect
57. #393C22semidirect
61. #392C22semidirect
64. #390C22semidirect
65. #390C22semidirect
66. #390C22semidirect
67. #389C22semidirect
68. #389C22semidirect
70. #388C22semidirect
71. #386C22semidirect
72. #386C22semidirect
73. #386C22semidirect
74. #386C22semidirect
75. #385C22semidirect
76. #383C22semidirect
79. C4×C4⋊C4C22semidirect
80. C4×C4⋊C4C22semidirect
81. C4×C4⋊C4C22semidirect
83. C4×C22⋊C4C22semidirect
84. C4×C22⋊C4C22semidirect
125. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#5182
#26312
#26242
#25982
#25402
#24722
#24522
#24332
#24282
#24242
#24082
#23952
#23922
#23662
#23632
#23472