Finite Group Databasebeta

← #37443

Subgroups — #37443

Order 256.

1299 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic, complemented
2C22×#123C211yesmaximal, characteristic, complemented
3#2814C212yesmaximal, complemented
4#2813C212yesmaximal, complemented
5#2663C212yesmaximal, complemented
6#2661C212yesmaximal, complemented
7#2630C214yesmaximal, complemented
8#2622C212yesmaximal, complemented
9#2613C212yesmaximal, complemented
10#2602C214yesmaximal, complemented
11#2484C211yesmaximal, characteristic, complemented
12#2482C211yesmaximal, characteristic, complemented
13#2413C212yesmaximal, complemented
14#2411C212yesmaximal, complemented
15#2406C211yesmaximal, characteristic, complemented
16#2404C211yesmaximal, characteristic, complemented
17#2280C211yesmaximal, characteristic, complemented
18C22×#41C2211yescharacteristic
19C23×D4C2211yescharacteristic
20C24×C4C2211yescharacteristic
21#525—28no—
22C2×#123—28no—
23C2×#123C2212yes—
24C2×#123C2212yes—
25C2×#123C2212yes—
26C2×#123C2212yes—
27C2×#123C2212yes—
28C2×#123C2212yes—
29C2×C22⋊Q8C2211yescharacteristic, complemented
30C2×C22⋊Q8C2211yescharacteristic, complemented
31C2×#121C2211yescharacteristic, complemented
32C2×#121C2211yescharacteristic, complemented
33C2×#120C2212yes—
34C2×#120C2212yes—
35C2×C4×D4C2212yescomplemented
36C2×C4×D4C2212yescomplemented
37C2×#117C2211yescharacteristic
38C2×#117C2212yes—
39C22×C4⋊C4C2211yescharacteristic
40C22×C4⋊C4C2211yescharacteristic
41C22×C22⋊C4C2211yescharacteristic
42C22×C22⋊C4C2211yescharacteristic
43C22×C22⋊C4C2211yescharacteristic
44#400C2212yescomplemented
45#399C2212yescomplemented
46#398C2214yescomplemented
47#398C2212yescomplemented
48#397C2214yescomplemented
49#397C2214yescomplemented
50#397C2212yescomplemented
51#396C2214yescomplemented
52#396C2214yescomplemented
53#395C2212yescomplemented
54#394C2214yes—
55#394C2214yes—
56#394C2214yescomplemented
57#394C2214yescomplemented
58#394C2214yescomplemented
59#394C2212yescomplemented
60#393C2212yescomplemented
61#392C2214yes—
62#392C2214yes—
63#392C2212yescomplemented
64#390C2214yescomplemented
65#389C2214yescomplemented
66#388C2214yescomplemented
67#388C2214yescomplemented
68#388C2214yescomplemented
69#387C2214yescomplemented
70#386C2212yes—
71#386C2212yes—
72#386C2214yescomplemented
73#385C2212yes—
74#385C2212yes—
75#385C2214yescomplemented
76#381C2212yes—
77#381C2211yescharacteristic
78#381C2211yescharacteristic
79#380C2212yescomplemented
80#380C2212yescomplemented
81#379C2212yes—
82#379C2212yes—
83C4×C22⋊C4C2212yescomplemented
84C4×C22⋊C4C2212yescomplemented
85C25C2311yescharacteristic
86C25C2311yescharacteristic
87C2×#41—24no—
88C2×#41—44no—
89C22×Q8C2311yescharacteristic
90C22×D4—24no—
91C22×D4—44no—
92C22×D4C2312yes—
93C22×D4C2312yes—
94C22×D4C2312yes—
95C22×D4C2311yescharacteristic
96C22×D4C2312yes—
97C22×D4C2312yes—
98C22×D4C2314yes—
99C23×C4—44no—
100C23×C4D418yescomplemented
101C23×C4C2311yescharacteristic
102C23×C4C2311yescharacteristic
103C23×C4C2311yescharacteristic
104C23×C4C2311yescharacteristic
105C23×C4C2311yescharacteristic
106C23×C4C2312yes—
107C23×C4C2311yescharacteristic
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110C23×C4C2311yescharacteristic
111#123—28no—
112#123—28no—
113#123—28no—
114#123—28no—
115#123—28no—
116#123—44no—
117C22⋊Q8—24no—
118C22⋊Q8—24no—
119#121—24no—
120#121—24no—
121#120—28no—
122#120—44no—
123C4×D4—28no—
124C4×D4—28no—
125#117—24no—
126#117—44no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2314yes—
138C2×C4⋊C4C2314yes—
139C2×C4⋊C4C2314yes—
140C2×C4⋊C4C2314yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
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⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2314yes—
158C2×C22⋊C4C2314yes—
159C2×C22⋊C4C2314yes—
160C2×C22⋊C4C2314yes—
161C2×C22⋊C4C2314yes—
162C2×C22⋊C4C2314yes—
163C2×C22⋊C4C2314yes—
164C2×C22⋊C4C2314yes—
165C2×C42C2312yes—
166C2×C42C2314yes—
167#95C2312yes—
168#95C2312yes—
169#95C2312yes—
170#95C2312yes—
171#95C2314yes—
172#95C2314yes—
173#95C2314yes—
174#95C2314yes—
175#95C2314yes—
176#95C2314yes—
177C24—24no—
178C24—22no—
179C24—24no—
180C24—42no—
181C24—42no—
182C24—42no—
183C24—42no—
184C24C2×D414yes—
185C24C2412yes—
186C24C2412yes—
187C24C2411yescharacteristic
188C24C2411yescharacteristic
189C24C2411yescharacteristic
190#41—416no—
191C2×Q8—22no—
192C2×Q8—42no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—416no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—416no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C2×D414yes—
227C22×C4C2×D414yes—
228C22×C4C2×D414yes—
229C22×C4C2×D414yes—
230C22×C4C2×D414yes—
231C22×C4C2×D414yes—
232C22×C4C2414yes—
233C22×C4C2414yes—
234C22×C4C2412yes—
235C22×C4C2412yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2411yescharacteristic
241C22×C4C2411yescharacteristic
242C22×C4C2411yescharacteristic
243C22×C4C2411yescharacteristic
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
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—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C42—24no—
278C42—44no—
279C23—22no—
280C23—22no—
281C23—28no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—22no—
294C23—24no—
295C23—24no—
296C23—24no—
297C23—22no—
298C23—44no—
299C23—44no—
300C23—416no—
301C23—44no—
302C23—44no—
303C23—44no—
304C23—88no—
305C23C22×D412yes—
306C23C22×D412yes—
307C23C22×D412yes—
308C23H(5,𝔽2)14yes—
309C23C2511yescharacteristic
310Q8—44no—
311D4—44no—
312D4—48no—
313D4—48no—
314D4—48no—
315D4—48no—
316D4—48no—
317D4—88no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
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—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4C22×D412yes—
353C2×C4C22×D412yes—
354C2×C4C22×D412yes—
355C2×C4C22×D412yes—
356C22—22no—
357C22—22no—
358C22—28no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—416no—
366C22—44no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—44no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22—88no—
379C22#53414yes—
380C22C23×D411yescharacteristic
381C22C2×H(5,𝔽2)12yes—
382C22#58512yes—
383C22#58512yes—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—44no—
395C4—44no—
396C2—24no—
397C2—42no—
398C2—42no—
399C2—82no—
400C2—82no—
401C2C2×#53411yescharacteristic
402C2#344911yescharacteristic
403C2#344911yescharacteristic
404C2#353314yes—
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. C22×#123C2semidirect
3. #2814C2semidirect
4. #2813C2semidirect
5. #2663C2semidirect
6. #2661C2semidirect
7. #2630C2semidirect
8. #2622C2semidirect
9. #2613C2semidirect
10. #2602C2semidirect
11. #2484C2semidirect
12. #2482C2semidirect
13. #2413C2semidirect
14. #2411C2semidirect
15. #2406C2semidirect
16. #2404C2semidirect
17. #2280C2semidirect
29. C2×C22⋊Q8C22semidirect
30. C2×C22⋊Q8C22semidirect
31. C2×#121C22semidirect
32. C2×#121C22semidirect
35. C2×C4×D4C22semidirect
36. C2×C4×D4C22semidirect
44. #400C22semidirect
45. #399C22semidirect
46. #398C22semidirect
47. #398C22semidirect
48. #397C22semidirect
49. #397C22semidirect
50. #397C22semidirect
51. #396C22semidirect
52. #396C22semidirect
53. #395C22semidirect
56. #394C22semidirect
57. #394C22semidirect
58. #394C22semidirect
59. #394C22semidirect
60. #393C22semidirect
63. #392C22semidirect
64. #390C22semidirect
65. #389C22semidirect
66. #388C22semidirect
67. #388C22semidirect
68. #388C22semidirect
69. #387C22semidirect
72. #386C22semidirect
75. #385C22semidirect
79. #380C22semidirect
80. #380C22semidirect
83. C4×C22⋊C4C22semidirect
84. C4×C22⋊C4C22semidirect
100. C23×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C22×#1232
#28142
#28132
#26632
#26612
#26302
#26222
#26132
#26022
#24842
#24822
#24132
#24112
#24062
#24042
#22802