Finite Group Databasebeta

← #25736

Subgroups — #25736

Order 256.

850 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#535C211yesmaximal, characteristic, complemented
2C2×#496C211yesmaximal, characteristic, complemented
3C2×#495C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#449C211yesmaximal, characteristic, complemented
6C2×#450C211yesmaximal, characteristic, complemented
7C2×#447C211yesmaximal, characteristic, complemented
8#2620C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#2012C211yesmaximal, characteristic, complemented
12#2011C211yesmaximal, characteristic, complemented
13#1959C211yesmaximal, characteristic, complemented
14#1957C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22C22×SD16C2211yescharacteristic, complemented
23C22×D8C2211yescharacteristic
24C22×D8C2211yescharacteristic
25C22×D8C2211yescharacteristic, complemented
26#535—24no—
27#535—24no—
28C2×#127C2211yescharacteristic
29C2×#127C2211yescharacteristic
30C2×#124C2211yescharacteristic
31C2×#124C2211yescharacteristic
32C2×#124C2211yescharacteristic, complemented
33C2×C22⋊Q8C2211yescharacteristic
34C2×#121—22no—
35C2×#121C2211yescharacteristic
36C2×#120—22no—
37C2×#117C2211yescharacteristic
38#496—24no—
39#495—24no—
40#452—24no—
41#450—24no—
42#449—24no—
43#447—24no—
44C2×#103C2211yescharacteristic, complemented
45C2×#103C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic, complemented
51C2×#102C2211yescharacteristic, complemented
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×#97C2211yescharacteristic, complemented
55C2×C4×C8C2211yescharacteristic, complemented
56#396—22no—
57#394—22no—
58#391C2211yescharacteristic, complemented
59#390C2211yescharacteristic, complemented
60#384C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—41no—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—22no—
100C2×SD16—41no—
101C2×SD16—41no—
102C2×SD16—41no—
103C2×SD16—41no—
104C2×D8—22no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—41no—
111C2×D8—41no—
112C2×D8—41no—
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C22×C8C2311yescharacteristic
116C22×C8C2311yescharacteristic
117#127—22no—
118#127—22no—
119#127—22no—
120#127D414yescomplemented
121#124—22no—
122#124—22no—
123#124—22no—
124#124—22no—
125#124—22no—
126#124D414yescomplemented
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129#121—216no—
130#121—22no—
131#121—22no—
132#120—216no—
133#117—22no—
134#117—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4D412yes—
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4D412yes—
147C2×C42C2311yescharacteristic
148C2×C42C2311yescharacteristic
149#103—22no—
150#103—22no—
151#102—22no—
152#102—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157C22⋊C8D414yescomplemented
158C22⋊C8D414yescomplemented
159#97—22no—
160C4×C8—22no—
161#95—22no—
162C24—24no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—42no—
168C24—42no—
169C24—82no—
170C24—82no—
171C24C2×D411yescharacteristic
172#41—44no—
173#41—44no—
174#41—44no—
175#41—44no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8C2×D412yes—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—28no—
191C2×D4—28no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C2×D4—82no—
226C2×D4—82no—
227C2×D4C2×D412yes—
228C2×D4C2×D412yes—
229C2×D4C2×D412yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C2×D412yes—
242C22×C4C2×D412yes—
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D412yes—
245C22×C4C2×D412yes—
246C22×C4C2411yescharacteristic
247Q16—44no—
248SD16—44no—
249SD16—44no—
250SD16—44no—
251SD16—44no—
252D8—44no—
253D8—44no—
254D8—44no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8C2×D412yes—
264C2×C8C2×D412yes—
265C2×C8C2×D412yes—
266C2×C8C2×D412yes—
267C4⋊C4—24no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C22⋊C4—28no—
275C22⋊C4—28no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—82no—
281C42—21no—
282C42—21no—
283C42C2×D412yes—
284C42C2×D412yes—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—48no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23—84no—
311C23—84no—
312C23—88no—
313C23—88no—
314C23#12712yes—
315C23C22×D411yescharacteristic
316Q8—22no—
317Q8—22no—
318Q8—41no—
319Q8—41no—
320Q8—41no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324D4—22no—
325D4—22no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4—22no—
330D4—48no—
331D4—48no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—88no—
343D4—88no—
344D4—84no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—48no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4#12014yes—
368C2×C4#12712yes—
369C2×C4C22×D411yescharacteristic
370C2×C4C22×D411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4H(5,𝔽2)11yescharacteristic
374C2×C4H(5,𝔽2)11yescharacteristic
375C8—22no—
376C8—22no—
377C8—22no—
378C8—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—816no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—82no—
404C22—82no—
405C22C2×#12011yescharacteristic
406C22C2×#12711yescharacteristic
407C22#53511yescharacteristic
408C22#57612yes—
409C22#57712yes—
410C4—22no—
411C4—24no—
412C4—42no—
413C4—42no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4D4211yescharacteristic
418C4D4211yescharacteristic
419C4D4211yescharacteristic
420C4D4211yescharacteristic
421C2—42no—
422C2—82no—
423C2—82no—
424C2—82no—
425C2—161no—
426C2#243311yescharacteristic
427C2#301212yes—
428C2#315411yescharacteristic
429C2#315511yescharacteristic
430C2#322412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#535C2semidirect
2. C2×#496C2semidirect
3. C2×#495C2semidirect
4. C2×#452C2semidirect
5. C2×#449C2semidirect
6. C2×#450C2semidirect
7. C2×#447C2semidirect
8. #2620C2semidirect
9. #2015C2semidirect
10. #2014C2semidirect
11. #2012C2semidirect
12. #2011C2semidirect
13. #1959C2semidirect
14. #1957C2semidirect
15. #1750C2semidirect
18. C22×Q16C22semidirect
21. C22×SD16C22semidirect
22. C22×SD16C22semidirect
25. C22×D8C22semidirect
32. C2×#124C22semidirect
44. C2×#103C22semidirect
45. C2×#103C22semidirect
50. C2×#102C22semidirect
51. C2×#102C22semidirect
54. C2×#97C22semidirect
55. C2×C4×C8C22semidirect
58. #391C22semidirect
59. #390C22semidirect
60. #384C22semidirect
61. #336C22semidirect
62. #336C22semidirect
120. #127D4semidirect
126. #124D4semidirect
157. C22⋊C8D4semidirect
158. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5352
C2×#4962
C2×#4952
C2×#4522
C2×#4492
C2×#4502
C2×#4472
#26202
#20152
#20142
#20122
#20112
#19592
#19572
#17502