Finite Group Databasebeta

← #22864

Subgroups — #22864

Order 256.

637 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic
4#2362C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic
6#1968C211yesmaximal, characteristic, complemented
7#1936C211yesmaximal, characteristic, complemented
8#1925C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic, complemented
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C22⋊C4—21no—
26#450C418yescomplemented
27#440—24no—
28C2×#107C2211yescharacteristic, complemented
29C2×#106C2211yescharacteristic, complemented
30#418—24no—
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic, complemented
41C2×C4×C8C2211yescharacteristic, complemented
42#391C2211yescharacteristic, complemented
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic, complemented
46#388—21no—
47#388—21no—
48#387—21no—
49#386—21no—
50#386—21no—
51#380C2211yescharacteristic
52#379—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4C2211yescharacteristic, complemented
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62C25—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2×C414yes—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16C2×C414yes—
81C2×SD16C2×C414yescomplemented
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22⋊Q8—22no—
90C22⋊Q8C2×C414yescomplemented
91C4×Q8—22no—
92C4×Q8—41no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—41no—
96C4×D4—41no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—44no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#106—41no—
126#103—22no—
127#103—41no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102C2×C414yes—
132#102C2×C414yescomplemented
133#102D414yes—
134C22⋊C8C2×C414yescomplemented
135#97—22no—
136C4×C8—41no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—24no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—81no—
152C24—81no—
153C24C2×D411yescharacteristic
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8C22×C412yes—
160C2×D4—28no—
161C2×D4—24no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4C22×C412yes—
176C2×D4C22×C412yes—
177C2×D4C2×D412yes—
178C2×D4#4112yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4C22×C412yes—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4#4112yes—
202C22×C4C2411yescharacteristic
203SD16—28no—
204SD16—44no—
205M16—44no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8C22×C412yes—
216C2×C8C22×C412yes—
217C2×C8C2×D412yes—
218C4⋊C4—24no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4C22×C412yes—
230C4⋊C4C2×D412yes—
231C22⋊C4—24no—
232C22⋊C4—22no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—42no—
247C42—42no—
248C42—42no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—48no—
260C23—48no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23C4×D412yes—
277C23C22×D411yescharacteristic
278Q8—22no—
279Q8—22no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—48no—
288D4—41no—
289D4—41no—
290D4C4×D414yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4C4×D412yes—
329C2×C4AGL(1,ℤ/8)12yes—
330C2×C4C23×C411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C8—22no—
337C8—22no—
338C8—22no—
339C8—41no—
340C22—22no—
341C22—24no—
342C22—22no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—816no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C2×C4×D411yescharacteristic
363C22D4211yescharacteristic
364C22#54611yescharacteristic
365C22C2×AGL(1,ℤ/8)11yescharacteristic
366C22#57511yescharacteristic
367C22#57711yescharacteristic
368C22#57711yescharacteristic
369C4—22no—
370C4—22no—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4C2×C4×D411yescharacteristic
381C4#51811yescharacteristic
382C4#54611yescharacteristic
383C4#55111yescharacteristic
384C2—42no—
385C2—44no—
386C2—82no—
387C2#235811yescharacteristic
388C2C4×AGL(1,ℤ/8)11yescharacteristic
389C2#294911yescharacteristic
390C2#327211yescharacteristic
391C2#327511yescharacteristic
392C2#331011yescharacteristic
393C2#331111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#440C2semidirect
4. #2362C2semidirect
6. #1968C2semidirect
7. #1936C2semidirect
8. #1925C2semidirect
9. #1894C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
12. #1864C2semidirect
13. #1854C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
21. C2×C4×Q8C22semidirect
26. #450C4semidirect
28. C2×#107C22semidirect
29. C2×#106C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
40. C2×#97C22semidirect
41. C2×C4×C8C22semidirect
42. #391C22semidirect
45. #389C22semidirect
54. C4×C4⋊C4C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #336C22semidirect
61. #336C22semidirect
81. C2×SD16C2×C4semidirect
90. C22⋊Q8C2×C4semidirect
132. #102C2×C4semidirect
134. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×#4402
C2×#4182
#23622
D4×C22⋊C42
#19682
#19362
#19252
#18942
#18802
#18772
#18642
#18542
C4×#1022
#17532