Finite Group Databasebeta

← #22874

Subgroups — #22874

Order 256.

657 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3C22×#102C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic
6#1968C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic, complemented
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C22⋊C4—21no—
26#450C418yescomplemented
27C4×SD16—24no—
28C2×#106C2211yescharacteristic, complemented
29C2×#106C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#102—24no—
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic, complemented
41C2×C4×C8C2211yescharacteristic, complemented
42#391C2211yescharacteristic, complemented
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#389C2211yescharacteristic, complemented
47#387—21no—
48#386—21no—
49#386—21no—
50#385—21no—
51#380C2211yescharacteristic
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54C4×C22⋊C4—21no—
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×C4—21no—
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16C2×C414yescomplemented
82C2×SD16C2×C414yescomplemented
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8C2×C414yescomplemented
92C4×Q8—22no—
93C4×Q8—41no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—41no—
97C4×D4—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
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—21no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42—21no—
121C2×C42—21no—
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#106—22no—
125#106—41no—
126#103—22no—
127#103—41no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102C2×C414yescomplemented
132#102C2×C414yescomplemented
133#102D414yes—
134C22⋊C8C2×C414yescomplemented
135C4×C8—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—42no—
194C22×C4—42no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4C22×C412yes—
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4#4112yes—
205C22×C4C2411yescharacteristic
206SD16—28no—
207SD16—44no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—44no—
218C2×C8C22×C412yes—
219C2×C8C22×C412yes—
220C2×C8C2×D412yes—
221C4⋊C4—24no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C4⋊C4C22×C412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—24no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4SD1614yescomplemented
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—42no—
250C42—81no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—48no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—48no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23C4×D412yes—
277C23C2×SD1612yes—
278C23#13512yes—
279C23C22×D411yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—22no—
288D4—22no—
289D4—48no—
290D4—41no—
291D4—41no—
292D4C4×D414yes—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
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—41no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4C4×D412yes—
331C2×C4C2×SD1612yes—
332C2×C4C2×SD1612yes—
333C2×C4C23×C411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C8—22no—
340C8—22no—
341C8—22no—
342C8—41no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—816no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22C4×SD1614yescomplemented
365C22C2×C4×D411yescharacteristic
366C22D4211yescharacteristic
367C22#54611yescharacteristic
368C22C22×SD1611yescharacteristic
369C22C2×#13511yescharacteristic
370C22#57611yescharacteristic
371C22#57711yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—41no—
378C4—42no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4C2×C4×D411yescharacteristic
384C4#51811yescharacteristic
385C4#54611yescharacteristic
386C4#55111yescharacteristic
387C2—24no—
388C2—44no—
389C2—82no—
390C2#235811yescharacteristic
391C2C2×C4×SD1611yescharacteristic
392C2#294611yescharacteristic
393C2#328011yescharacteristic
394C2D4×SD1611yescharacteristic
395C2#329511yescharacteristic
396C2#329611yescharacteristic
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×C4×SD16C2semidirect
4. #2360C2semidirect
6. #1968C2semidirect
7. #1934C2semidirect
8. #1926C2semidirect
9. #1892C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
12. #1864C2semidirect
13. #1853C2semidirect
14. C4×#102C2semidirect
15. C8×C22⋊C4C2semidirect
21. C2×C4×Q8C22semidirect
26. #450C4semidirect
28. C2×#106C22semidirect
29. C2×#106C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
40. C2×C4×C8C22semidirect
41. C2×C4×C8C22semidirect
42. #391C22semidirect
46. #389C22semidirect
53. C4×C4⋊C4C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #336C22semidirect
61. #336C22semidirect
81. C2×SD16C2×C4semidirect
82. C2×SD16C2×C4semidirect
91. C22⋊Q8C2×C4semidirect
131. #102C2×C4semidirect
132. #102C2×C4semidirect
134. C22⋊C8C2×C4semidirect
244. C22⋊C4SD16semidirect
364. C22C4×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×C4×SD162
C22×#1022
#23602
D4×C22⋊C42
#19682
#19342
#19262
#18922
#18802
#18772
#18642
#18532
C4×#1022
C8×C22⋊C42