Finite Group Databasebeta

← #25958

Subgroups — #25958

Order 256.

603 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic
3#2440C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic, complemented
10#1864C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic, complemented
12#1850C211yesmaximal, characteristic, complemented
13#1807C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic, complemented
19C2×C22⋊Q8—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4—21no—
24#476—24no—
25C2×#106C2211yescharacteristic, complemented
26C2×#106C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#106C2211yescharacteristic, complemented
29C2×C4⋊C8C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#102—24no—
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#398—21no—
42#395—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic, complemented
46#389C2211yescharacteristic, complemented
47#386—21no—
48#386—21no—
49#385—21no—
50#384C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#379—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—22no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4Q814yescomplemented
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—41no—
85C22⋊Q8—44no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#106—22no—
118#106—22no—
119#106—41no—
120#106—41no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130#102D414yes—
131C22⋊C8—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24—21no—
140C24—24no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—81no—
146C24—81no—
147C24C2×D411yescharacteristic
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×D4—28no—
152C2×D4—24no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4C2×D412yes—
167C2×D4C2×Q812yes—
168C2×D4C2×Q812yes—
169C2×D4#4112yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×Q812yes—
195C22×C4#4112yes—
196C22×C4C2411yescharacteristic
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—44no—
209C2×C8C2×D412yes—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4C2×D412yes—
228C22⋊C4—22no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4SD1614yes—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—48no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—48no—
262C23—84no—
263C23—84no—
264C23—84no—
265C23—84no—
266C23—84no—
267C23C22⋊Q812yes—
268C23C2×SD1612yes—
269C23AGL(1,ℤ/8)12yes—
270C23C22×D411yescharacteristic
271Q8—84no—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—22no—
277D4—48no—
278D4—41no—
279D4—41no—
280D4C22⋊Q814yescomplemented
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4C22⋊Q812yes—
319C2×C4C2×SD1612yes—
320C2×C4C2×SD1612yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×Q811yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—816no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#47614yescomplemented
353C22C2×C22⋊Q811yescharacteristic
354C22D4211yescharacteristic
355C22#54611yescharacteristic
356C22C22×SD1611yescharacteristic
357C22C2×AGL(1,ℤ/8)11yescharacteristic
358C22#57611yescharacteristic
359C22#57711yescharacteristic
360C4—22no—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—42no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4C2×C22⋊Q811yescharacteristic
372C4#54311yescharacteristic
373C4#54611yescharacteristic
374C4#55111yescharacteristic
375C2—24no—
376C2—44no—
377C2—82no—
378C2#243711yescharacteristic
379C2C2×#47611yescharacteristic
380C2#308111yescharacteristic
381C2#326411yescharacteristic
382C2D4×SD1611yescharacteristic
383C2#329511yescharacteristic
384C2#330611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×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
1. C2×#476C2semidirect
3. #2440C2semidirect
5. #2094C2semidirect
6. #2065C2semidirect
7. #2065C2semidirect
8. #2023C2semidirect
9. #1864C2semidirect
10. #1864C2semidirect
11. #1853C2semidirect
12. #1850C2semidirect
13. #1807C2semidirect
14. #1802C2semidirect
18. C2×C4⋊Q8C22semidirect
25. C2×#106C22semidirect
26. C2×#106C22semidirect
27. C2×#106C22semidirect
28. C2×#106C22semidirect
29. C2×C4⋊C8C22semidirect
30. C2×C4⋊C8C22semidirect
45. #389C22semidirect
46. #389C22semidirect
50. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
69. C22×D4Q8semidirect
280. D4C22⋊Q8semidirect
352. C22#476semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C22×#1022
#24402
D4×C22⋊C42
#20942
#20652
#20232
#18642
#18532
#18502
#18072
#18022
#17862