Finite Group Databasebeta

← #24560

Subgroups — #24560

Order 256.

581 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic, complemented
3#2847C211yesmaximal, characteristic
4#2280C211yesmaximal, characteristic, complemented
5#2077C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1895C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1854C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#120—21no—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#482—24no—
26C2×#107C2211yescharacteristic, complemented
27C2×#107C2211yescharacteristic, complemented
28C2×#106C2211yescharacteristic, complemented
29C2×#106C2211yescharacteristic, complemented
30C2×#102C2211yescharacteristic, complemented
31C2×#102C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic, complemented
34#407—24no—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#386—21no—
40#386—21no—
41#386—21no—
42#381—21no—
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#379—21no—
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25D411yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4D411yescharacteristic
73C23×C4D411yescharacteristic
74C23×C4D411yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88#121—22no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—41no—
93#121—41no—
94#120—44no—
95#117—22no—
96#117—22no—
97#117—41no—
98#117—41no—
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
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—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C42—21no—
119C2×C42—21no—
120#107—22no—
121#107—41no—
122#106—22no—
123#106—41no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8D414yescomplemented
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24—21no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24C2×D411yescharacteristic
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
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—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D412yes—
202C22×C4C2×D411yescharacteristic
203C22×C4#4112yes—
204C22×C4#4112yes—
205C22×C4#4112yes—
206C22×C4C2411yescharacteristic
207M16—44no—
208M16—44no—
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—21no—
218C2×C8—21no—
219C2×C8C2×D412yes—
220C2×C8C2×D412yes—
221C4⋊C4—21no—
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—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C42—42no—
248C42—42no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—44no—
268C23—44no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23—44no—
273C23—44no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#12112yes—
282C23#12312yes—
283C23AGL(1,ℤ/8)12yes—
284C23C22×D411yescharacteristic
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—84no—
290D4—84no—
291D4—84no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—41no—
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—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4#12112yes—
336C2×C4#12312yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C2×#4111yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C8—22no—
344C8—22no—
345C8—41no—
346C8—41no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—416no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22C2×#12111yescharacteristic
377C22C2×#12311yescharacteristic
378C22#53411yescharacteristic
379C22C2×AGL(1,ℤ/8)11yescharacteristic
380C22#57511yescharacteristic
381C22#57511yescharacteristic
382C22#57511yescharacteristic
383C4—22no—
384C4—22no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4#54611yescharacteristic
394C4#54611yescharacteristic
395C4#55111yescharacteristic
396C4#55111yescharacteristic
397C2—42no—
398C2—42no—
399C2—42no—
400C2—161no—
401C2#241211yescharacteristic
402C2#305411yescharacteristic
403C2#305511yescharacteristic
404C2#309311yescharacteristic
405C2#309411yescharacteristic
406C2#319411yescharacteristic
407C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. C2×#407C2semidirect
4. #2280C2semidirect
5. #2077C2semidirect
6. #2075C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1895C2semidirect
10. #1895C2semidirect
11. #1875C2semidirect
12. #1854C2semidirect
13. #1825C2semidirect
14. #1809C2semidirect
15. #1809C2semidirect
26. C2×#107C22semidirect
27. C2×#107C22semidirect
28. C2×#106C22semidirect
29. C2×#106C22semidirect
30. C2×#102C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
33. C2×#102C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
131. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4072
#28472
#22802
#20772
#20752
#20342
#20332
#18952
#18752
#18542
#18252
#18092