Finite Group Databasebeta

← #24890

Subgroups — #24890

Order 256.

577 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic, complemented
3#2408C211yesmaximal, characteristic
4#2354C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#120—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic, complemented
24#482—24no—
25C2×#107C2211yescharacteristic, complemented
26C2×#107C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#106C2211yescharacteristic, complemented
29#418—24no—
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic, complemented
41#394—21no—
42#392—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#388—21no—
47#388—21no—
48#384C2211yescharacteristic, complemented
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic, complemented
52#379—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4D411yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#121—22no—
84#121—22no—
85#121—41no—
86#121—41no—
87#120—44no—
88C4×D4—22no—
89C4×D4—22no—
90#117—22no—
91#117—22no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
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—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120#107—22no—
121#107—41no—
122#106—22no—
123#106—41no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—41no—
130#102—41no—
131#102D414yescomplemented
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—22no—
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—21no—
162C2×D4—21no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4C2×D412yes—
177C2×D4#4112yes—
178C22×C4—21no—
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—21no—
193C22×C4—21no—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4#4112yes—
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
201M16—44no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8C2×D412yes—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4C2×D412yes—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—42no—
246C42—42no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—48no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23—84no—
280C23—84no—
281C23#12312yes—
282C23C22×D411yescharacteristic
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—44no—
288D4—48no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—84no—
294D4—84no—
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—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
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#12312yes—
332C2×C4AGL(1,ℤ/8)12yes—
333C2×C4C22×D411yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C8—22no—
340C8—41no—
341C8—41no—
342C8—41no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—42no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—816no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C2×#12311yescharacteristic
373C22D4211yescharacteristic
374C22#54611yescharacteristic
375C22C2×AGL(1,ℤ/8)11yescharacteristic
376C22#57511yescharacteristic
377C22#57611yescharacteristic
378C22#57711yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4#52511yescharacteristic
390C4#53811yescharacteristic
391C4#54611yescharacteristic
392C4#55111yescharacteristic
393C2—42no—
394C2—81no—
395C2—82no—
396C2—82no—
397C2#241811yescharacteristic
398C2#309411yescharacteristic
399C2#309611yescharacteristic
400C2#326711yescharacteristic
401C2#327411yescharacteristic
402C2#331911yescharacteristic
403C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ 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×#418C2semidirect
4. #2354C2semidirect
5. #2093C2semidirect
6. #2071C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1926C2semidirect
10. #1926C2semidirect
11. #1875C2semidirect
12. #1870C2semidirect
13. #1854C2semidirect
14. #1809C2semidirect
15. #1786C2semidirect
23. C2×#117C22semidirect
25. C2×#107C22semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
28. C2×#106C22semidirect
39. C2×C22⋊C8C22semidirect
40. C2×C22⋊C8C22semidirect
48. #384C22semidirect
51. #380C22semidirect
54. C4×C4⋊C4C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
131. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4182
#24082
#23542
#20932
#20712
#20342
#20332
#19262
#18752
#18702
#18542
#18092
#17862