Finite Group Databasebeta

← #29896

Subgroups — #29896

Order 256.

528 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2847C211yesmaximal, characteristic
2#2280C211yesmaximal, characteristic, complemented
3#2077C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1779C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#120—21no—
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#107C2211yescharacteristic, complemented
24C2×#107C2211yescharacteristic, complemented
25C2×#106C2211yescharacteristic, complemented
26C2×#106C2211yescharacteristic, complemented
27C2×#102C2211yescharacteristic, complemented
28C2×#102C2211yescharacteristic, complemented
29C2×#102C2211yescharacteristic, complemented
30C2×#102C2211yescharacteristic, complemented
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#386—21no—
38#386—21no—
39#386—21no—
40#381—21no—
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#379—21no—
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51#340C2211yescharacteristic, complemented
52#340C2211yescharacteristic, complemented
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59C25D411yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—41no—
70C23×C4D411yescharacteristic
71C23×C4D411yescharacteristic
72C23×C4D411yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#121—22no—
81#121—22no—
82#121—41no—
83#121—41no—
84#121—41no—
85#121—41no—
86#120—44no—
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
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×C42—21no—
111C2×C42—21no—
112#107—41no—
113#107—41no—
114#106—41no—
115#106—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24—21no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
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—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4#4112yes—
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
201C2×C8—21no—
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—
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—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—42no—
240C42—42no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23—44no—
265C23—44no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#12312yes—
274C23#13512yes—
275C23AGL(1,ℤ/8)12yes—
276C23C22×D411yescharacteristic
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—84no—
282D4—84no—
283D4—84no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
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—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#12312yes—
329C2×C4C2×#4111yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C8—41no—
336C8—41no—
337C8—41no—
338C8—41no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—416no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#12311yescharacteristic
369C22#53411yescharacteristic
370C22#53411yescharacteristic
371C22C2×#13511yescharacteristic
372C22C2×AGL(1,ℤ/8)11yescharacteristic
373C22#57511yescharacteristic
374C22#57511yescharacteristic
375C4—22no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4#53811yescharacteristic
386C4#53811yescharacteristic
387C4#53811yescharacteristic
388C4#53811yescharacteristic
389C2—42no—
390C2—42no—
391C2—42no—
392C2—161no—
393C2#262211yescharacteristic
394C2#309111yescharacteristic
395C2#309411yescharacteristic
396C2#318811yescharacteristic
397C2#319111yescharacteristic
398C2#319411yescharacteristic
399C2#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
2. #2280C2semidirect
3. #2077C2semidirect
4. #2077C2semidirect
5. #2076C2semidirect
6. #2075C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #2033C2semidirect
10. #2033C2semidirect
11. #1875C2semidirect
12. #1875C2semidirect
13. #1854C2semidirect
14. #1854C2semidirect
15. #1779C2semidirect
23. C2×#107C22semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
26. C2×#106C22semidirect
27. C2×#102C22semidirect
28. C2×#102C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
51. #340C22semidirect
52. #340C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect

Maximal subgroups

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

subgroupindex
#28472
#22802
#20772
#20762
#20752
#20342
#20332
#18752
#18542
#17792