Finite Group Databasebeta

← #31676

Subgroups — #31676

Order 256.

516 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2485C211yesmaximal, characteristic, complemented
2#2408C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2075C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#120—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic, complemented
22C2×#107C2211yescharacteristic, complemented
23C2×#106C2211yescharacteristic, complemented
24C2×C4⋊C8C2211yescharacteristic, complemented
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic, complemented
36C2×C22⋊C8C2211yescharacteristic, complemented
37C2×#97C2211yescharacteristic, complemented
38#397—21no—
39#394—21no—
40#394—21no—
41#392—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic, complemented
45#389C2211yescharacteristic, complemented
46#388—21no—
47#387—21no—
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic, complemented
52#379—21no—
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59C25—21no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4D411yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#121—22no—
79#121—22no—
80#121—41no—
81#121—41no—
82#120—44no—
83C4×D4—41no—
84C4×D4—41no—
85#117—41no—
86#117—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#107—41no—
112#106—41no—
113C4⋊C8—41no—
114#102—22no—
115#102—22no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120#102—41no—
121#102—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126#97—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24—21no—
135C24—42no—
136C24—42no—
137C24—42no—
138C24—42no—
139C24—42no—
140C24—81no—
141C24—81no—
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4#4112yes—
171C2×D4#4112yes—
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—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2411yescharacteristic
195C2×C8—21no—
196C2×C8—21no—
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—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
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—41no—
220C4⋊C4—42no—
221C4⋊C4—81no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C42—21no—
236C42—21no—
237C42—42no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—48no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—84no—
267C23—84no—
268C23—84no—
269C23—84no—
270C23—84no—
271C23—84no—
272C23AGL(1,ℤ/8)12yes—
273C23C22×D411yescharacteristic
274D4—22no—
275D4—22no—
276D4—22no—
277D4—22no—
278D4—44no—
279D4—48no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—84no—
285D4—84no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
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—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×C4C2×#4111yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C8—41no—
331C8—41no—
332C8—41no—
333C8—41no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—816no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#53411yescharacteristic
364C22#54611yescharacteristic
365C22#54611yescharacteristic
366C22C2×AGL(1,ℤ/8)11yescharacteristic
367C22#57511yescharacteristic
368C22#57611yescharacteristic
369C22#57711yescharacteristic
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4#53811yescharacteristic
381C4#53811yescharacteristic
382C4#55111yescharacteristic
383C4#56011yescharacteristic
384C2—42no—
385C2—81no—
386C2—82no—
387C2—82no—
388C2#269711yescharacteristic
389C2#319411yescharacteristic
390C2#319811yescharacteristic
391C2#331011yescharacteristic
392C2#331111yescharacteristic
393C2#331811yescharacteristic
394C2#331911yescharacteristic
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. #2485C2semidirect
3. #2077C2semidirect
4. #2075C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1955C2semidirect
10. #1936C2semidirect
11. #1936C2semidirect
12. #1875C2semidirect
13. #1875C2semidirect
14. #1786C2semidirect
15. #1786C2semidirect
21. C2×#117C22semidirect
22. C2×#107C22semidirect
23. C2×#106C22semidirect
24. C2×C4⋊C8C22semidirect
35. C2×C22⋊C8C22semidirect
36. C2×C22⋊C8C22semidirect
37. C2×#97C22semidirect
44. #389C22semidirect
45. #389C22semidirect
51. #380C22semidirect
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
#24852
#24082
#20772
#20752
#20672
#20652
#20342
#20332
#19552
#19362
#18752
#17862