Finite Group Databasebeta

← #31310

Subgroups — #31310

Order 256.

537 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2494C211yesmaximal, characteristic, complemented
2#2433C211yesmaximal, characteristic
3#2092C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1961C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1926C211yesmaximal, characteristic, complemented
13#1924C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#120—21no—
21C2×#120—21no—
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×#102C2211yescharacteristic, complemented
34C2×#102C2211yescharacteristic, complemented
35C2×#102C2211yescharacteristic, complemented
36C2×#102C2211yescharacteristic, complemented
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic, complemented
40#401—21no—
41#397—21no—
42#392—21no—
43#392—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic, complemented
47#389C2211yescharacteristic, complemented
48#388—21no—
49#387—21no—
50#382C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic, complemented
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#336C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59C25—21no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#127—22no—
81#127—22no—
82#127—41no—
83#127—41no—
84#121—41no—
85#121—41no—
86#120—44no—
87#120—44no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—41no—
108#106—41no—
109C4⋊C8—41no—
110#102—22no—
111#102—22no—
112#102—41no—
113#102—41no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120#102—41no—
121#102—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124#97—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C24—21no—
137C24—42no—
138C24—42no—
139C24—42no—
140C24—42no—
141C24—42no—
142C24—81no—
143C24—81no—
144C24C2×D411yescharacteristic
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—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4#4112yes—
184C2×D4#4112yes—
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×C4—21no—
195C22×C4—21no—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
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×C8—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—42no—
225C4⋊C4—81no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—81no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—81no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—48no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
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—41no—
267C23—41no—
268C23—84no—
269C23—84no—
270C23—84no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23AGL(1,ℤ/8)12yes—
275C23C22×D411yescharacteristic
276D4—22no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—48no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—84no—
290D4—84no—
291D4—84no—
292D4—84no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
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×C4AGL(1,ℤ/8)12yes—
328C2×C4C2×#4111yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C8—41no—
335C8—41no—
336C8—41no—
337C8—41no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—816no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#53511yescharacteristic
368C22#54611yescharacteristic
369C22#54611yescharacteristic
370C22C2×AGL(1,ℤ/8)11yescharacteristic
371C22C2×AGL(1,ℤ/8)11yescharacteristic
372C22#57611yescharacteristic
373C22#57711yescharacteristic
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4#53811yescharacteristic
385C4#53811yescharacteristic
386C4#55111yescharacteristic
387C4#56011yescharacteristic
388C2—42no—
389C2—82no—
390C2—82no—
391C2—161no—
392C2#268511yescharacteristic
393C2#321711yescharacteristic
394C2#322411yescharacteristic
395C2#331011yescharacteristic
396C2#331811yescharacteristic
397C2#331811yescharacteristic
398C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
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. #2494C2semidirect
3. #2092C2semidirect
4. #2067C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1982C2semidirect
10. #1961C2semidirect
11. #1936C2semidirect
12. #1926C2semidirect
13. #1924C2semidirect
14. #1786C2semidirect
15. #1786C2semidirect
22. C2×#107C22semidirect
23. C2×#106C22semidirect
24. C2×C4⋊C8C22semidirect
33. C2×#102C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
39. C2×#97C22semidirect
46. #389C22semidirect
47. #389C22semidirect
50. #382C22semidirect
53. C4×C4⋊C4C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #336C22semidirect
58. #336C22semidirect

Maximal subgroups

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

subgroupindex
#24942
#24332
#20922
#20672
#20412
#20342
#20332
#19822
#19612
#19362
#19262
#19242
#17862