Finite Group Databasebeta

← #27254

Subgroups — #27254

Order 256.

636 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2440C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic
5#2071C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1957C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1893C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic, complemented
19C2×#127C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×#120—21no—
25C22×C4⋊C4C2211yescharacteristic
26#486—24no—
27C2×#102—24no—
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic, complemented
42C2×C4×C8C2211yescharacteristic, complemented
43#398—21no—
44#395—21no—
45#392—21no—
46#392—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic, complemented
50#389C2211yescharacteristic, complemented
51#385—21no—
52#384C2211yescharacteristic, complemented
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—22no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C23×C4—21no—
79C23×C4C2311yescharacteristic
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C4⋊Q8—22no—
88C4⋊Q8—41no—
89#127—22no—
90#127—22no—
91#127—41no—
92#127—41no—
93C22⋊Q8—44no—
94#121—22no—
95#121—22no—
96#120—44no—
97#120—44no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130#102—41no—
131#102—41no—
132#102D414yescomplemented
133C22⋊C8—22no—
134C4×C8—22no—
135C4×C8—41no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95D411yescharacteristic
140#95D411yescharacteristic
141C24—21no—
142C24—21no—
143C24—21no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24—81no—
150C24—81no—
151C24C2×D411yescharacteristic
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×D4—28no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4C2×D412yes—
194C2×D4#4112yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4#4112yes—
213C22×C4#4112yes—
214C22×C4C2411yescharacteristic
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—44no—
227C2×C8C2×D412yes—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—81no—
243C4⋊C4C2×D412yes—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4D814yes—
251C22⋊C4SD1614yes—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
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—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—84no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23#12412yes—
289C23C2×D812yes—
290C23C2×SD1612yes—
291C23C22×D411yescharacteristic
292Q8—84no—
293D4—22no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—48no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—84no—
307D4—84no—
308D4—84no—
309D4—84no—
310D4—84no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
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—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—84no—
341C2×C4#12412yes—
342C2×C4C2×D812yes—
343C2×C4C2×D812yes—
344C2×C4C2×SD1612yes—
345C2×C4C2×SD1612yes—
346C2×C4C22×D411yescharacteristic
347C2×C4C2×#4111yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C8—22no—
353C8—41no—
354C8—41no—
355C8—41no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—816no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#48614yescomplemented
385C22C2×#12411yescharacteristic
386C22D4211yescharacteristic
387C22#54611yescharacteristic
388C22C22×D811yescharacteristic
389C22C22×SD1611yescharacteristic
390C22#57611yescharacteristic
391C22#57711yescharacteristic
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#52511yescharacteristic
403C4#53811yescharacteristic
404C4#54611yescharacteristic
405C4#55111yescharacteristic
406C2—24no—
407C2—82no—
408C2—82no—
409C2—161no—
410C2#247211yescharacteristic
411C2C2×#48611yescharacteristic
412C2#313611yescharacteristic
413C2D4×D811yescharacteristic
414C2D4×SD1611yescharacteristic
415C2#329411yescharacteristic
416C2#329511yescharacteristic
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. C2×#486C2semidirect
2. C22×#102C2semidirect
3. #2440C2semidirect
5. #2071C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2023C2semidirect
9. #1957C2semidirect
10. #1934C2semidirect
11. #1934C2semidirect
12. #1893C2semidirect
13. #1874C2semidirect
14. #1786C2semidirect
15. C8×C22⋊C4C2semidirect
18. C2×C4⋊Q8C22semidirect
36. C2×#102C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
41. C2×C4×C8C22semidirect
42. C2×C4×C8C22semidirect
49. #389C22semidirect
50. #389C22semidirect
52. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
132. #102D4semidirect
384. C22#486semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
C22×#1022
#24402
#24332
#20712
#20412
#20232
#19572
#19342
#18932
#18742
#17862
C8×C22⋊C42