Finite Group Databasebeta

← #26563

Subgroups — #26563

Order 256.

489 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#442C211yesmaximal, characteristic, complemented
3#2422C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic
9#1948C211yesmaximal, characteristic, complemented
10#1943C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#123—21no—
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#481—24no—
25#442—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#397—21no—
42#394—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#386—21no—
49#385—21no—
50#385—21no—
51#384C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C2×D8—22no—
70C2×D8—22no—
71C2×D8—22no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#123—44no—
77#121—22no—
78#121—22no—
79#121—22no—
80#121—41no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4—44no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—22no—
114#106—22no—
115#106—22no—
116#106—22no—
117#106—41no—
118C4⋊C8—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—41no—
124#102D414yescomplemented
125C22⋊C8—22no—
126C22⋊C8—22no—
127#97—22no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C24C2×D411yescharacteristic
137C2×D4—24no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—81no—
153C2×D4C2×D412yes—
154C2×D4#4112yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2411yescharacteristic
182D8—28no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8C2×D412yes—
194C2×C8#4112yes—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4C2×D412yes—
211C4⋊C4#4112yes—
212C22⋊C4—24no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23AGL(1,ℤ/8)12yes—
249C23C22×D411yescharacteristic
250D4—22no—
251D4—22no—
252D4—22no—
253D4—22no—
254D4—48no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—84no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4AGL(1,ℤ/8)12yes—
300C2×C4C22×D411yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C42−1+411yescharacteristic
306C8—22no—
307C8—22no—
308C8—41no—
309C8—41no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#52511yescharacteristic
332C22D4211yescharacteristic
333C22#54711yescharacteristic
334C22C2×AGL(1,ℤ/8)11yescharacteristic
335C22C2×AGL(1,ℤ/8)11yescharacteristic
336C22#57711yescharacteristic
337C22#57711yescharacteristic
338C4—22no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4#52511yescharacteristic
350C4#53911yescharacteristic
351C4#54711yescharacteristic
352C4#55911yescharacteristic
353C2—42no—
354C2—82no—
355C2—82no—
356C2#245511yescharacteristic
357C2#310911yescharacteristic
358C2#312511yescharacteristic
359C2#326511yescharacteristic
360C2#327211yescharacteristic
361C2#334511yescharacteristic
362C2#334511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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×#481C2semidirect
2. C2×#442C2semidirect
3. #2422C2semidirect
4. #2361C2semidirect
5. #2094C2semidirect
6. #2054C2semidirect
7. #2033C2semidirect
9. #1948C2semidirect
10. #1943C2semidirect
11. #1925C2semidirect
12. #1924C2semidirect
13. #1864C2semidirect
14. #1864C2semidirect
15. #1853C2semidirect
124. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4422
#24222
#23612
#20942
#20542
#20332
#19992
#19482
#19432
#19252
#19242
#18642
#18532