Finite Group Databasebeta

← #25288

Subgroups — #25288

Order 256.

482 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3#2541C211yesmaximal, characteristic
4#2370C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1982C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic, complemented
14#1925C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C2×#127C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×#121—21no—
21C2×#121—21no—
22C2×C4×D4C2211yescharacteristic
23#490—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#97C2211yescharacteristic
43#392—21no—
44#392—21no—
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#387—21no—
51#387—21no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#127—22no—
76#127—22no—
77#127—22no—
78#127—22no—
79#127—41no—
80#127—41no—
81#125—22no—
82#125D414yescomplemented
83#121—44no—
84#121—44no—
85C4×D4—22no—
86C4×D4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
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×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#106—22no—
111C4⋊C8—22no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—41no—
125#102—41no—
126#97—22no—
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—
137C2×D4—21no—
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—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4#4112yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2411yescharacteristic
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4C2×D412yes—
214C4⋊C4C2×D412yes—
215C4⋊C4#4112yes—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—42no—
226C42—81no—
227C42—81no—
228C42C2×D412yes—
229C42#4112yes—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23C22×D411yescharacteristic
252D4—22no—
253D4—22no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—84no—
265D4—84no—
266D4—84no—
267D4—84no—
268D4—84no—
269D4—84no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—84no—
300C2×C4#12112yes—
301C2×C4#12112yes—
302C2×C4AGL(1,ℤ/8)12yes—
303C2×C4AGL(1,ℤ/8)12yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C42−1+411yescharacteristic
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C2×#12111yescharacteristic
336C22#54611yescharacteristic
337C22#54811yescharacteristic
338C22C2×AGL(1,ℤ/8)11yescharacteristic
339C22C2×AGL(1,ℤ/8)11yescharacteristic
340C22#57611yescharacteristic
341C22#57711yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4#52911yescharacteristic
354C4#54211yescharacteristic
355C4#54811yescharacteristic
356C4#55011yescharacteristic
357C2—82no—
358C2—161no—
359C2—161no—
360C2#242611yescharacteristic
361C2#303811yescharacteristic
362C2#304511yescharacteristic
363C2#331011yescharacteristic
364C2#331811yescharacteristic
365C2#336611yescharacteristic
366C2#336811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#490C2semidirect
2. C2×#478C2semidirect
4. #2370C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2041C2semidirect
8. #2041C2semidirect
9. #1982C2semidirect
10. #1982C2semidirect
11. #1982C2semidirect
12. #1940C2semidirect
13. #1926C2semidirect
14. #1925C2semidirect
15. #1762C2semidirect
82. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4782
#25412
#23702
#20672
#20652
#20412
#19822
#19402
#19262
#19252
#17622