Finite Group Databasebeta

← #23231

Subgroups — #23231

Order 256.

453 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic
2#2370C211yesmaximal, characteristic, complemented
3#2366C211yesmaximal, characteristic
4#1989C211yesmaximal, characteristic, complemented
5#1982C211yesmaximal, characteristic
6#1936C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1925C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1864C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic, complemented
12#1776C211yesmaximal, characteristic, complemented
13#1771C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C2×#127C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#490C418yescomplemented
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#390—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389—21no—
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#387—21no—
43#387—21no—
44#386—21no—
45#386—21no—
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70#127—22no—
71#127C2×C414yes—
72#125—22no—
73#125C2×C414yescomplemented
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—41no—
77C4×D4—41no—
78C4×D4—44no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#106—22no—
106C4⋊C8—22no—
107#102—22no—
108#102—22no—
109#102—41no—
110#102—41no—
111#102C2×C414yescomplemented
112#102C2×C414yescomplemented
113#102C2×C414yescomplemented
114#102C2×C414yescomplemented
115#97—41no—
116#97C2×C414yescomplemented
117C4×C8—41no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24—21no—
128C2×D4—24no—
129C2×D4—24no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—81no—
143C2×D4—81no—
144C2×D4C22×C412yes—
145C2×D4C22×C412yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8C22×C412yes—
176C2×C8C22×C412yes—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4C22×C412yes—
192C4⋊C4C22×C412yes—
193C4⋊C4#4112yes—
194C4⋊C4#4112yes—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—42no—
206C42—42no—
207C42—81no—
208C42—81no—
209C42C22×C412yes—
210C42#4112yes—
211C23—22no—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23C22×D411yescharacteristic
226D4—22no—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—48no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4C4×D412yes—
272C2×C4C4×D412yes—
273C2×C4AGL(1,ℤ/8)12yes—
274C2×C4C23×C411yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C42−1+411yescharacteristic
280C8—22no—
281C8—22no—
282C8—41no—
283C8—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C2×C4×D411yescharacteristic
299C22#54611yescharacteristic
300C22#54811yescharacteristic
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22#57511yescharacteristic
303C22#57611yescharacteristic
304C22#57711yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4C4×#4111yescharacteristic
318C4#52011yescharacteristic
319C4#55611yescharacteristic
320C4#55911yescharacteristic
321C2—82no—
322C2—82no—
323C2#236711yescharacteristic
324C2C4×AGL(1,ℤ/8)11yescharacteristic
325C2#294611yescharacteristic
326C2#330911yescharacteristic
327C2#331011yescharacteristic
328C2#336611yescharacteristic
329C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C2×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
2. #2370C2semidirect
4. #1989C2semidirect
6. #1936C2semidirect
7. #1934C2semidirect
8. #1925C2semidirect
9. #1925C2semidirect
10. #1864C2semidirect
11. #1864C2semidirect
12. #1776C2semidirect
13. #1771C2semidirect
14. #1762C2semidirect
15. C4×#102C2semidirect
21. #490C4semidirect
73. #125C2×C4semidirect
111. #102C2×C4semidirect
112. #102C2×C4semidirect
113. #102C2×C4semidirect
114. #102C2×C4semidirect
116. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
#23702
#23662
#19892
#19822
#19362
#19342
#19252
#18642
#17762
#17712
#17622
C4×#1022