Finite Group Databasebeta

← #22978

Subgroups — #22978

Order 256.

435 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic
2#2370C211yesmaximal, characteristic, complemented
3D4×C4⋊C4C211yesmaximal, characteristic
4#1992C211yesmaximal, characteristic, complemented
5#1948C211yesmaximal, characteristic, complemented
6#1943C211yesmaximal, characteristic, complemented
7#1941C211yesmaximal, characteristic, complemented
8#1920C211yesmaximal, characteristic, complemented
9#1918C211yesmaximal, characteristic, complemented
10#1864C211yesmaximal, characteristic
11#1864C211yesmaximal, characteristic
12#1771C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic
14#1762C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16C2×#125C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4—21no—
21#478C418yescomplemented
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#387—21no—
40#387—21no—
41#385—21no—
42#385—21no—
43#384C2211yescharacteristic
44#380—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#125—22no—
69#125C2×C414yescomplemented
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—41no—
74C4×D4—41no—
75C4×D4C2×C414yes—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107C2×C414yescomplemented
104#106—41no—
105#106—41no—
106#106C2×C414yescomplemented
107C4⋊C8C2×C414yescomplemented
108#102—22no—
109#102C2×C414yes—
110#102C2×C414yes—
111#102Q814yes—
112#97—22no—
113#97—41no—
114C4×C8—41no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95D411yescharacteristic
123#95D411yescharacteristic
124C24—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4C22×C412yes—
129C2×D4C2×Q812yes—
130C2×D4#4112yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8C22×C412yes—
164C2×C8C22×C412yes—
165C2×C8C2×Q812yes—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4C22×C412yes—
187C4⋊C4C22×C412yes—
188C4⋊C4C22×C412yes—
189C4⋊C4C2×Q812yes—
190C22⋊C4—24no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—81no—
199C42—81no—
200C42C22×C412yes—
201C42#4112yes—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23C22×D411yescharacteristic
210D4—22no—
211D4—22no—
212D4—22no—
213D4C4×Q814yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—48no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4C4×D412yes—
254C2×C4C4×D412yes—
255C2×C4AGL(1,ℤ/8)12yes—
256C2×C4C23×C411yescharacteristic
257C2×C4C22×Q811yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C42−1+411yescharacteristic
262C8—22no—
263C8—22no—
264C8—22no—
265C8—41no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22C2×C4×D411yescharacteristic
274C22#54611yescharacteristic
275C22D4×Q811yescharacteristic
276C22C2×AGL(1,ℤ/8)11yescharacteristic
277C22#57511yescharacteristic
278C22#57711yescharacteristic
279C22#57711yescharacteristic
280C4—22no—
281C4—22no—
282C4—22no—
283C4—42no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4C2×C4×Q811yescharacteristic
294C4#52011yescharacteristic
295C4#55211yescharacteristic
296C4#55411yescharacteristic
297C2—44no—
298C2#236011yescharacteristic
299C2C4×AGL(1,ℤ/8)11yescharacteristic
300C2#295211yescharacteristic
301C2#330711yescharacteristic
302C2#331011yescharacteristic
303C2#338411yescharacteristic
304C2#338811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. #1992C2semidirect
5. #1948C2semidirect
6. #1943C2semidirect
7. #1941C2semidirect
8. #1920C2semidirect
9. #1918C2semidirect
12. #1771C2semidirect
21. #478C4semidirect
69. #125C2×C4semidirect
103. #107C2×C4semidirect
106. #106C2×C4semidirect
107. C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
#23702
D4×C4⋊C42
#19922
#19482
#19432
#19412
#19202
#19182
#18642
#17712
#17622
C4×#1022