Finite Group Databasebeta

← #26414

Subgroups — #26414

Order 256.

409 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2440C211yesmaximal, characteristic
2#2360C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2087C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1921C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic
10#1872C211yesmaximal, characteristic
11#1863C211yesmaximal, characteristic, complemented
12#1852C211yesmaximal, characteristic
13#1850C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32#398—21no—
33#395—21no—
34#391C2211yescharacteristic
35#389—21no—
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#387—21no—
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46C4×C22⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C22⋊Q8—22no—
74C22⋊Q8—44no—
75C22⋊Q8Q814yescomplemented
76C4×Q8—22no—
77C4×Q8—41no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#103—22no—
112#103—41no—
113#103—41no—
114#103—41no—
115C22⋊C8—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24C2×D411yescharacteristic
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—81no—
134C2×Q8C2×Q812yes—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×Q812yes—
159C22×C4#4112yes—
160C22×C4C2411yescharacteristic
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
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—44no—
174C4⋊C4—24no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4C2×Q812yes—
196C4⋊C4#4112yes—
197C4⋊C4#4112yes—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4Q1614yes—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—81no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23C22⋊Q812yes—
213C23C2×Q1612yes—
214C23AGL(1,ℤ/8)12yes—
215C23C22×D411yescharacteristic
216Q8—22no—
217Q8—22no—
218Q8—41no—
219Q8—41no—
220Q8—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4C22⋊Q812yes—
264C2×C4C2×Q1612yes—
265C2×C4C2×Q1612yes—
266C2×C4C22×Q811yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22#47714yes—
283C22C2×C22⋊Q811yescharacteristic
284C22#54611yescharacteristic
285C22#54611yescharacteristic
286C22C22×Q1611yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22#57711yescharacteristic
289C22#57711yescharacteristic
290C4—22no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—42no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#52911yescharacteristic
304C4#53811yescharacteristic
305C4#55411yescharacteristic
306C4#55411yescharacteristic
307C2—24no—
308C2#245011yescharacteristic
309C2C2×#47711yescharacteristic
310C2#308411yescharacteristic
311C2#329911yescharacteristic
312C2#329911yescharacteristic
313C2#330611yescharacteristic
314C2#331011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
3. #2100C2semidirect
4. #2087C2semidirect
5. #2068C2semidirect
6. #2024C2semidirect
7. #1987C2semidirect
8. #1921C2semidirect
11. #1863C2semidirect
14. #1803C2semidirect
75. C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#24402
#23602
#21002
#20872
#20682
#20242
#19872
#19212
#18922
#18722
#18632
#18522
#18502
#18032
C2×#3402