Finite Group Databasebeta

← #28199

Subgroups — #28199

Order 256.

403 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#446C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic
3#2581C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic
8#2055C211yesmaximal, characteristic
9#1936C211yesmaximal, characteristic
10#1935C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic
12#1933C211yesmaximal, characteristic, complemented
13#1836C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic
15#1803C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C22⋊Q8—21no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#446—24no—
22#440—24no—
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#400—21no—
37#396—21no—
38#396—21no—
39#395—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#387—21no—
49#387—21no—
50#384C2211yescharacteristic
51#382C2211yescharacteristic
52#380—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8D411yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—22no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22⋊Q8—44no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×D4—22no—
78C4×D4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108#103—22no—
109#103—22no—
110#103—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114#97—22no—
115#97D414yes—
116C4×C8—22no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127C24—21no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—41no—
136C2×Q8#4112yes—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4#4112yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160SD16—28no—
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×C8C2×D412yes—
172C2×C8C2×D412yes—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
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—21no—
187C4⋊C4—21no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C42C2×D412yes—
206C42#4112yes—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23C22×D411yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—44no—
218Q8—41no—
219Q8—41no—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—41no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
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—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—84no—
261C2×C4#12112yes—
262C2×C4#12112yes—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4C22×D411yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—22no—
271C8—22no—
272C8—41no—
273C8—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#12111yescharacteristic
282C22#54811yescharacteristic
283C22#54811yescharacteristic
284C22C2×AGL(1,ℤ/8)11yescharacteristic
285C22#57511yescharacteristic
286C22#57711yescharacteristic
287C22#57811yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#52511yescharacteristic
302C4#53611yescharacteristic
303C4#55211yescharacteristic
304C4#55211yescharacteristic
305C2—82no—
306C2#253611yescharacteristic
307C2#305511yescharacteristic
308C2#306311yescharacteristic
309C2#336711yescharacteristic
310C2#336811yescharacteristic
311C2#337611yescharacteristic
312C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#446C2semidirect
3. #2581C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
10. #1935C2semidirect
12. #1933C2semidirect
13. #1836C2semidirect
15. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4462
C2×#4402
#25812
#25382
#20872
#20862
#20552
#19362
#19352
#19342
#19332
#18362
#18042
#18032