Finite Group Databasebeta

← #25990

Subgroups — #25990

Order 256.

401 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2440C211yesmaximal, characteristic
2#2344C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2088C211yesmaximal, characteristic, complemented
5#2088C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1948C211yesmaximal, characteristic, complemented
9#1920C211yesmaximal, characteristic, complemented
10#1920C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic
12#1850C211yesmaximal, characteristic
13#1823C211yesmaximal, characteristic
14#1807C211yesmaximal, characteristic
15C2×#340C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×#117—21no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#398—21no—
30#395—21no—
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#387—21no—
36#387—21no—
37#385—21no—
38#384C2211yescharacteristic
39#381—21no—
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C22⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
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—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C22⋊Q8—44no—
73#117—44no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
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—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4Q814yescomplemented
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#106—22no—
104#106—22no—
105#106—41no—
106#106—41no—
107C4⋊C8—22no—
108C4⋊C8—41no—
109C22⋊C8—22no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24C2×D411yescharacteristic
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—81no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—81no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×Q812yes—
148C22×C4#4112yes—
149C22×C4C2411yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—44no—
162C2×C8#4112yes—
163C4⋊C4—24no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
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—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4C2×Q812yes—
186C4⋊C4C2×Q812yes—
187C4⋊C4#4112yes—
188C22⋊C4—22no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4SD1614yes—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—42no—
197C42—42no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23C22⋊Q812yes—
203C23C2×SD1612yes—
204C23AGL(1,ℤ/8)12yes—
205C23C22×D411yescharacteristic
206Q8—84no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—48no—
233C2×C4—41no—
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—84no—
249C2×C4C22⋊Q812yes—
250C2×C4C2×SD1612yes—
251C2×C4C2×SD1612yes—
252C2×C4C22×Q811yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4H(5,𝔽2)11yescharacteristic
257C2×C42−1+411yescharacteristic
258C8—22no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22#47614yes—
269C22C2×C22⋊Q811yescharacteristic
270C22#54611yescharacteristic
271C22#54711yescharacteristic
272C22C22×SD1611yescharacteristic
273C22C2×AGL(1,ℤ/8)11yescharacteristic
274C22#57711yescharacteristic
275C22#57811yescharacteristic
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4#53311yescharacteristic
290C4#54411yescharacteristic
291C4#55111yescharacteristic
292C4#55211yescharacteristic
293C2—24no—
294C2#243811yescharacteristic
295C2C2×#47611yescharacteristic
296C2#308211yescharacteristic
297C2#329811yescharacteristic
298C2#331011yescharacteristic
299C2#333511yescharacteristic
300C2#334311yescharacteristic
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. #2088C2semidirect
5. #2088C2semidirect
6. #2058C2semidirect
7. #1987C2semidirect
8. #1948C2semidirect
9. #1920C2semidirect
10. #1920C2semidirect
86. C2×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#24402
#23442
#21002
#20882
#20582
#19872
#19482
#19202
#18532
#18502
#18232
#18072
C2×#3402