Finite Group Databasebeta

← #30000

Subgroups — #30000

Order 256.

351 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2864C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2088C211yesmaximal, characteristic, complemented
4#2088C211yesmaximal, characteristic, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2067C211yesmaximal, characteristic
9#2065C211yesmaximal, characteristic
10#2065C211yesmaximal, characteristic
11#1949C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1842C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic
15#1804C211yesmaximal, characteristic
16C2×C4×D4C2211yescharacteristic
17C2×#117—21no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#387—21no—
43#387—21no—
44#387—21no—
45#387—21no—
46#382C2211yescharacteristic
47#382C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C43C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C4×D4—22no—
66C4×D4Q814yes—
67#117—44no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87C2×C42D411yescharacteristic
88C2×C42D411yescharacteristic
89C2×C42D411yescharacteristic
90C2×C42D411yescharacteristic
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94#107—41no—
95#107—41no—
96#106—41no—
97#106—41no—
98C4⋊C8—22no—
99C4⋊C8—22no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#102—41no—
105#102—41no—
106#102—41no—
107#102—41no—
108#95—21no—
109#95—21no—
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#95—21no—
119C24—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4C2×Q812yes—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4C2×Q812yes—
175C22⋊C4—24no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C42—22no—
181C42—22no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—41no—
187C42—41no—
188C42—42no—
189C42C2×Q812yes—
190C42#4112yes—
191C23—22no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23C22×D411yescharacteristic
199D4—22no—
200D4—22no—
201D4—41no—
202D4—41no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—48no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4C22⋊Q812yes—
240C2×C4C22⋊Q812yes—
241C2×C4#13512yes—
242C2×C4AGL(1,ℤ/8)12yes—
243C2×C4C22×Q811yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C2×C42−1+411yescharacteristic
248C2×C42−1+411yescharacteristic
249C8—41no—
250C8—41no—
251C8—41no—
252C8—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22C2×C22⋊Q811yescharacteristic
261C22#53711yescharacteristic
262C22#53711yescharacteristic
263C22C2×#13511yescharacteristic
264C22C2×AGL(1,ℤ/8)11yescharacteristic
265C22#57511yescharacteristic
266C22#57511yescharacteristic
267C4—22no—
268C4—22no—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—41no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4#54011yescharacteristic
281C4#54111yescharacteristic
282C4#54411yescharacteristic
283C4#54411yescharacteristic
284C2—82no—
285C2#262911yescharacteristic
286C2#307611yescharacteristic
287C2#307711yescharacteristic
288C2#324711yescharacteristic
289C2#325011yescharacteristic
290C2#325111yescharacteristic
291C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2864C2semidirect
3. #2088C2semidirect
4. #2088C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
11. #1949C2semidirect
12. #1949C2semidirect
13. #1842C2semidirect

Maximal subgroups

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

subgroupindex
#28642
#22932
#20882
#20872
#20862
#20672
#20652
#19492
#18422
#18042