Finite Group Databasebeta

← #26007

Subgroups — #26007

Order 256.

370 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2440C211yesmaximal, characteristic
2#2347C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2088C211yesmaximal, characteristic, complemented
5#2086C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1948C211yesmaximal, characteristic, complemented
9#1921C211yesmaximal, characteristic, complemented
10#1919C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic
12#1851C211yesmaximal, characteristic
13#1825C211yesmaximal, characteristic
14#1809C211yesmaximal, characteristic
15#1736C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#398—21no—
29#395—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#388—21no—
36#385—21no—
37#384C2211yescharacteristic
38#383—21no—
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41C4×C4⋊C4—21no—
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×C4C2311yescharacteristic
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71C22⋊Q8—44no—
72#117—22no—
73#117Q814yescomplemented
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
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—41no—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4D411yescharacteristic
96C2×C22⋊C4D411yescharacteristic
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—22no—
103#106—22no—
104#106—41no—
105#106—41no—
106C4⋊C8—22no—
107C4⋊C8—41no—
108C22⋊C8—22no—
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#95D411yescharacteristic
118#95D411yescharacteristic
119C24C2×D411yescharacteristic
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—81no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—81no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×Q812yes—
144C22×C4#4112yes—
145C22×C4C2411yescharacteristic
146M16—44no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
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#4112yes—
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—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4C2×Q812yes—
181C4⋊C4C2×Q812yes—
182C4⋊C4#4112yes—
183C22⋊C4—24no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4—81no—
188C42—24no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—81no—
194C42—81no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23C22⋊Q812yes—
202C23C22×D411yescharacteristic
203Q8—84no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
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×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4C22⋊Q812yes—
246C2×C4AGL(1,ℤ/8)12yes—
247C2×C4#13712yes—
248C2×C4C22×Q811yescharacteristic
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C2×C42−1+411yescharacteristic
254C8—22no—
255C8—41no—
256C8—41no—
257C8—41no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22C2×C22⋊Q811yescharacteristic
266C22#54611yescharacteristic
267C22#54711yescharacteristic
268C22C2×AGL(1,ℤ/8)11yescharacteristic
269C22C2×#13711yescharacteristic
270C22#57711yescharacteristic
271C22#57711yescharacteristic
272C4—22no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—42no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4#53311yescharacteristic
286C4#54411yescharacteristic
287C4#55111yescharacteristic
288C4#55211yescharacteristic
289C2—42no—
290C2#243811yescharacteristic
291C2#307911yescharacteristic
292C2#308411yescharacteristic
293C2#332011yescharacteristic
294C2#332411yescharacteristic
295C2#334511yescharacteristic
296C2#335611yescharacteristic
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. #2099C2semidirect
4. #2088C2semidirect
5. #2086C2semidirect
6. #2058C2semidirect
7. #1987C2semidirect
8. #1948C2semidirect
9. #1921C2semidirect
10. #1919C2semidirect
73. #117Q8semidirect

Maximal subgroups

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

subgroupindex
#24402
#23472
#20992
#20882
#20862
#20582
#19872
#19482
#19212
#19192
#18532
#18512
#18252
#18092
#17362