Finite Group Databasebeta

← #26418

Subgroups — #26418

Order 256.

387 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2442C211yesmaximal, characteristic
2#2360C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2026C211yesmaximal, characteristic, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic
10#1872C211yesmaximal, characteristic
11#1863C211yesmaximal, characteristic, complemented
12#1852C211yesmaximal, characteristic
13#1850C211yesmaximal, characteristic
14#1805C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31#400—21no—
32#398—21no—
33#391C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#387—21no—
39#385—21no—
40#385—21no—
41#382C2211yescharacteristic
42#380—21no—
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×Q8C2311yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8—22no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70#125—22no—
71#125—41no—
72C22⋊Q8—22no—
73C22⋊Q8Q814yes—
74C4×Q8—22no—
75C4×Q8—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—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⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108#103—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112C22⋊C8—22no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95D411yescharacteristic
123#95D411yescharacteristic
124C24C2×D411yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×Q812yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
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—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×Q812yes—
155C22×C4#4112yes—
156C22×C4C2411yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
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—44no—
170C4⋊C4—24no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4C2×Q812yes—
191C4⋊C4#4112yes—
192C4⋊C4#4112yes—
193C22⋊C4—24no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—42no—
202C42—81no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23C22⋊Q812yes—
208C23#13512yes—
209C23#13712yes—
210C23C22×D411yescharacteristic
211Q8—22no—
212Q8—22no—
213Q8—41no—
214Q8—41no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
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—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—84no—
258C2×C4—84no—
259C2×C4C22⋊Q812yes—
260C2×C4C22×Q811yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22#47914yes—
277C22C2×C22⋊Q811yescharacteristic
278C22#54611yescharacteristic
279C22#54611yescharacteristic
280C22C2×#13511yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57611yescharacteristic
283C22#57811yescharacteristic
284C4—22no—
285C4—42no—
286C4—42no—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—42no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4#52911yescharacteristic
298C4#53811yescharacteristic
299C4#55411yescharacteristic
300C4#55411yescharacteristic
301C2—24no—
302C2#245011yescharacteristic
303C2C2×#47911yescharacteristic
304C2#308311yescharacteristic
305C2#329711yescharacteristic
306C2#330011yescharacteristic
307C2#331211yescharacteristic
308C2#331411yescharacteristic
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. #2101C2semidirect
4. #2086C2semidirect
5. #2068C2semidirect
6. #2026C2semidirect
7. #1989C2semidirect
8. #1918C2semidirect
11. #1863C2semidirect
14. #1805C2semidirect

Maximal subgroups

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

subgroupindex
#24422
#23602
#21012
#20862
#20682
#20262
#19892
#19182
#18922
#18722
#18632
#18522
#18502
#18052
C2×#3402