Finite Group Databasebeta

← #26705

Subgroups — #26705

Order 256.

408 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2445C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic, complemented
3#2088C211yesmaximal, characteristic, complemented
4#2041C211yesmaximal, characteristic
5#2023C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#1982C211yesmaximal, characteristic
8#1936C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1920C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#123—21no—
19C2×C4×Q8—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#394—21no—
34#394—21no—
35#391—21no—
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#387—21no—
43#387—21no—
44#386—21no—
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71#127—22no—
72#127—41no—
73#123—44no—
74C4×Q8—44no—
75C4×D4—22no—
76C4×D4—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#106—22no—
103#106—41no—
104C4⋊C8—41no—
105#102—22no—
106#102—22no—
107#102—22no—
108#102—22no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—41no—
113#97—41no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95D411yescharacteristic
121#95D411yescharacteristic
122C24—21no—
123C24—21no—
124C2×Q8—24no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—81no—
143C2×D4—81no—
144C2×D4#4112yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
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×C8—21no—
172C2×C8—21no—
173C4⋊C4—24no—
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—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4#4112yes—
189C4⋊C4#4112yes—
190C4⋊C4#4112yes—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C42—42no—
206C42—42no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23C22×D411yescharacteristic
222Q8—48no—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—84no—
232D4—84no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
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—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—84no—
268C2×C4AGL(1,ℤ/8)12yes—
269C2×C4AGL(1,ℤ/8)12yes—
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C2×C42−1+411yescharacteristic
276C8—41no—
277C8—41no—
278C8—41no—
279C8—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#52511yescharacteristic
295C22#54611yescharacteristic
296C22#54811yescharacteristic
297C22C2×AGL(1,ℤ/8)11yescharacteristic
298C22C2×AGL(1,ℤ/8)11yescharacteristic
299C22#57711yescharacteristic
300C22#57711yescharacteristic
301C4—22no—
302C4—22no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4#52911yescharacteristic
314C4#54211yescharacteristic
315C4#55211yescharacteristic
316C4#55911yescharacteristic
317C2—82no—
318C2—161no—
319C2#245811yescharacteristic
320C2#310911yescharacteristic
321C2#312611yescharacteristic
322C2#330611yescharacteristic
323C2#332411yescharacteristic
324C2#336811yescharacteristic
325C2#337311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2365C2semidirect
3. #2088C2semidirect
5. #2023C2semidirect
6. #2023C2semidirect
8. #1936C2semidirect
9. #1926C2semidirect
10. #1926C2semidirect
11. #1926C2semidirect
12. #1920C2semidirect
13. #1917C2semidirect
14. #1864C2semidirect
15. #1762C2semidirect

Maximal subgroups

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

subgroupindex
#24452
#23652
#20882
#20412
#20232
#19822
#19362
#19262
#19202
#19172
#18642
#17622