Finite Group Databasebeta

← #32298

Subgroups — #32298

Order 256.

355 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2504C211yesmaximal, characteristic
2#2496C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic
8#2033C211yesmaximal, characteristic
9#1949C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1847C211yesmaximal, characteristic, complemented
16C2×#126—21no—
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#400—21no—
34#397—21no—
35#397—21no—
36#397—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#387—21no—
42#386—21no—
43#386—21no—
44#384C2211yescharacteristic
45#382C2211yescharacteristic
46#382C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#126—44no—
68#121—41no—
69#121—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4D411yescharacteristic
92C2×C22⋊C4D411yescharacteristic
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—41no—
97#107—41no—
98#106—41no—
99#106—41no—
100#106—41no—
101#106—41no—
102C4⋊C8—22no—
103#102—22no—
104#102—22no—
105#102—41no—
106#102—41no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109#97—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#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121C24—21no—
122C24C2×D411yescharacteristic
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
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—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8#4112yes—
165C2×C8#4112yes—
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—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C22⋊C4—22no—
186C22⋊C4—41no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—42no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23AGL(1,ℤ/8)12yes—
213C23C22×D411yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—84no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
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—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—41no—
258C2×C4—41no—
259C2×C4—84no—
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C2×C42−1+411yescharacteristic
265C2×C42−1+411yescharacteristic
266C8—22no—
267C8—22no—
268C8—41no—
269C8—41no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22#53511yescharacteristic
285C22#54711yescharacteristic
286C22#54711yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22#57511yescharacteristic
289C22#57711yescharacteristic
290C22#57711yescharacteristic
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#53911yescharacteristic
304C4#54211yescharacteristic
305C4#55311yescharacteristic
306C4#56311yescharacteristic
307C2—42no—
308C2—161no—
309C2#271811yescharacteristic
310C2#321311yescharacteristic
311C2#322211yescharacteristic
312C2#334511yescharacteristic
313C2#334511yescharacteristic
314C2#335011yescharacteristic
315C2#335111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. #2496C2semidirect
3. #2101C2semidirect
4. #2094C2semidirect
5. #2085C2semidirect
6. #2065C2semidirect
9. #1949C2semidirect
10. #1936C2semidirect
11. #1926C2semidirect
12. #1924C2semidirect
13. #1853C2semidirect
14. #1853C2semidirect
15. #1847C2semidirect

Maximal subgroups

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

subgroupindex
#25042
#24962
#21012
#20942
#20852
#20652
#20332
#19492
#19362
#19262
#19242
#18532
#18472