Finite Group Databasebeta

← #31577

Subgroups — #31577

Order 256.

368 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2570C211yesmaximal, characteristic, complemented
2#2445C211yesmaximal, characteristic
3#2092C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic
9#1957C211yesmaximal, characteristic
10#1941C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1916C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C2×#127C2211yescharacteristic
17C2×#123—21no—
18C2×C4×D4C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#400—21no—
34#398—21no—
35#394—21no—
36#394—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#387—21no—
44#387—21no—
45#386—21no—
46#384C2211yescharacteristic
47#382C2211yescharacteristic
48#382C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#127—22no—
69#127—41no—
70#123—44no—
71C4×D4—22no—
72C4×D4—41no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—41no—
98#107—41no—
99#106—22no—
100#106—41no—
101C4⋊C8—41no—
102#102—22no—
103#102—22no—
104#102—22no—
105#102—41no—
106#102—41no—
107#102—41no—
108#102—41no—
109#102—41no—
110C4×C8—41no—
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—
122C24—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—81no—
139C2×D4—81no—
140C2×D4#4112yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—81no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
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#4112yes—
169C4⋊C4—22no—
170C4⋊C4—22no—
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—21no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—81no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23C22×D411yescharacteristic
217D4—22no—
218D4—22no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—84no—
227C2×C4—21no—
228C2×C4—21no—
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—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4#13512yes—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—22no—
271C8—41no—
272C8—41no—
273C8—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#53611yescharacteristic
289C22#54611yescharacteristic
290C22#55011yescharacteristic
291C22C2×#13511yescharacteristic
292C22C2×AGL(1,ℤ/8)11yescharacteristic
293C22#57611yescharacteristic
294C22#57711yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4#53911yescharacteristic
308C4#53911yescharacteristic
309C4#55211yescharacteristic
310C4#56211yescharacteristic
311C2—82no—
312C2—161no—
313C2#269411yescharacteristic
314C2#322811yescharacteristic
315C2#324111yescharacteristic
316C2#329711yescharacteristic
317C2#331011yescharacteristic
318C2#339411yescharacteristic
319C2#339911yescharacteristic
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
1. #2570C2semidirect
3. #2092C2semidirect
4. #2092C2semidirect
5. #2083C2semidirect
6. #2073C2semidirect
7. #2067C2semidirect
10. #1941C2semidirect
11. #1926C2semidirect
12. #1925C2semidirect
13. #1916C2semidirect
14. #1864C2semidirect
15. #1804C2semidirect

Maximal subgroups

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

subgroupindex
#25702
#24452
#20922
#20832
#20732
#20672
#20412
#19572
#19412
#19262
#19252
#19162
#18642
#18042