Finite Group Databasebeta

← #29429

Subgroups — #29429

Order 256.

400 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#2860C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic, complemented
3#2087C211yesmaximal, characteristic, complemented
4#2065C211yesmaximal, characteristic, complemented
5#2065C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic
7#1970C211yesmaximal, characteristic
8#1941C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1921C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic
14#1864C211yesmaximal, characteristic, complemented
15C4×C4⋊C8C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#126—21no—
18C2×C4×D4C2211yescharacteristic
19C2×#117—21no—
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#388—21no—
46#388—21no—
47#388—21no—
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C43C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C2×D8—22no—
63C2×D8—22no—
64C2×D8—41no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#126—44no—
70C4×D4—22no—
71C4×D4—22no—
72#117—44no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
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⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42D411yescharacteristic
97C2×C42D411yescharacteristic
98C2×C42D411yescharacteristic
99C2×C42D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—22no—
104#106—22no—
105#106—41no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108#102—22no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—41no—
113#102—41no—
114C4×C8—41no—
115C4×C8—41no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122C24—21no—
123C24—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—21no—
135C2×D4#4112yes—
136C22×C4—24no—
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×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2411yescharacteristic
152D8—44no—
153C2×C8—21no—
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#4112yes—
164C2×C8#4112yes—
165C4⋊C4—21no—
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—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4#4112yes—
182C22⋊C4—24no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—42no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23C22×D411yescharacteristic
219D4—22no—
220D4—22no—
221D4—41no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
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—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—48no—
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#13512yes—
261C2×C4#13512yes—
262C2×C4AGL(1,ℤ/8)12yes—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—22no—
271C8—22no—
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#52511yescharacteristic
289C22#53211yescharacteristic
290C22#53711yescharacteristic
291C22C2×#13511yescharacteristic
292C22C2×#13511yescharacteristic
293C22C2×AGL(1,ℤ/8)11yescharacteristic
294C22C2×AGL(1,ℤ/8)11yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—41no—
301C4—41no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4#55111yescharacteristic
308C4#55211yescharacteristic
309C4#55311yescharacteristic
310C4#55611yescharacteristic
311C2—82no—
312C2—161no—
313C2#260111yescharacteristic
314C2#310511yescharacteristic
315C2#310611yescharacteristic
316C2#316811yescharacteristic
317C2#318411yescharacteristic
318C2#324711yescharacteristic
319C2#324711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2860C2semidirect
2. #2293C2semidirect
3. #2087C2semidirect
4. #2065C2semidirect
5. #2065C2semidirect
8. #1941C2semidirect
9. #1934C2semidirect
10. #1934C2semidirect
11. #1926C2semidirect
12. #1921C2semidirect
14. #1864C2semidirect
15. C4×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#28602
#22932
#20872
#20652
#20542
#19702
#19412
#19342
#19262
#19212
#18792
#18642
C4×C4⋊C82