Finite Group Databasebeta

← #27874

Subgroups — #27874

Order 256.

361 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#2624C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#1925C211yesmaximal, characteristic, complemented
8#1925C211yesmaximal, characteristic, complemented
9#1921C211yesmaximal, characteristic, complemented
10#1919C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic
12#1893C211yesmaximal, characteristic
13#1809C211yesmaximal, characteristic, complemented
14#1808C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28#399—21no—
29#397—21no—
30#394—21no—
31#394—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#388—21no—
38#387—21no—
39#387—21no—
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#121—22no—
73#121—22no—
74#117—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—22no—
104#107—41no—
105#107—41no—
106#106—22no—
107#102—22no—
108#102—22no—
109#102—41no—
110#102—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#95—21no—
120C24—21no—
121C24C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—81no—
130C2×D4—81no—
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×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4#4112yes—
148C22×C4#4112yes—
149C22×C4C2411yescharacteristic
150M16—44no—
151C2×C8—21no—
152C2×C8—21no—
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—44no—
162C2×C8#4112yes—
163C2×C8#4112yes—
164C4⋊C4—22no—
165C4⋊C4—22no—
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—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—81no—
199C42—81no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23#12312yes—
214C23C22×D411yescharacteristic
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—84no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
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—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
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#12312yes—
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
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—
284C22C2×#12311yescharacteristic
285C22#53211yescharacteristic
286C22#53711yescharacteristic
287C22#57611yescharacteristic
288C22#57611yescharacteristic
289C22#57711yescharacteristic
290C22#57811yescharacteristic
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#55111yescharacteristic
304C4#55111yescharacteristic
305C4#55211yescharacteristic
306C4#55211yescharacteristic
307C2—42no—
308C2—161no—
309C2#249811yescharacteristic
310C2#309611yescharacteristic
311C2#309811yescharacteristic
312C2#317511yescharacteristic
313C2#317911yescharacteristic
314C2#325811yescharacteristic
315C2#325911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2314C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2083C2semidirect
6. #2071C2semidirect
7. #1925C2semidirect
8. #1925C2semidirect
9. #1921C2semidirect
10. #1919C2semidirect
13. #1809C2semidirect
14. #1808C2semidirect
15. #1729C2semidirect

Maximal subgroups

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

subgroupindex
#26242
#23142
#21012
#20922
#20832
#20712
#19252
#19212
#19192
#18952
#18932
#18092
#18082
#17292