Finite Group Databasebeta

← #32289

Subgroups — #32289

Order 256.

337 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2500C211yesmaximal, characteristic
2#2495C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2033C211yesmaximal, characteristic
9#1948C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1846C211yesmaximal, characteristic, complemented
16C2×#121C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#400—21no—
33#397—21no—
34#394—21no—
35#394—21no—
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#388—21no—
43#384C2211yescharacteristic
44#383—21no—
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×C4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66#121—41no—
67#121—41no—
68#117—41no—
69#117—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4D411yescharacteristic
90C2×C22⋊C4D411yescharacteristic
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#107—41no—
96#107—41no—
97#107—41no—
98#106—41no—
99#106—41no—
100#106—41no—
101C4⋊C8—22no—
102#102—22no—
103#102—22no—
104#102—41no—
105#102—41no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108C4×C8—22no—
109#95—21no—
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#95D411yescharacteristic
119#95D411yescharacteristic
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×C4—21no—
143C22×C4—21no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2411yescharacteristic
150C2×C8—21no—
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#4112yes—
161C2×C8#4112yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
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—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—22no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23C22×D411yescharacteristic
211D4—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—84no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
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—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
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#13512yes—
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C2×C42−1+411yescharacteristic
261C2×C42−1+411yescharacteristic
262C8—22no—
263C8—22no—
264C8—41no—
265C8—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22#53511yescharacteristic
281C22#54711yescharacteristic
282C22#54711yescharacteristic
283C22C2×#13511yescharacteristic
284C22#57511yescharacteristic
285C22#57611yescharacteristic
286C22#57711yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4#53911yescharacteristic
300C4#54211yescharacteristic
301C4#55311yescharacteristic
302C4#56311yescharacteristic
303C2—81no—
304C2—161no—
305C2#271811yescharacteristic
306C2#321211yescharacteristic
307C2#322411yescharacteristic
308C2#333711yescharacteristic
309C2#334111yescharacteristic
310C2#334811yescharacteristic
311C2#335511yescharacteristic
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. #2495C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2083C2semidirect
6. #2065C2semidirect
9. #1948C2semidirect
10. #1934C2semidirect
11. #1925C2semidirect
12. #1923C2semidirect
13. #1854C2semidirect
14. #1854C2semidirect
15. #1846C2semidirect

Maximal subgroups

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

subgroupindex
#25002
#24952
#21012
#20922
#20832
#20652
#20342
#20332
#19482
#19342
#19252
#19232
#18542
#18462