Finite Group Databasebeta

← #32278

Subgroups — #32278

Order 256.

386 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2501C211yesmaximal, characteristic
2#2496C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic
8#2033C211yesmaximal, characteristic
9#1947C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1845C211yesmaximal, characteristic, complemented
16C2×#126—21no—
17C2×#123—21no—
18C2×#123—21no—
19C2×#121C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#400—21no—
36#399—21no—
37#399—21no—
38#397—21no—
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#387—21no—
44#384C2211yescharacteristic
45#384C2211yescharacteristic
46#384C2211yescharacteristic
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#123—44no—
69#123—44no—
70#121—41no—
71#121—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—41no—
103#107—41no—
104#107—41no—
105#107—41no—
106#107—41no—
107#107—41no—
108C4⋊C8—22no—
109#102—22no—
110#102—22no—
111#102—41no—
112#102—41no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C4×C8—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95D411yescharacteristic
122#95D411yescharacteristic
123C24—21no—
124C24C2×D411yescharacteristic
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
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—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
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—21no—
165C2×C8—21no—
166C2×C8#4112yes—
167C2×C8#4112yes—
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—21no—
181C4⋊C4—21no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4D814yes—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23C2×D812yes—
219C23C22×D411yescharacteristic
220D4—41no—
221D4—41no—
222D4—41no—
223D4—41no—
224D4—84no—
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—21no—
235C2×C4—21no—
236C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4C2×D812yes—
264C2×C4C2×D812yes—
265C2×C4AGL(1,ℤ/8)12yes—
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C2×C42−1+411yescharacteristic
271C2×C42−1+411yescharacteristic
272C8—22no—
273C8—22no—
274C8—41no—
275C8—41no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22#53511yescharacteristic
291C22#54711yescharacteristic
292C22#54711yescharacteristic
293C22C22×D811yescharacteristic
294C22C2×AGL(1,ℤ/8)11yescharacteristic
295C22#57811yescharacteristic
296C22#57811yescharacteristic
297C4—22no—
298C4—22no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4#53911yescharacteristic
310C4#54211yescharacteristic
311C4#55311yescharacteristic
312C4#56311yescharacteristic
313C2—42no—
314C2—161no—
315C2#271811yescharacteristic
316C2#320511yescharacteristic
317C2#322311yescharacteristic
318C2#333411yescharacteristic
319C2#333411yescharacteristic
320C2#334311yescharacteristic
321C2#335311yescharacteristic
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. #2099C2semidirect
4. #2093C2semidirect
5. #2083C2semidirect
6. #2067C2semidirect
9. #1947C2semidirect
10. #1934C2semidirect
11. #1925C2semidirect
12. #1923C2semidirect
13. #1852C2semidirect
14. #1852C2semidirect
15. #1845C2semidirect

Maximal subgroups

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

subgroupindex
#25012
#24962
#20992
#20932
#20832
#20672
#20332
#19472
#19342
#19252
#19232
#18522
#18452