Finite Group Databasebeta

← #27443

Subgroups — #27443

Order 256.

413 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2452C211yesmaximal, characteristic, complemented
2#2354C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic
6#2023C211yesmaximal, characteristic, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1921C211yesmaximal, characteristic, complemented
11#1917C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic
13#1879C211yesmaximal, characteristic
14#1862C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#97C2211yescharacteristic
31#396—21no—
32#393—21no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#390C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#388—21no—
41#385—21no—
42#384C2211yescharacteristic
43#380—21no—
44C4×C4⋊C4C2211yescharacteristic
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×Q8—21no—
60C22×D4C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C2×D8—22no—
65C2×D8—22no—
66C2×D8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C22⋊Q8—44no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—41no—
78#117—44no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108C4⋊C8—41no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114#102—41no—
115#97—41no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95D411yescharacteristic
121#95D411yescharacteristic
122C24—21no—
123C24—21no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—81no—
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—21no—
136C2×D4—21no—
137C2×D4#4112yes—
138C2×D4#4112yes—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2411yescharacteristic
158D8—44no—
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—21no—
169C2×C8—21no—
170C2×C8—21no—
171C4⋊C4—22no—
172C4⋊C4—22no—
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—42no—
187C4⋊C4—81no—
188C4⋊C4#4112yes—
189C4⋊C4#4112yes—
190C22⋊C4—24no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—42no—
208C23—22no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223Q8—84no—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—48no—
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—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—84no—
268C2×C4AGL(1,ℤ/8)12yes—
269C2×C4AGL(1,ℤ/8)12yes—
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C2×C42−1+411yescharacteristic
276C8—41no—
277C8—41no—
278C8—41no—
279C8—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#52511yescharacteristic
295C22#54611yescharacteristic
296C22#54811yescharacteristic
297C22C2×AGL(1,ℤ/8)11yescharacteristic
298C22C2×AGL(1,ℤ/8)11yescharacteristic
299C22#57611yescharacteristic
300C22#57811yescharacteristic
301C4—22no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4#52911yescharacteristic
314C4#54211yescharacteristic
315C4#55111yescharacteristic
316C4#55211yescharacteristic
317C2—82no—
318C2—82no—
319C2#247711yescharacteristic
320C2#310911yescharacteristic
321C2#312411yescharacteristic
322C2#330811yescharacteristic
323C2#332511yescharacteristic
324C2#336611yescharacteristic
325C2#336711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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. #2452C2semidirect
2. #2354C2semidirect
3. #2100C2semidirect
4. #2071C2semidirect
6. #2023C2semidirect
7. #1987C2semidirect
8. #1936C2semidirect
9. #1925C2semidirect
10. #1921C2semidirect
11. #1917C2semidirect
14. #1862C2semidirect
15. #1762C2semidirect

Maximal subgroups

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

subgroupindex
#24522
#23542
#21002
#20712
#20542
#20232
#19872
#19362
#19252
#19212
#19172
#18792
#18622
#17622