Finite Group Databasebeta

← #26744

Subgroups — #26744

Order 256.

427 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2491C211yesmaximal, characteristic, complemented
2#2363C211yesmaximal, characteristic, complemented
3#2054C211yesmaximal, characteristic, complemented
4#2034C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic, complemented
6#1999C211yesmaximal, characteristic
7#1988C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1924C211yesmaximal, characteristic, complemented
11#1919C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1825C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#394—21no—
35#393—21no—
36#390—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#386—21no—
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C22⋊C4—21no—
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C2×D8—22no—
67C2×D8—41no—
68C2×D8—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8—41no—
75#124—44no—
76#121—22no—
77#121—22no—
78#121—22no—
79#121—41no—
80C4×D4—22no—
81C4×D4—41no—
82#117—22no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
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×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#106—22no—
111C4⋊C8—41no—
112#102—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—41no—
117#102—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120#97—41no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24—21no—
129C24C2×D411yescharacteristic
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—81no—
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—81no—
150C2×D4#4112yes—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172D8—44no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4#4112yes—
197C4⋊C4#4112yes—
198C4⋊C4#4112yes—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—42no—
217C42—42no—
218C42—81no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C22×D411yescharacteristic
241Q8—84no—
242D4—22no—
243D4—22no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—84no—
251D4—84no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—84no—
289C2×C4AGL(1,ℤ/8)12yes—
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C8—41no—
297C8—41no—
298C8—41no—
299C8—41no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#52511yescharacteristic
322C22#54611yescharacteristic
323C22#54611yescharacteristic
324C22C2×AGL(1,ℤ/8)11yescharacteristic
325C22#57511yescharacteristic
326C22#57611yescharacteristic
327C22#57711yescharacteristic
328C4—22no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4#52911yescharacteristic
340C4#53811yescharacteristic
341C4#55511yescharacteristic
342C4#55911yescharacteristic
343C2—81no—
344C2—82no—
345C2—161no—
346C2#245911yescharacteristic
347C2#311511yescharacteristic
348C2#311811yescharacteristic
349C2#331811yescharacteristic
350C2#331911yescharacteristic
351C2#332111yescharacteristic
352C2#333211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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
1. #2491C2semidirect
2. #2363C2semidirect
3. #2054C2semidirect
4. #2034C2semidirect
5. #2023C2semidirect
7. #1988C2semidirect
8. #1936C2semidirect
9. #1926C2semidirect
10. #1924C2semidirect
11. #1919C2semidirect
12. #1879C2semidirect
13. #1875C2semidirect
14. #1825C2semidirect
15. #1762C2semidirect

Maximal subgroups

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

subgroupindex
#24912
#23632
#20542
#20342
#20232
#19992
#19882
#19362
#19262
#19242
#19192
#18792
#18752
#18252
#17622