Finite Group Databasebeta

← #25204

Subgroups — #25204

Order 256.

470 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#483C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3#2501C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic
13#1850C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#483—24no—
26#465—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#399—21no—
40#399—21no—
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic
46#387—21no—
47#385—21no—
48#384C2211yescharacteristic
49#384C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—41no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—41no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#123—44no—
78#123—44no—
79C22⋊Q8—22no—
80C22⋊Q8D414yes—
81#121—22no—
82#121—22no—
83C4×Q8—22no—
84C4×Q8—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#106—22no—
117#106—22no—
118#106—22no—
119C4⋊C8—22no—
120C4⋊C8—41no—
121#103—22no—
122#103—22no—
123#103—22no—
124#102—22no—
125#102—22no—
126#102—41no—
127C22⋊C8—22no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24C2×D411yescharacteristic
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8C2×D412yes—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
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—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D412yes—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4#4112yes—
175C22×C4C2411yescharacteristic
176SD16—44no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—44no—
189C2×C8#4112yes—
190C4⋊C4—24no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—81no—
209C4⋊C4C2×D412yes—
210C4⋊C4#4112yes—
211C22⋊C4—24no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4SD1614yescomplemented
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—42no—
224C42—81no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23#12112yes—
237C23C2×SD1612yes—
238C23AGL(1,ℤ/8)12yes—
239C23C22×D411yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—41no—
243Q8—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—84no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
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—41no—
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—84no—
287C2×C4—84no—
288C2×C4#12112yes—
289C2×C4C2×SD1612yes—
290C2×C4C2×SD1612yes—
291C2×C4C22×D411yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—22no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22#46014yes—
315C22C2×#12111yescharacteristic
316C22#54611yescharacteristic
317C22#54711yescharacteristic
318C22C22×SD1611yescharacteristic
319C22C2×AGL(1,ℤ/8)11yescharacteristic
320C22#57711yescharacteristic
321C22#57811yescharacteristic
322C4—22no—
323C4—22no—
324C4—22no—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—42no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#52911yescharacteristic
335C4#53911yescharacteristic
336C4#54611yescharacteristic
337C4#54711yescharacteristic
338C2—24no—
339C2—161no—
340C2#242511yescharacteristic
341C2C2×#46011yescharacteristic
342C2#304611yescharacteristic
343C2#329811yescharacteristic
344C2#330611yescharacteristic
345C2#333511yescharacteristic
346C2#334311yescharacteristic
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
1. C2×#483C2semidirect
4. #2360C2semidirect
5. #2084C2semidirect
6. #2072C2semidirect
7. #2065C2semidirect
8. #2055C2semidirect
9. #1948C2semidirect
10. #1926C2semidirect
11. #1880C2semidirect
13. #1850C2semidirect
14. #1803C2semidirect
15. C2×#340C2semidirect
218. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4652
#25012
#23602
#20842
#20722
#20652
#20552
#19482
#19262
#18802
#18742
#18502
#18032
C2×#3402