Finite Group Databasebeta

← #25221

Subgroups — #25221

Order 256.

481 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic
3#2501C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1851C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#123—21no—
19C2×#123—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#481—24no—
26#469—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#399—21no—
40#399—21no—
41#390—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#386—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×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×D8—22no—
69C2×D8—22no—
70C2×D8—41no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#123—44no—
79#123—44no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121D414yes—
84C4×D4—22no—
85C4×D4—41no—
86#117—22no—
87#117—41no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
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⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
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⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4D411yescharacteristic
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#106—22no—
119#106—22no—
120#106—22no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129C22⋊C8—22no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C24C2×D411yescharacteristic
137C2×D4—24no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—81no—
153C2×D4—81no—
154C2×D4C2×D412yes—
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—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D412yes—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4#4112yes—
178C22×C4C2411yescharacteristic
179D8—44no—
180M16—44no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8#4112yes—
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—21no—
202C4⋊C4—21no—
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—22no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—81no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#12112yes—
250C23C22×D411yescharacteristic
251D4—22no—
252D4—22no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—84no—
260D4—84no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4#12112yes—
298C2×C4AGL(1,ℤ/8)12yes—
299C2×C4AGL(1,ℤ/8)12yes—
300C2×C4C22×D411yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C42−1+411yescharacteristic
306C8—22no—
307C8—41no—
308C8—41no—
309C8—41no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22C2×#12111yescharacteristic
332C22#54611yescharacteristic
333C22#54711yescharacteristic
334C22C2×AGL(1,ℤ/8)11yescharacteristic
335C22C2×AGL(1,ℤ/8)11yescharacteristic
336C22#57711yescharacteristic
337C22#57711yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4#52911yescharacteristic
350C4#53911yescharacteristic
351C4#54611yescharacteristic
352C4#54711yescharacteristic
353C2—42no—
354C2—82no—
355C2—161no—
356C2#242511yescharacteristic
357C2#303811yescharacteristic
358C2#304211yescharacteristic
359C2#330611yescharacteristic
360C2#333211yescharacteristic
361C2#334511yescharacteristic
362C2#335411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
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×#481C2semidirect
3. #2501C2semidirect
4. #2363C2semidirect
5. #2084C2semidirect
6. #2071C2semidirect
7. #2065C2semidirect
8. #2054C2semidirect
9. #1948C2semidirect
10. #1926C2semidirect
11. #1879C2semidirect
12. #1875C2semidirect
13. #1851C2semidirect
14. #1802C2semidirect
15. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4692
#25012
#23632
#20842
#20712
#20652
#20542
#19482
#19262
#18792
#18752
#18512
#18022
#17362