Finite Group Databasebeta

← #26130

Subgroups — #26130

Order 256.

504 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C22×#106C211yesmaximal, characteristic, complemented
3#2416C211yesmaximal, characteristic
4D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2077C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1919C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#121—21no—
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#481—24no—
26C2×#106—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#396—21no—
41#394—21no—
42#390C2211yescharacteristic
43#389—21no—
44#389C2211yescharacteristic
45#386—21no—
46#385—21no—
47#385—21no—
48#384C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4D411yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#121—22no—
78#121—22no—
79#121—22no—
80#121—41no—
81#121—44no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—44no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
116#106—22no—
117#106—22no—
118#106—22no—
119#106—22no—
120#106—41no—
121#106D414yescomplemented
122#102—22no—
123#102—22no—
124#102—22no—
125#102—41no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134C24—21no—
135C24C2×D411yescharacteristic
136C24C2×D411yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
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—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—81no—
151C2×D4SD1614yes—
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—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4#4112yes—
183C22×C4#4112yes—
184C22×C4C2411yescharacteristic
185C2×C8—28no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8C2×D412yes—
197C2×C8#4112yes—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4C2×D412yes—
218C4⋊C4C2×D412yes—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C42—21no—
229C42—21no—
230C42—42no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#12312yes—
249C23C2×SD1612yes—
250C23C2×SD1612yes—
251C23AGL(1,ℤ/8)12yes—
252C23C22×D411yescharacteristic
253D4—28no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—84no—
259D4—84no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4#12312yes—
303C2×C4C2×SD1612yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C42−1+411yescharacteristic
310C8—22no—
311C8—22no—
312C8—41no—
313C8—41no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#48114yescomplemented
335C22C2×#12311yescharacteristic
336C22D4211yescharacteristic
337C22#54711yescharacteristic
338C22C22×SD1611yescharacteristic
339C22C2×AGL(1,ℤ/8)11yescharacteristic
340C22#57711yescharacteristic
341C22#57711yescharacteristic
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4#52511yescharacteristic
354C4#53911yescharacteristic
355C4#54811yescharacteristic
356C4#55311yescharacteristic
357C2—24no—
358C2—42no—
359C2—161no—
360C2#244311yescharacteristic
361C2C2×#48111yescharacteristic
362C2#309911yescharacteristic
363C2#326511yescharacteristic
364C2D4×SD1611yescharacteristic
365C2#333511yescharacteristic
366C2#334511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2416C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#481C2semidirect
2. C22×#106C2semidirect
4. D4×C4⋊C4C2semidirect
5. #2094C2semidirect
6. #2084C2semidirect
7. #2077C2semidirect
8. #2034C2semidirect
9. #1926C2semidirect
10. #1919C2semidirect
11. #1893C2semidirect
12. #1874C2semidirect
13. #1869C2semidirect
14. #1853C2semidirect
15. #1853C2semidirect
121. #106D4semidirect
334. C22#481semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C22×#1062
#24162
D4×C4⋊C42
#20942
#20842
#20772
#20342
#19262
#19192
#18932
#18742
#18692
#18532