Finite Group Databasebeta

← #26178

Subgroups — #26178

Order 256.

490 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3#2445C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2073C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#1936C211yesmaximal, characteristic, complemented
10#1917C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#123—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#459—24no—
25#442—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#394—21no—
39#394—21no—
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#386—21no—
47#385—21no—
48#385—21no—
49#382C2211yescharacteristic
50#380—21no—
51C4×C4⋊C4C2211yescharacteristic
52C2×#95—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8D414yes—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#127—22no—
81#127—41no—
82#123—44no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—41no—
89C4×D4—41no—
90C4×D4—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
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×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—41no—
115#106—22no—
116C4⋊C8—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—41no—
125#97—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24—21no—
135C24—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—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—81no—
155C2×D4—81no—
156C2×D4C2×D412yes—
157C2×D4C2×D412yes—
158C2×D4#4112yes—
159C2×D4#4112yes—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2411yescharacteristic
184D8—28no—
185D8—44no—
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×C8—21no—
197C2×C8C2×D412yes—
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—41no—
209C4⋊C4—42no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4#4112yes—
213C22⋊C4—24no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23—41no—
235C23—41no—
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—41no—
250C23C22×D411yescharacteristic
251D4—22no—
252D4—22no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—84no—
263D4—84no—
264D4#12314yescomplemented
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
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—84no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4AGL(1,ℤ/8)12yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C42−1+411yescharacteristic
308C8—22no—
309C8—41no—
310C8—41no—
311C8—41no—
312C22—24no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#52511yescharacteristic
334C22D4211yescharacteristic
335C22#54811yescharacteristic
336C22C2×AGL(1,ℤ/8)11yescharacteristic
337C22#57511yescharacteristic
338C22#57611yescharacteristic
339C22#57611yescharacteristic
340C4—22no—
341C4—41no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4C2×#12311yescharacteristic
352C4#53911yescharacteristic
353C4#54711yescharacteristic
354C4#55311yescharacteristic
355C2—44no—
356C2—82no—
357C2—161no—
358C2#244311yescharacteristic
359C2#311511yescharacteristic
360C2#312211yescharacteristic
361C2#326411yescharacteristic
362C2#327311yescharacteristic
363C2#336611yescharacteristic
364C2#337011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
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
2. C2×#442C2semidirect
3. #2445C2semidirect
4. #2349C2semidirect
5. #2092C2semidirect
6. #2088C2semidirect
7. #2073C2semidirect
8. #2041C2semidirect
9. #1936C2semidirect
10. #1917C2semidirect
11. #1879C2semidirect
12. #1879C2semidirect
13. #1879C2semidirect
14. #1862C2semidirect
15. #1862C2semidirect
264. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4592
C2×#4422
#24452
#23492
#20922
#20882
#20732
#20412
#19362
#19172
#18792
#18622