Finite Group Databasebeta

← #26244

Subgroups — #26244

Order 256.

485 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3#2540C211yesmaximal, characteristic
4D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1983C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1955C211yesmaximal, characteristic, complemented
12#1947C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#484—24no—
26#480—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#399—21no—
46#392—21no—
47#390—21no—
48#390—21no—
49#389—21no—
50#385—21no—
51#385—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#124—22no—
75#124—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80#121—22no—
81#121—22no—
82#121—41no—
83#121—41no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—44no—
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—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×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
117#107—22no—
118#107—22no—
119#107D414yescomplemented
120#106—22no—
121C4⋊C8—22no—
122#103—22no—
123#103—22no—
124#103—41no—
125#103—41no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134#97—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—81no—
162C2×D4—81no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8C2×D412yes—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4C2×D412yes—
222C4⋊C4C2×D412yes—
223C4⋊C4#4112yes—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—42no—
235C42—81no—
236C42#4112yes—
237C42#4112yes—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23AGL(1,ℤ/8)12yes—
258C23#13712yes—
259C23C22×D411yescharacteristic
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264D4—28no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—84no—
270D4—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
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—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4#12312yes—
313C2×C4#12312yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C8—22no—
321C8—41no—
322C8—41no—
323C8—41no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#12311yescharacteristic
346C22D4211yescharacteristic
347C22#54611yescharacteristic
348C22C2×AGL(1,ℤ/8)11yescharacteristic
349C22C2×#13711yescharacteristic
350C22#57611yescharacteristic
351C22#57811yescharacteristic
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4#53211yescharacteristic
364C4#54011yescharacteristic
365C4#55011yescharacteristic
366C4#55911yescharacteristic
367C2—42no—
368C2—42no—
369C2—161no—
370C2#244511yescharacteristic
371C2#309211yescharacteristic
372C2#309811yescharacteristic
373C2#326411yescharacteristic
374C2#327011yescharacteristic
375C2#330811yescharacteristic
376C2#331411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
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×#484C2semidirect
2. C2×#480C2semidirect
4. D4×C4⋊C4C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1983C2semidirect
10. #1959C2semidirect
11. #1955C2semidirect
12. #1947C2semidirect
13. #1874C2semidirect
14. #1872C2semidirect
15. #1776C2semidirect
119. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4802
#25402
D4×C4⋊C42
#20362
#20352
#20342
#20332
#19832
#19592
#19552
#19472
#18742
#18722
#17762